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Switching Control Strategies

  • Xinzhi Liu
  • Peter Stechlinski
Chapter
  • 1k Downloads
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 19)

Abstract

This chapter is motivated by the application of control strategies to eradicate epidemics. In part, the previous switched epidemic models are reintroduced with continuous (e.g., vaccination of newborns continuously in time) or switching control (i.e., piecewise continuous application of vaccination or treatment schemes) for evaluation and optimization. As discussed earlier, infectious disease models are a crucial component in designing and implementing detection, prevention, and control programs (e.g., the World Health Organization’s program against smallpox, leading to its global eradication by 1977). The switched SIR model is first returned to for consideration and analysis of vaccination of the susceptible group (e.g., newborns or the entire cohort). Subsequently, the developed theoretical methods are applied to the switched SIR model with a treatment program in effect. Common Lyapunov functions are used to provide controlled eradication of diseases modeled by the so-called SEIR (Susceptible-Exposed-Infected-Recovered) model with seasonal variations captured by switching. A screening process, where traveling individuals are examined for infection, is proposed and studied for the switched multi-city model of the previous chapter. Switching control of diseases such as Dengue and Chikungunya which are spread via mosquito–human interactions, is investigated.

This chapter is motivated by the application of control strategies to eradicate epidemics. The previous switched epidemic models are reintroduced with continuous control (e.g., vaccination of newborns continuously in time) or switching control (i.e., piecewise continuous application of vaccination or treatment schemes) for evaluation and optimization. As discussed earlier, infectious disease models are a crucial component in designing and implementing detection, prevention, and control programs (e.g., WHO’s program against smallpox, leading to its global eradication by 1977). The switched SIR model is first returned to analyze vaccination of the susceptible group (e.g., newborns or the entire cohort). Subsequently, the developed theoretical methods are applied to the switched SIR model with a treatment program in effect. Common Lyapunov functions are used to provide controlled eradication of diseases modeled by the so-called SEIR (Susceptible-Exposed-Infected-Recovered) model with seasonal variations captured by switching. A screening process, where traveling individuals are examined for infection, is proposed and studied for the switched multi-city model of the previous chapter. Switching control of diseases such as dengue and chikungunya, which are spread via mosquito–human interactions, is also investigated.

5.1 Vaccination of the Susceptible Group

The majority of developed countries have in place cohort immunization programs (also called time-constant immunization or vaccination programs here) for a number of diseases with varying degrees of success [1]. For example, measles immunization in many areas of the Western world recommends vaccinations at 15 months of age and 6 years of age [139]. Studies analyzing this type of program mathematically can be found in, for example, [4, 69, 75, 83, 92, 100, 101, 102, 110, 138, 147, 173].

The mathematical formulation of a newborn continuous vaccination strategy takes the following form [69, 138, 173]: assume that a fraction ρ ∈ [0, 1] of susceptible newborns are vaccinated, moving them to the recovered class R, continuously in time. In this model, natural and vaccine-acquired immunity are viewed as the same. Applied to the classical endemic model SIR model ( 3.9) gives
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& = (1-\rho )\mu -\beta S(t)I(t) -\mu S(t), \\ \dot{I}(t)& =\beta S(t)I(t) - (g+\mu )I(t), \\ \dot{R}(t)& =\mu \rho +gI(t) -\mu R(t).\end{array} }$$
(5.1)
Newborn vaccinations reduce the birth rate μ of the susceptible population to (1 −ρ)μ. Equation (5.1) admits the following equilibria: a disease-free solution (1 −ρ, 0, ρ) ≡ QDFS(5. 1) and an endemic solution
$$\displaystyle{ Q_{\mathrm{ES}}^{(5.1)} \equiv \left (\frac{\mu +g} {\beta }, \frac{\mu } {\beta }(\mathrm{R}_{0}^{(5.1)} - 1), \frac{g} {\beta } (\mathrm{R}_{0}^{(5.1)} - 1)+\rho \right ), }$$
(5.2)
where
$$\displaystyle{ \mathrm{R}_{0}^{(5.1)} \equiv \frac{\beta (1-\rho )} {\mu +g} }$$
is the basic reproduction number of (5.1). The underlying mechanics of the newborn vaccination can be translated into something more familiar by the following change of variables [69]: let \(S \equiv \widehat{ S}(1-\rho )\), \(I \equiv \widehat{ I}(1-\rho )\), and \(R \equiv \widehat{ R}(1-\rho )+\rho\). Then (5.1) is equivalently written as
$$\displaystyle{ \begin{array}{rl} \frac{d\widehat{S}} {dt} (t)& =\mu -\beta (1-\rho )\widehat{S}(t)\widehat{I}(t) -\mu \widehat{ S}(t), \\ \frac{d\widehat{I}} {dt} (t)& =\beta (1-\rho )\widehat{S}(t)\widehat{I}(t) - (g+\mu )\widehat{I}(t), \\ \frac{d\widehat{R}} {dt} (t)& = g\widehat{I}(t) -\mu \widehat{ R}(t). \end{array} }$$
(5.3)
This control strategy therefore has the effect of transforming the contact rate from β to β(1 −ρ). This is most clearly reflected in the basic reproduction number R0(5. 1), which dictates the usual threshold for long-term behavior (i.e., disease eradication versus endemicity). The condition R0(5. 1) < 1 yields a critical vaccination rate to achieve herd immunity [63]:
$$\displaystyle{ \rho _{\mbox{ crit}} \equiv 1 - 1/\mathrm{R}_{0}^{(3.8)} \in [0,1). }$$
With seasonality modeled by a switched contact rate β σ (where \(\sigma \in \mathcal{S}_{\mbox{ dwell}}\)), the model is given by
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu (1-\rho ) -\beta _{\sigma }S(t)I(t) -\mu S(t), \\ \dot{I}(t)& =\beta _{\sigma }S(t)I(t) - (g+\mu )I(t), \\ \dot{R}(t)& = gI(t) -\mu R(t)+\rho \mu, \\ (S(0),I(0),R(0))& = (S_{0},I_{0},R_{0}), \end{array} }$$
(5.4)
with physical domain
$$\displaystyle{ D_{(5.4)} \equiv \{ (S,I,R) \in \mathbb{R}_{+}^{3}: S + I + R = 1\} = D_{ (3.8)} }$$
(which is positively invariant to (5.4)). See Fig. 5.1 for the flow diagram associated with (5.4).
Fig. 5.1

Flow of the switched SIR system with newborn vaccinations (5.4). The red line represents the horizontal transmission and the blue line represents the vaccination scheme

Although the present focus is on disease eradication by control, we mention that each mode admits an endemic equilibria of the form
$$\displaystyle{ Q_{\mathrm{ES}}^{(5.4),i} \equiv \left (\frac{\mu +g} {\beta _{i}}, \frac{\mu } {\beta _{i}}(\mathrm{R}_{0}^{(5.4),i} - 1), \frac{g} {\beta _{i}} (\mathrm{R}_{0}^{(5.4),i} - 1)+\rho \right ),\quad \forall i \in \mathcal{M}, }$$
(5.5)
with mode basic reproduction numbers
$$\displaystyle{ \mathrm{R}_{0}^{(5.4),i} \equiv \frac{\beta _{i}} {\mu +g}(1-\rho ) = (1-\rho )\mathrm{R}_{0}^{(3.8),i},\quad \forall i \in \mathcal{M}. }$$
(5.6)
Recall Theorem  3.1, in which the switched SIR model ( 3.8) was shown to achieve eradication if
$$\displaystyle{ \mathrm{R}_{0}^{(3.8)} = \frac{1} {\omega } \sum _{i=1}^{m}\mathrm{R}_{ 0}^{(3.8),i}\tau _{ i} = \frac{\sum _{i=1}^{m}\beta _{i}\tau _{i}} {\omega (\mu +g)} < 1 }$$
whenever \(\sigma \in \mathcal{S}_{\mbox{ periodic}}(\omega )\) (in such a way that the disease-free solution is globally asymptotically I-stable). Moreover, R0(3. 8) > 1 implies persistence of the disease (see Theorem  3.4). In contrast, consider the following theorem.

Theorem 5.1

If \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and
$$\displaystyle{ \mathrm{R}_{0}^{(5.4)} \equiv \frac{1} {\omega } \sum _{i=1}^{m}\mathrm{R}_{ 0}^{(5.4),i}\tau _{ i} = \frac{\sum _{i=1}^{m}\beta _{i}(1-\rho )\tau _{i}} {\omega (\mu +g)} < 1, }$$
then the disease-free solution \(Q_{\mathrm{DFS}}^{(5.4)} \equiv (1-\rho,0,\rho )\) of the switched SIR system with newborn vaccination (5.4) is globally attractive and globally asymptotically I-stable in the meaningful domain.

Proof

By (5.4),
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& =& \mu (1-\rho ) -\beta _{\sigma }S(t)I(t) -\mu S(t), {}\\ & \leq & \mu (1-\rho ) -\mu S(t). {}\\ \end{array}$$
Consider the comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}(t)& =\mu (1-\rho ) -\mu x(t), \\ x(0)& = S_{0}, \end{array} }$$
(5.7)
which has unique solution x(t) ≡ (S0 − (1 −ρ))exp(−μ t) + (1 −ρ) that satisfies
$$\displaystyle{ \lim _{t\rightarrow \infty }x(t) = 1 -\rho. }$$
By the comparison theorem, for any ε > 0 there exists a time t > 0 such that S(t) ≤ x(t) ≤ 1 −ρ +ε for t ≥ t, and so
$$\displaystyle\begin{array}{rcl} \dot{I}(t)& =& \beta _{\sigma }S(t)I(t) - (\mu +g)I(t), {}\\ & \leq & (\beta _{\sigma }([1 -\rho +\epsilon ] -\mu -g)I(t), {}\\ & \equiv & \lambda _{\sigma,\epsilon }I(t), {}\\ \end{array}$$
where λi, ε ≡ β i (1 −ρ) − gμ +ε β i for each \(i \in \mathcal{M}\). Choose N to be the smallest integer such that N ω > t. Then, as in the proof of Theorem  3.1,
$$\displaystyle\begin{array}{rcl} I((N + 1)\omega )& \leq & I(N\omega )\exp \left (\sum _{i=1}^{m}\lambda _{ i,\epsilon }\tau _{i}\right ), {}\\ & =& \eta (\epsilon )I(N\omega ), {}\\ \end{array}$$
where
$$\displaystyle{ \eta (\epsilon ) \equiv \exp \left (\sum _{i=1}^{m}\lambda _{ i,\epsilon }\tau _{i}\right ). }$$
Now, R0(5. 4) < 1 gives that i = 1 m λ i τ i  < 0. Then it holds that i = 1 m λ i τ i  < −δ for some δ > 0, and
$$\displaystyle{ \sum _{i=1}^{m}\lambda _{ i,\epsilon }\tau _{i} =\sum _{ i=1}^{m}\lambda _{ i}\tau _{i} +\epsilon \sum _{ i=1}^{m}\beta _{ i}\tau _{i} < -\delta +\epsilon \sum _{ i=1}^{m}\beta _{ i}\tau _{i}. }$$
Choosing
$$\displaystyle{ \epsilon = \frac{\delta } {2\sum _{i=1}^{m}\beta _{i}\tau _{i}} }$$
implies that η(ε) < 1. It can be similarly shown that I((N + h + 1)ω) ≤ η I((N + h)ω) for any integer \(h \in \mathbb{N}\) and the rest of the proof of Theorem  3.1 may be applied to produce the result.
The threshold condition R0(5. 4) < 1 defines a critical newborn vaccination rate:
$$\displaystyle{ \mathrm{R}_{0}^{(5.4)} = \frac{\sum _{i=1}^{m}\beta _{ i}(1-\rho )\tau _{i}} {\omega (\mu +g)} < 1 }$$
implies that
$$\displaystyle{ (1-\rho )\frac{\sum _{i=1}^{m}\beta _{i}\tau _{i}} {\omega (\mu +g)} = (1-\rho )\mathrm{R}_{0}^{(3.8)} < 1, }$$
and hence the critical rate is given as
$$\displaystyle{ \rho _{\mbox{ crit}} \equiv 1 - 1/\mathrm{R}_{0}^{(3.8)} = 1 - \frac{\omega (\mu +g)} {\sum _{i=1}^{m}\beta _{i}\tau _{i}} \in [0,1), }$$
which guarantees disease eradication. That is, if the disease persists in the switched SIR model (R0(3. 8) > 1), then disease eradication can be achieved by newborn vaccinations as long as ρ ≥ ρcrit. If R0(3. 8) = 1 then ρcrit = 0 and as R0(3. 8) →  then ρcrit → 1. Other controlled eradication results can also be shown under different classes of switching rules, as in Sect.  3.4 (i.e., if \(\sigma \in \mathcal{S}_{\mbox{ dwell}}\) according to Theorems  3.2 and  3.3).

Example 5.1

Consider (5.4) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), and initial conditions (S0, I0, R0) = (0. 75, 0. 25, 0). Motivated by the measles parameters of [138], let β1 = 18, β2 = 3, g = 1, μ = 0. 1, and ρ = 0 which give that R0(5. 4) = 6. 136 and persistence of the disease, i.e., by Theorem  3.4 (see Fig. 5.2 for an illustration; the solution I oscillates approximately between the endemic minimum and maximum, Imin = 0. 0576 and Imax = 0. 0854). With ρ = 0. 85 (ρcrit = 0. 84), R0(5. 4) = 0. 920 and the disease is eradicated according to Theorem 5.1 (see Fig. 5.3).
Fig. 5.2

Simulation of Example 5.1. (a ) ρ = 0. (b ) The black lines represent Imin = 0. 0576 and Imax = 0. 0854

Fig. 5.3

Simulation of Example 5.1 with ρ = 0. 85

Instead of a newborn vaccination strategy, consider an immunization strategy applied to the entire susceptible cohort in an SIR model ( 3.8). Mathematically, suppose that the susceptible population is vaccinated at a rate v ≥ 0 per unit time and again assume permanent immunity is acquired through vaccination (which is indistinguishable from naturally acquired immunity). Thus, the dynamics of the model
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -\beta _{\sigma }S(t)I(t) -\mu S(t) - vS(t), \\ \dot{I}(t)& =\beta _{\sigma }S(t)I(t) - (g+\mu )I(t), \\ \dot{R}(t)& = gI(t) + vS(t) -\mu R(t), \\ (S(0),I(0),R(0))& = (S_{0},I_{0},R_{0}), \end{array} }$$
(5.8)
are investigated. The flow diagram for (5.8) is shown in Fig. 5.4.
Fig. 5.4

Flow of the switched SIR system with susceptible vaccinations (5.8). The red line represents the horizontal transmission and the blue line represents the vaccination scheme

As before, the meaningful domain is unchanged (D(5. 8) = D(3. 8)) and positively invariant (thus giving a global unique solution for appropriate initial conditions). However, the set of mode basic reproduction numbers of (5.8) is changed from the uncontrolled switched SIR model (as expected):
$$\displaystyle{ \mathrm{R}_{0}^{(5.8),i} \equiv \frac{\beta _{i}} {\mu +g} \frac{\mu } {\mu +v},\quad \forall i \in \mathcal{M}, }$$
(5.9)
while the disease-free solution is calculated as
$$\displaystyle{ Q_{\mathrm{DFS}}^{(5.8)} \equiv \left ( \frac{\mu } {\mu +v},0,1 - \frac{\mu } {\mu +v}\right ). }$$
(5.10)
Each mode \(i \in \mathcal{M}\) admits an endemic equilibrium:
$$\displaystyle{ Q_{\mathrm{ES}}^{(5.8),i} \equiv \left (\frac{\mu +g} {\beta _{i}}, \frac{\mu } {\mu +g}\left (1 - \frac{1} {\mathrm{R}_{0}^{(5.8),i}}\right ), \frac{\mu } {\mu +g}\left (1 - \frac{1} {\mathrm{R}_{0}^{(5.8),i}}\right ) + \frac{v} {\mu } \frac{\mu +g} {\beta _{i}} \right ). }$$
(5.11)
Different from the newborn vaccination scheme, (5.8) gives that
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& =& \mu -\beta _{\sigma }S(t)I(t) -\mu S(t) - vS(t), {}\\ & \leq & \mu -(\mu +v)S(t), {}\\ \end{array}$$
which motivates analyzing the comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}(t)& =\mu -(\mu +v)x(t), \\ x(0)& = S_{0},\end{array} }$$
(5.12)
that has unique solution
$$\displaystyle{ x(t) \equiv \left (S_{0} - \frac{\mu } {\mu +v}\right )\exp (-(\mu +v)t) + \frac{\mu } {\mu +v}, }$$
satisfying
$$\displaystyle{ \lim _{t\rightarrow \infty }x(t) = \frac{\mu } {\mu +v} }$$
(the first component of the disease-free solution). Thus, the same analysis as in Theorem 5.1 yields that the solution of (5.8) converges to the disease-free solution QDFS(5. 8) (which is globally asymptotically I-stable in the meaningful domain) if
$$\displaystyle{ \mathrm{R}_{0}^{(5.8)} \equiv \frac{1} {\omega } \sum _{i=1}^{m}\mathrm{R}_{ 0}^{(5.8),i}\tau _{ i} = \frac{\sum _{i=1}^{m}\beta _{i}\tau _{i}} {\omega (\mu +g)} \frac{\mu } {\mu +v}; }$$
the critical cohort immunization rate as
$$\displaystyle{ v_{\mbox{ crit}} \equiv \mu (\mathrm{R}_{0}^{(3.8)} - 1) =\mu \left (\frac{\sum _{i=1}^{m}\beta _{ i}\tau _{i}} {\omega (\mu +g)} - 1\right ) \in \mathbb{R}_{+} }$$
(assuming that R0(3. 8) ≥ 1).

Example 5.2

Consider (5.8) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), and initial conditions (S0, I0, R0) = (0. 75, 0. 25, 0). Motivated by the measles parameters of [138], let β1 = 18, β2 = 3, g = 1, μ = 0. 1, and v = 0. 57 (vcrit = 0. 51). Then R0(5. 8) = 0. 92 (see Fig. 5.5 for an illustration).
Fig. 5.5

Simulation of Example 5.2 with v = 0. 57

The vaccination models thus far assume immediate movement from susceptible to vaccinated. This ignores the time it takes to obtain immunity by completing a vaccination program. The following assumptions are made [101]:
  1. 1.

    The mean period of vaccine-induced immunity is 1∕γ for some γ > 0.

     
  2. 2.

    Individuals in the vaccinated class contract the disease at a reduced rate β σ V (i.e., β i V  < β i for each \(i \in \mathcal{M}\) since individuals may have partial immunity during the vaccination process).

     
Under these assumptions, the SVIR model with switching is written as
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -\beta _{\sigma }S(t)I(t) -\mu S(t) - vS(t), \\ \dot{V }(t)& = vS(t) -\beta _{\sigma }^{V }V (t)I(t) -\gamma V (t) -\mu V (t), \\ \dot{I}(t)& =\beta _{\sigma }S(t)I(t) +\beta _{ \sigma }^{V }V (t)I(t) - gI(t) -\mu I(t), \\ \dot{R}(t)& = gI(t) +\gamma V (t) -\mu R(t), \\ (S(0),V (0),I(0),R(0))& = (S_{0},V _{0},I_{0},R_{0}).\end{array} }$$
(5.13)
For (5.13), the set of mode basic reproduction numbers can be calculated as follows:
$$\displaystyle{ \mathrm{R}_{0}^{(5.13),i} \equiv \left ( \frac{\beta _{i}} {\mu +g} + \frac{\beta _{i}^{V }} {\mu +g} \frac{v} {\mu +\gamma }\right ) \frac{\mu } {\mu +v},\quad \forall i \in \mathcal{M}. }$$
(5.14)
The flow diagram of (5.13) is illustrated in Fig. 5.6.
Fig. 5.6

Flow diagram of the switched SVIR system (5.13). The red line represents the transmission of the disease and the blue line represents the vaccination

Observe that as the efficacy of the vaccine is increased (i.e., β i V decreases or γ increases), the mode reproduction numbers reduce to those of the SIR model ( 3.8) (and are equal in the limit β i V  → 0 for each i or γ → ). However, as noted in [101], increasing the efficacy of the vaccine is usually more difficult than controlling the vaccination rate v. There is a single disease-free equilibrium point [101]:
$$\displaystyle{ Q_{\mathrm{DFS}}^{(5.13)} \equiv (\bar{S},\bar{V },\bar{I},\bar{R}) \equiv \left ( \frac{\mu } {\mu +v}, \frac{v\mu } {(\mu +\gamma )(\mu +v)},0, \frac{v\gamma } {(\mu +\gamma )(\mu +v)}\right ) }$$
(5.15)
and, as per usual, any mode for which R0(5. 8), i ≥ 1 admits an endemic equilibrium
$$\displaystyle{ Q_{\mathrm{ES}}^{(5.13),i} \equiv (S_{ i}^{{\ast}},V _{ i}^{{\ast}},I_{ i}^{{\ast}},R_{ i}^{{\ast}}),\quad \forall i \in \mathcal{M}, }$$
where I i is the positive root of the function IA1I2 + A2I + A3(1 − R0(5. 8), i) where
$$\displaystyle\begin{array}{rcl} A_{1}& \equiv & (\mu +g)\beta _{i}\beta _{i}^{V } > 0,\quad \forall i \in \mathcal{M}, {}\\ A_{2}& \equiv & (\mu +g)[(\mu +v)\beta _{i}^{V } + (\mu +\gamma )\beta _{ i}] -\beta _{i}^{V }\beta _{ i}\mu,\quad \forall i \in \mathcal{M}, {}\\ A_{3}& \equiv & (\mu +g)(\mu +v)(\mu +\gamma ) > 0,\quad \forall i \in \mathcal{M}, {}\\ \end{array}$$
and
$$\displaystyle\begin{array}{rcl} S_{i}^{{\ast}}& \equiv & \frac{\mu } {\mu +v +\beta _{i}I_{i}^{{\ast}}},\quad \forall i \in \mathcal{M}, {}\\ V _{i}^{{\ast}}& \equiv & \frac{v\mu } {(\mu +v +\beta _{i}I_{i}^{{\ast}})(\mu +\gamma +\beta _{ i}^{V }I_{i}^{{\ast}})},\quad \forall i \in \mathcal{M}, {}\\ R_{i}^{{\ast}}& \equiv & 1 - S_{ i}^{{\ast}}- I_{ i}^{{\ast}}- V _{ i}^{{\ast}},\quad \forall i \in \mathcal{M}. {}\\ \end{array}$$

Theorem 5.2

If \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and
$$\displaystyle{ \mathrm{R}_{0}^{(5.13)} \equiv \frac{1} {\omega } \sum _{i=1}^{m}\mathrm{R}_{ 0}^{(5.13),i}\tau _{ i} < 1, }$$
then the disease-free solution \(Q_{\mathrm{DFS}}^{(5.13)}\) of the switched SIR system with progressive immunity (5.13) is globally attractive and globally asymptotically I-stable in the meaningful domain.

Proof

Observe from (5.8) that
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& =& \mu -\beta _{\sigma }S(t)I(t) -\mu S(t) - vS(t), {}\\ & \leq & \mu -(\mu +v)S(t). {}\\ \end{array}$$
Similarly,
$$\displaystyle\begin{array}{rcl} \dot{V }(t)& =& vS(t) -\beta _{\sigma }^{V }V (t)I(t) -\gamma V (t) -\mu V (t), {}\\ & \leq & vS(t) -\gamma V (t) -\mu V (t). {}\\ \end{array}$$
The comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}(t)& =\mu -(\mu +v)x(t), \\ \dot{y}(t)& = vx(t) - (\gamma +\mu )y(t), \\ (x(0),y(0))& = (S_{0},V _{0}), \end{array} }$$
(5.16)
gives the appropriately needed result; for any ε > 0, there exists t > 0 such that \(S(t) \leq x(t) \leq \bar{ S}+\epsilon\) and \(V (t) \leq y(t) \leq \bar{ V }+\epsilon\) for t ≥ t. Returning to the differential equation for I,
$$\displaystyle\begin{array}{rcl} \dot{I}(t)& =& \beta _{\sigma }S(t)I(t) +\beta _{ \sigma }^{V }V (t)I(t) - gI(t) -\mu I(t), {}\\ & \leq & (\beta _{\sigma }[\bar{S}+\epsilon ] +\beta _{ \sigma }^{V }[\bar{V }+\epsilon ] -\mu -g)I(t), {}\\ & =& \lambda _{\sigma,\epsilon }I(t), {}\\ \end{array}$$
where \(\lambda _{i,\epsilon } \equiv \beta _{\sigma }[\bar{S}+\epsilon ] +\beta _{ \sigma }^{V }[\bar{V }+\epsilon ] -\mu -g\) for each \(i \in \mathcal{M}\). The condition R0(5. 4) < 1 gives that
$$\displaystyle{ \sum _{i=1}^{m}(\beta _{ i}\bar{S} +\beta _{ \sigma }^{V }\bar{V } -\mu -g)\tau _{ i} < 0. }$$
As in the proof of Theorem 5.1, ε > 0 can be chosen sufficiently small so that i = 1 m λi, ετ i  < 0 and it follows that limt → I(t) = 0. The limiting equation for S is \(\dot{S}(t) =\mu -\mu S(t) - vS(t)\); S converges to \(\bar{S} =\mu /(\mu +v)\), and the limiting equation for V is \(\dot{V }(t) = v\mu /(\mu +v) -\gamma V (t) -\mu V (t)\), from which it follows that V converges to \(\bar{V }\). Finally, the limiting equation for R is \(\dot{R}(t) =\gamma v\mu /[(\mu +v)(\gamma +\mu )] -\mu R(t)\), from which convergence of R to \(\bar{R}\) follows. Therefore, the solution of system (5.13) converges to the disease-free equilibrium \(Q_{\mathrm{DFS}}^{(5.13)}\). Asymptotic I-stability follows as usual.
The critical vaccination rate in the case of progressive immunity is calculated by setting
$$\displaystyle{ \mathrm{R}_{0}^{(5.13)} = \frac{1} {\omega } \left (\frac{\sum _{i=1}^{m}\beta _{i}\tau _{i}} {\mu +g} + \frac{\sum _{i=1}^{m}\beta _{i}^{V }\tau _{i}} {\mu +g} \frac{v} {\mu +\gamma }\right ) \frac{\mu } {\mu +v} = 1. }$$
Namely,
$$\displaystyle{ v_{\mbox{ crit}} \equiv \mu \left (\frac{\sum _{i=1}^{m}\beta _{i}\tau _{i}} {\omega (\mu +g)} - 1\right )\left (1 -\mu \frac{\sum _{i=1}^{m}\beta _{i}^{V }\tau _{i}} {\omega (\mu +g)} \right )^{-1}. }$$

Example 5.3

Consider (5.13) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), and initial conditions (S0, V0, I0, R0) = (0. 75, 0, 0. 25, 0). Given β1 = 18, β2 = 3, g = 1, μ = 0. 1, γ = 1 and vaccine-reduced contact rates β1 V  = 1 and β2 V  = 0. 17. Then v = 0. 8 (vcrit = 0. 51) implies that \(\mathrm{R}_{0}^{(5.13)} = 0.580\) (see Fig. 5.7 for an illustration).
Fig. 5.7

Simulation of Example 5.3 with v = 0. 8

5.2 Treatment Schedules for Classes of Infected

The control strategy of treating infections is investigated. More specifically, a piecewise constant switching control is presented. Assume that p i  ≥ 0, \(i \in \mathcal{M}\), are treatment rates, per unit time, of the infected population which may be applied to the infected population. The value p i can be broken down as p i  = ν i q where 1∕q > 0 is average treatment period and ν i  > 0 is the treatment success rate. Assuming movement to the recovered class from the treatment process, the switched system is written as follows:
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -\beta _{\sigma }S(t)I(t) -\mu S(t), \\ \dot{I}(t)& =\beta _{\sigma }S(t)I(t) - gI(t) -\mu I(t) - p_{\sigma }I(t), \\ \dot{R}(t)& = gI(t) -\mu R(t) + p_{\sigma }I(t).\end{array} }$$
(5.17)
The variables here have been normalized by the total population (since S + I + R = 1 is an invariant of (5.17)). Indeed, the physically meaningful domain of (5.17) is equal to
$$\displaystyle{ D_{(5.17)} \equiv \{ (S,I,R) \in \mathbb{R}_{+}^{3}: S + I + R = 1\} = D_{ (3.8)}. }$$
See Fig. 5.8 for an illustration of the flow diagram for (5.17).
Fig. 5.8

Flow of the switched SIR system with treatment (5.17). The red line represents the horizontal transmission and the blue line represents the treatment strategy

The treatment rate acts to reduce the average infectious period (from an average of 1∕(μ + g) to 1∕(μ + g + p i )); the set of mode basic reproduction numbers are reduced as
$$\displaystyle{ \mathrm{R}_{0}^{(5.17),i} \equiv \frac{\beta _{i}} {\mu +g + p_{i}} \leq \frac{\beta _{i}} {\mu +g} =\mathrm{ R}_{0}^{(3.8),i},\quad \forall i \in \mathcal{M}. }$$
(5.18)
Disease eradication by switching treatment can immediately be proved from the techniques of Sect.  3.4 by making the following observation:
$$\displaystyle{ \dot{I}(t) =\beta _{\sigma }S(t)I(t) - gI(t) -\mu I(t) - p_{\sigma }I(t) \leq \lambda _{i}I(t), }$$
where λ i  ≡ β i gμp i for each i. By repeating the standard attractivity and partial stability switched systems methods already used, the following result is provided.

Theorem 5.3

Consider the switched SIR model with switching treatment (5.17) . Global attractivity of the disease-free solution \(Q_{\mathrm{DFS}}^{(5.17)} \equiv (1,0,0)\) holds under any of the following conditions:
  1. (i)

    \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and \(\mathrm{R}_{0}^{(5.17)} \equiv \frac{1} {\omega } \sum _{i=1}^{m}\mathrm{R}_{ 0}^{(5.17),i}\tau _{ i} < 1;\)

     
  2. (ii)
    \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) and there exists h > 0 such that
    $$\displaystyle{ \left < \mathrm{R}_{0}^{(5.17)}\right > \equiv \sup _{ t\geq h} \frac{\sum _{i=1}^{m}\beta _{i}T_{i}(t)} {t(\mu +g) +\sum _{ i=1}^{m}p_{i}T_{i}(t)} < 1; }$$
    (5.19)
     
  3. (iii)
    \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) satisfies T+ ≤ N0 + qT(t) for some q ∈ (0,1) and N0 ≥ 0 such that
    $$\displaystyle{ \max \{\mathrm{R}_{0}^{(5.17),i}: i \in \mathcal{M}^{-}\}- 1 < q(\max \{\mathrm{R}_{ 0}^{(5.17),i}: i \in \mathcal{M}^{+}\} - 1), }$$
    where \(\mathcal{M}^{-}\equiv \{ i \in \mathcal{M}:\mathrm{ R}_{0}^{(5.17),i} < 1\}\) and \(\mathcal{M}^{+} \equiv \{ i \in \mathcal{M}:\mathrm{ R}_{0}^{(5.17),i} \geq 1\}\).
     

On the other hand, if \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and \(\mathrm{R}_{0}^{(5.17)} > 1\), then the disease persists uniformly in (5.17).

In the setting of Theorem 5.3, case (i) also implies asymptotic I-stability of QDFS(5. 17) in the meaningful domain. Cases (ii)-(iii) give exponential I-stability.

Example 5.4

Consider (5.17) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), initial conditions (S0, I0, R0) = (0. 75, 0. 25, 0), and model parameters β1 = 18, β2 = 3, g = 1, μ = 0. 1. Given p = 1 (recall v = 0. 57 ensured disease eradication in the cohort immunization scheme (5.8)), then \(\mathrm{R}_{0}^{(5.17)} = 3.21\) and the scheme is ineffective (see Fig. 5.9 for an illustration).
Fig. 5.9

Simulation of Example 5.4 with p = 1

This treatment strategy can be extended to generalized forces of infections (recall the formulation in the switched SIR model with general switched incidence rates ( 3.29)): suppose that the incidence rate takes the form (t, S, I) ↦ h σ (I)S to give the system
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -h_{\sigma }(I(t))S(t) -\mu S(t), \\ \dot{I}(t)& = h_{\sigma }(I(t))S(t) - (g +\mu +p_{\sigma })I(t), \\ \dot{R}(t)& = (g + p_{\sigma })I(t) -\mu R(t), \\ (S(0),I(0),R(0))& = (S_{0},I_{0},R_{0}), \end{array} }$$
(5.20)
where the forces of infection h i are assumed to satisfy necessary physical conditions (i.e., so that (a)–(d) in Sect.  3.5 are satisfied by f i (S, I) ≡ h i (I)S). The treatment rate can be used to control the disease to eradication, via the set of mode reproduction numbers
$$\displaystyle{ \mathrm{R}_{0}^{(5.20),i} \equiv \frac{1} {\mu +g + p_{i}} \frac{dh_{i}} {dI} (0),\quad \forall i \in \mathcal{M}, }$$
as follows.

Theorem 5.4

Assume that \(h_{i} \in C^{2}([0,1], \mathbb{R}_{+})\) satisfies \(\frac{d^{2}h_{ i}} {dI^{2}} (I) \leq 0\) for all I ∈ [0,1], \(i \in \mathcal{M}\) . If \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) and
$$\displaystyle{ p_{i} > \frac{dh_{i}} {dI} (0) - (\mu +g),\quad \forall i \in \mathcal{M}, }$$
(5.21)
then \(Q_{\mathrm{DFS}}^{(5.20)} \equiv (1,0,0)\) is globally asymptotically stable in the meaningful domain
$$\displaystyle{ D_{(5.20)} \equiv \{ (S,I,R) \in \mathbb{R}_{+}^{3}: S + I + R = 1\}; }$$
the disease is eradicated by the switching treatment control.

Proof

Define the mapping
$$\displaystyle{ V (S,I) \equiv S -\ln (S) + I - 1 }$$
which is continuously differentiable on
$$\displaystyle{ \varOmega _{SI}^{\epsilon } \equiv \{ (S,I) \in \mathbb{R}_{ +}^{2}: S + I \leq 1\} \cap \{ (S,I): S \geq \epsilon \}=\{ (S,I): S \geq \epsilon,S + I \leq 1\}, }$$
for ε > 0 [73]. Observe that V (1, 0) = 0, V > 0 for (S, I) ∈ Ω S I ε {(1, 0)},
$$\displaystyle{ \frac{\partial V } {\partial S} (S,I) = 1 - 1/S,\quad \frac{\partial ^{2}V } {\partial S^{2}} (S,I) = 1/S^{2},\quad \frac{\partial V } {\partial I} (S,I) = 1,\quad \frac{\partial ^{2}V } {\partial I^{2}} (S,I) = 0, }$$
implying that (S, I) = (1, 0) is the unique (global) minimum of the Lyapunov function in Ω S I ε . The time-derivative of V along trajectories of (5.20) yields that
$$\displaystyle\begin{array}{rcl} \dot{V }_{(1.2)}(t,S,I,R)& =& \left (1 - 1/S\right )(\mu -h_{\sigma }(I)S -\mu S) + h_{\sigma }(I)S - (\mu +g + p_{\sigma })I, {}\\ & =& \mu \left [\left (1 - 1/S\right )\left (1 - S\right )\right ] + (\mu +g + p_{\sigma })I\left ( \frac{h_{\sigma }(I)} {(\mu +g + p_{\sigma })I} - 1\right ). {}\\ \end{array}$$
Proceed by arguments in [73]: observe that (1 − 1∕S)(1 − S) < 0 for ε ≤ S < 1; (1 − 1∕S)(1 − S) = 0 if S = 1. The concavity condition on the set of functions h i implies that \(h_{i}(I)/I \leq \frac{dh_{i}} {dI} (0)\) for all I > 0. It follows that
$$\displaystyle{ \frac{h_{i}(I)} {(\mu +g + p_{i})I} \leq \frac{1} {\mu +g + p_{i}} \frac{dh_{i}} {dI} (0) \leq \mathrm{ R}_{0}^{(5.20),i},\quad \forall i \in \mathcal{M}. }$$
The condition (5.21) implies that
$$\displaystyle{ \mathrm{R}_{0}^{(5.20),i} < 1,\quad \forall i \in \mathcal{M}. }$$
Moreover, h i (I)∕[(μ i + g i )I] − 1 < 0 for each \(i \in \mathcal{M}\), so that \(\dot{V }_{(5.20)}(t,S,I,R) < 0\) holds unless (S, I) = (1, 0) and the arbitrary choice of ε yields global asymptotic stability of (1, 0) in Ω S I ε → 0. The equation R = 1 − IS implies the conclusion holds in D(5. 20).

Example 5.5

Consider (5.20) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), initial conditions (S0, I0, R0) = (0. 75, 0. 25, 0), and h i (I) = β i sin(π I∕2) for i = 1, 2. Let β1 = 4, β2 = 1. 6, g = 1. 9, μ = 0. 1. Observe that f i (S, I) ≡ h i (I)S satisfies f i (t, S, I) > 0 for S, I ≠ 0, f i (t, S, 0) = f i (t, 0, I) = 0, \(\frac{\partial f_{i}(t,S,I)} {\partial I} > 0\) for 0 ≤ I < 1. Then p1 = 5 and p2 = 1 imply that (5.21) holds and global asymptotic stability of the disease-free solution by Theorem 5.4. On the other hand, if p1 = p2 = 0 then, since β i sin(π I∕2)S ≥ β i S I for each i, the disease persists by a straightforward calculation of
$$\displaystyle{ \overline{\mathrm{R}_{0}}^{(3.29)} = 1.1 }$$
(where \(\overline{\mathrm{R}_{0}}^{(3.29)}\) is outlined in Sect.  3.5). The situation is illustrated in Fig. 5.10.
Fig. 5.10

Simulations of Example 5.5. (a ) p1 = p2 = 0. (b ) p1 = 5, p2 = 1

5.3 Introduction of the Exposed: A Controlled SEIR Model

A number of diseases exhibit a period of latency where individuals have been infected but are not yet infectious. (An incubation period is the time between infection and clinical onset of the disease; i.e., appearance of symptoms.) Motivated by this fact, we re-examine the assumption made earlier of a negligible latency period; assume that once a susceptible individual makes an adequate contact with an infected individual they enter a latent period before becoming infectious. Let E denote the class of individuals who have been exposed but are not yet infectious. Assume that individuals who have been exposed become infectious at a rate a > 0 (i.e., average incubating period of 1∕a). With other physiological and epidemiological assumptions matching those of the classical endemic model (i.e., ( 3.9)), the model is given by the following dynamic system:
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -\beta S(t)I(t) -\mu S(t), \\ \dot{E}(t)& =\beta S(t)I(t) - aE(t) -\mu E(t), \\ \dot{I}(t)& = aE(t) - gI(t) -\mu I(t), \\ \dot{R}(t)& = gI(t) -\mu R(t), \\ (S(0),E(0),I(0),R(0))& = (S_{0},E_{0},I_{0},R_{0}), \end{array} }$$
(5.22)
where
$$\displaystyle{ (S_{0},I_{0},E_{0},R_{0}) \in D_{(5.22)} \equiv \{ (S,E,I,R) \in \mathbb{R}_{+}^{4}: S + E + I + R = 1\}, }$$
which is invariant to (5.22); \(\{\dot{S} +\dot{ E} +\dot{ I} +\dot{ R}\}\vert _{S+I+E+R=1} = 0\), \(\dot{S}\vert _{S=0} =\mu > 0\), \(\dot{E}\vert _{E=0} =\beta SI \geq 0\), \(\dot{I}\vert _{I=0} = 0\), and \(\dot{R}\vert _{R=0} = gI \geq 0\). The basic reproduction number of (5.22) is calculated as
$$\displaystyle{ \mathrm{R}_{0}^{(5.22)} \equiv \frac{\beta a} {(\mu +g)(\mu +a)}; }$$
(5.23)
the average number of new cases is the product of the contact rate, β, the average fraction surviving the latent period, a∕(a +μ), and the average infectious period 1∕(μ + g) [65]. There is a single disease-free equilibrium
$$\displaystyle{ Q_{\mathrm{DFS}}^{(5.22)} \equiv (1,0,0,0) }$$
and an endemic equilibrium:
$$\displaystyle\begin{array}{rcl} Q_{\mathrm{ES}}^{(5.22)} \equiv \left ( \frac{1} {\mathrm{R}_{0}^{(5.22)}}, \frac{\mu (\mu +g)} {\beta a} (\mathrm{R}_{0}^{(5.22)} - 1), \frac{\mu } {\beta }(\mathrm{R}_{0}^{(5.22)} - 1), \frac{g} {\beta } (\mathrm{R}_{0}^{(5.22)} - 1)\right ).& & {}\\ \end{array}$$
The invariant S + E + I + R = 1 implies that the equation for R can be omitted (i.e., (5.22) is intrinsically three-dimensional).
Recall the basic reproduction number of the classical endemic SIR model (see Eq. ( 3.11)) and note that for the SEIR model, the reproduction number
$$\displaystyle{ \mathrm{R}_{0}^{(5.22)} =\mathrm{ R}_{ 0}^{(3.9)} \frac{a} {\mu +a}, }$$
which implies that R0(5. 22) ≤ R0(3. 9). Moreover, since the mean lifetime of an individual is much greater than the average latency period (i.e., 1∕μ ≫ 1∕a) then a ≫ μ so that a∕(a +μ) ≈ 1 [69]. Thus, R0(5. 22) ≈ R0(3. 9) in most cases. If the latent period is small compared to the infectious period (i.e., ag ≫ 1), which is often the case, the latent period can be ignored [103], which justifies the assumption made for the classical endemic model. The dynamics of (5.22) are again dictated by the basic reproduction number:
$$\displaystyle{ \mathrm{R}_{0}^{(5.22)} \leq 1 }$$
implies asymptotic stability of the disease-free equilibrium QDFS(5. 22) in D(5. 22);
$$\displaystyle{ \mathrm{R}_{0}^{(5.22)} > 1 }$$
implies asymptotic stability of the endemic equilibrium QES(5. 22) in D(5. 22) (see, e.g., [81]) and is approached in a damped oscillatory fashion [69]. In fact, the period of oscillations is approximately equal to
$$\displaystyle{ 2\pi \sqrt{ \frac{1} {\mu (\mathrm{R}_{0}^{(5.22)} - 1)}\left ( \frac{1} {\mu +g} + \frac{1} {\mu +a}\right )} }$$
where [69]:
  1. 1.

    The term 1∕(μ(R0(5. 22) − 1)) is the average age of infection.

     
  2. 2.

    The term 1∕(μ + g) + 1∕(μ + a) is the average period of host’s infectivity.

     

In effect, the SEIR model (5.22) admits a slower rate of growth of the disease after its introduction because the latent period delays an exposed person from becoming infectious [69].

With seasonal variations in (5.22), and a treatment of infected by the switching rate p σ (p1, , p m  ≥ 0), the model is given by
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu -\beta _{\sigma }S(t)I(t) -\mu S(t), \\ \dot{E}(t)& =\beta _{\sigma }S(t)I(t) - aE(t) -\mu E(t), \\ \dot{I}(t)& = aE(t) - gI(t) -\mu I(t) - p_{\sigma }I(t), \\ \dot{R}(t)& = gI(t) + p_{\sigma }I(t) -\mu R(t), \\ (S(0),E(0),I(0),R(0))& = (S_{0},E_{0},I_{0},R_{0}). \end{array} }$$
(5.24)
Here, it is assumed that infected individuals seek treatment but those who have been exposed and are in the latent period (possibly asymptomatic) do not seek treatment. See Fig. 5.11 for the flow of individuals in the population. The mode basic reproduction numbers are thus
$$\displaystyle{ \mathrm{R}_{0}^{(5.24),i} \equiv \frac{\beta _{i}a} {(\mu +g + p_{i})(\mu +a)},\quad \forall i \in \mathcal{M}. }$$
(5.25)
Intuitively, the basic reproduction number in each isolated mode equal to β i ∕(μ + g + p i ) (average contact rate times average period of infection) multiplied by 1∕(μ + a) (average latent period). Equation (5.24) still admits a common disease-free equilibrium
$$\displaystyle{ Q_{\mathrm{DFS}}^{(5.24)} \equiv (1,0,0,0) = Q_{\mathrm{ DFS}}^{(5.22)}, }$$
while each mode admits an endemic equilibrium:
$$\displaystyle\begin{array}{rcl} Q_{\mathrm{ES}}^{(5.24),i} \equiv \left ( \frac{1} {\mathrm{R}_{0}^{(5.24),i}}, \frac{\mu (\mu +g + p_{i})} {\beta _{i}a} (\mathrm{R}_{0}^{(5.24),i} - 1), \frac{\mu } {\beta _{i}}(\mathrm{R}_{0}^{(5.24),i} - 1), \frac{g + p_{i}} {\beta _{i}} (\mathrm{R}_{0}^{(5.24),i} - 1)\right ),& & {}\\ \end{array}$$
for each \(i \in \mathcal{M}\). The long-term behavior of (5.24) is characterized via common Lyapunov function techniques and the switching invariance principle.
Fig. 5.11

Flow of the switched SEIR system with treatment (5.24). The red line represents the horizontal transmission and the blue line represents the treatment strategy

Theorem 5.5

If \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) and
$$\displaystyle{ p_{i} > \frac{\beta _{i}a} {\mu +a} - (\mu +g),\quad \forall i \in \mathcal{M}, }$$
(5.26)
then \(Q_{\mathrm{DFS}}^{(5.24)}\) is globally attractive in the meaningful domain; the disease is eradicated by the switching treatment control.

Proof

Define the mapping V (E, I) ≡ a E + (a +μ)I (similar to the one from [140]) and define the following set:
$$\displaystyle{ \varOmega _{EI} =\{ (E,I) \in \mathbb{R}_{+}^{2}: E + I \leq 1\}. }$$
Observe that V (0, 0) = 0 and V (E, I) > 0 for (E, I) ∈ Ω E I {(0, 0)}. The time-derivative of V along trajectories of (5.24) is given by:
$$\displaystyle\begin{array}{rcl} \dot{V }_{(5.24)}(t,S,E,I,R)& =& a(\beta _{\sigma }SI - aE -\mu E) + (a+\mu )(aE - gI -\mu I - p_{\sigma }I), {}\\ & =& \beta _{\sigma }aSI - (\mu +g + p_{\sigma })(\mu +a)I, {}\\ & =& (\mathrm{R}_{0}^{(5.24)}S - 1)(\mu +g + p_{\sigma })(\mu +a)I. {}\\ \end{array}$$
Condition (5.26) precisely implies that R0(5. 24), i < 1 for all \(i \in \mathcal{M}\). From this it follows that \(\dot{V }_{(5.24)}(t,S,E,I,R) \leq 0\); V (E, I) is a common weak Lyapunov function of (5.24). The set
$$\displaystyle{ \{(E,I) \in \varOmega _{EI}:\dot{ V }_{(5.24)} = 0\} =\{ (E,I) \in \varOmega _{EI}: (E,I) = (c,0),\quad \forall 0 \leq c \leq 1\} }$$
and, by inspection of the limiting equations of (5.24) with I = 0, the solution converges to the disease-free equilibrium QDFS(5. 24) by Theorem  2.2.

Example 5.6

Consider (5.24) with \(\mathcal{M} =\{ 1,2\}\), σ defined as in ( 3.37), initial conditions (S0, E0, I0, R0) = (0. 9, 0, 0. 1, 0), and β1 = 8, β2 = 1. 6, g = 0. 9, μ = 0. 1. Let the latent period equal 1∕a = 1∕0. 3 from [103]. Given p1 = 6 and p2 = 1, then
$$\displaystyle{ 6 = p_{1} > \frac{\beta _{1}a} {\mu +a} - (\mu +g) = 5, }$$
and
$$\displaystyle{ 1 = p_{2} > \frac{\beta _{2}a} {\mu +a} - (\mu +g) = 0.2; }$$
the conditions of Theorem 5.5 are satisfied and QDFS(5. 24) is globally attractive in the meaningful domain. See Fig. 5.12 for an illustration.
Fig. 5.12

Simulations of Example 5.5. (a ) p1 = p2 = 0. (b ) p1 = 6, p2 = 1

Motivated by the number of infectious diseases transmitted by both horizontal and vertical modes (e.g., rubella, herpes simplex, hepatitis B, Chagas’ disease [140]), consider (5.24) with the additional assumption of vertical transmission:
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu (1 -\rho E(t) - qI(t)) -\beta _{\sigma }S(t)I(t) -\mu S(t), \\ \dot{E}(t)& =\mu (\rho E(t) + qI(t)) +\beta _{\sigma }S(t)I(t) - aE(t) -\mu E(t), \\ \dot{I}(t)& = aE(t) - gI(t) -\mu I(t) - p_{\sigma }I(t), \\ \dot{R}(t)& = gI(t) + p_{\sigma }I(t) -\mu R(t), \\ (S(0),E(0),I(0),R(0))& = (S_{0},E_{0},I_{0},R_{0}),\end{array} }$$
(5.27)
where ρ ∈ [0, 1] and q ∈ [0, 1] represent vertical transmission via exposed and infected individuals, respectively. The set
$$\displaystyle{ D_{(5.27<)} \equiv \{ (S,E,I,R) \in \mathbb{R}_{+}^{4}: S + E + I + R = 1\} }$$
is the meaningful domain (which is positively invariant). The mode basic reproduction numbers are (from the case [140]):
$$\displaystyle{ \mathrm{R}_{0}^{(5.27<),i} \equiv \frac{\beta _{i}a} {(\mu +g + p_{i})(\mu (1-\rho ) + a) -\mu qa},\quad \forall i \in \mathcal{M}, }$$
(5.28)
which can be interpreted via a Taylor expansion of the transmission of diseases through the generations of offspring in each mode [140] (where the authors also present the endemic equilibria and stability results for the time-invariant and uncontrolled version of (5.27)). Equation (5.27) admits a disease-free equilibrium QDFS(5. 27) ≡ (1, 0, 0, 0) and mode-dependent endemic equilibria QES, i(5. 27) ≡ (S i , E i , I i , R i ) where, for each \(i \in \mathcal{M}\),
$$\displaystyle\begin{array}{rcl} S_{i}^{{\ast}}& \equiv & \frac{1} {\mathrm{R}_{0}^{(5.27),i}}, {}\\ E_{i}^{{\ast}}& \equiv & 1 - S_{ i}^{{\ast}}- I_{ i}^{{\ast}}- R_{ i}^{{\ast}}, {}\\ I_{i}^{{\ast}}& \equiv & \frac{a\mu \mathrm{R}_{0}^{(5.27),i}} {\beta _{i}a +\rho \mu (g +\mu +p_{i})\mathrm{R}_{0}^{(5.27),i} + q\mu a\mathrm{R}_{0}^{(5.27),i}}(1 - 1/\mathrm{R}_{0}^{(5.27),i}), {}\\ R_{i}^{{\ast}}& \equiv & \frac{a(g + p_{i})\mathrm{R}_{0}^{(5.27),i}} {\beta _{i}a +\rho \mu (g +\mu +p_{i})\mathrm{R}_{0}^{(5.27),i} + q\mu a\mathrm{R}_{0}^{(5.27),i}}(1 - 1/\mathrm{R}_{0}^{(5.27),i}). {}\\ \end{array}$$
The flow of (5.27) is illustrated in Fig. 5.13.
Fig. 5.13

Flow of the switched SEIR system with vertical transmission and treatment (5.27). The red lines represent the horizontal and vertical transmission and the blue line represents the treatment strategy

Eradication is established as follows.

Theorem 5.6

If \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) and
$$\displaystyle{ p_{i} > \frac{(\beta _{i} +\mu q)a} {\mu (1-\rho ) + a} - (\mu +g),\quad \forall i \in \mathcal{M}, }$$
(5.29)
then \(Q_{\mathrm{DFS}}^{(5.27<)}\) is globally attractive in the meaningful domain; the disease is eradicated by the switching treatment control.

Proof

The proof proceeds similar to the proof of Theorem 5.5 by replacing the Lyapunov candidate function with
$$\displaystyle{ V (E,I) \equiv aE + (a +\mu -\rho \mu )I }$$
(adopted from one in [140]). Observe that V (0, 0) = 0 and V (E, I) > 0 when (E, I) ∈ Ω E I {(0, 0)} (where Ω E I is defined in the proof of Theorem 5.5). The time-derivative of V along trajectories of (5.27) is given by
$$\displaystyle\begin{array}{rcl} \dot{V }_{(5.27)}(t,S,E,I,R)& =& a(\beta _{\sigma }SI +\rho \mu E + q\mu I - aE -\mu E) {}\\ & & \quad + (a +\mu -\rho \mu )(aE - gI - p_{\sigma }I -\mu I), {}\\ & =& \beta _{\sigma }aSI - [(\mu +g + p_{\sigma })(\mu +a-\rho \mu ) -\mu qa]I, {}\\ & =& (\mathrm{R}_{0}^{(5.27)}S - 1)(\mu +g + p_{\sigma })(\mu +a -\rho \mu -\mu qa)I. {}\\ \end{array}$$
Hence, if R0(5. 27), i < 1 for all i, then \(\dot{V }_{(5.27<)}(t,S,E,I,R) \leq 0\). The remaining part follows by similar arguments to the proof of Theorem 5.5.
Appropriate for a disease like AIDS [140], suppose that the assumptions on the population dynamics and disease-induced mortality are relaxed; assume that the birth rate is b > 0, the natural death rate is d > 0 and the disease-induced death rate is α > 0. Applied to (5.24) yields the following dynamic system:
$$\displaystyle{ \begin{array}{rl} \dot{S_{c}}(t)& = b -\beta _{\sigma }\frac{S_{c}(t)I_{c}(t)} {N(t)} - dS_{c},(t) \\ \dot{E_{c}}(t)& =\beta _{\sigma }\frac{S_{c}(t)I_{c}(t)} {N(t)} - aE_{c}(t) - dE_{c}(t), \\ \dot{I_{c}}(t)& = aE_{c}(t) - gI_{c}(t) - dI_{c}(t) -\alpha I_{c}(t) - p_{\sigma }I_{c}(t), \\ \dot{R_{c}}(t)& = gI_{c}(t) + p_{\sigma }I_{c}(t) - dR_{c}(t),\end{array} }$$
(5.30)
where S c , E c , I c , R c represent the number of individuals in the susceptible, exposed, infectious, and removed classes, respectively. The total population N ≡ S c + E c + I c + R c satisfies the differential equation
$$\displaystyle{ \dot{N}(t) = (b - d)N(t) -\alpha I_{c}(t). }$$
Normalizing the equations via S ≡ S c N, E ≡ E c N, I ≡ I c N, R ≡ R c N yields the following switching system:
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& = b -\beta _{\sigma }S(t)I(t) - bS(t) +\alpha S(t)I(t), \\ \dot{E}(t)& =\beta _{\sigma }S(t)I(t) - aE(t) - bE(t) +\alpha E(t)I(t), \\ \dot{I}(t)& = aE(t) - gI(t) - bI(t) -\alpha I(t) - p_{\sigma }I(t) +\alpha I^{2}(t), \\ \dot{R}(t)& = gI(t) + p_{\sigma }I(t) - bR(t) +\alpha R(t)I(t), \\ (S(0),E(0),I(0),R(0))& = (S_{0},E_{0},I_{0},R_{0}), \end{array} }$$
(5.31)
where the normalized variables satisfy S(t) + E(t) + I(t) + R(t) = 1. The initial conditions satisfy (S0, E0, I0, R0) ∈ D(5. 31) = D(5. 24), which is invariant to (5.31); \(\{\dot{S} +\dot{ E} +\dot{ I} +\dot{ R}\}\left \vert \right._{S+I+E+R=1} = 0\), \(\dot{S}\left \vert \right._{S=0} = b > 0,\dot{E}\left \vert \right._{E=0} =\beta _{i}SI \geq 0,\dot{I}\left \vert \right._{I=0} = aE \geq 0\) and \(\dot{R}\left \vert \right._{R=0} = gI \geq 0\). A consequence of the disease-induced mortality, the terms α S I, α E I, α I R, and α I2 act as positive feedback in the dynamic system. The mode basic reproduction numbers are
$$\displaystyle{ \mathrm{R}_{0}^{(5.31),i} \equiv \frac{\beta _{i}a} {(b + g +\alpha +p_{i})(b + a)},\quad \forall i \in \mathcal{M}, }$$
(5.32)
which reflect the time-invariant case [81]. Comparing (5.32) to (5.25) reveals a reduction in the mode-dependent basic reproduction numbers; the disease-induced mortality reduces the infectious period and therefore the rate of transmission. As in the previously studied SEIR models, (5.31) again admits a disease-free equilibrium QDFS(5. 31) ≡ (1, 0, 0, 0) and m endemic equilibria QES(5. 31), i ≡ (S i , E i , I i , R i ). Here, I i satisfies the following cubic equation [81]:
$$\displaystyle{ \left (1 - \frac{\alpha } {a + b}I_{i}^{{\ast}}\right )\left (1 - \frac{\alpha } {\alpha +g + b + p_{i}}I_{i}^{{\ast}}\right )\left (1 + \frac{\beta _{i}-\alpha } {b} I_{i}^{{\ast}}\right ) =\mathrm{ R}_{ 0}^{(5.31),i}, }$$
(5.33)
for each \(i \in \mathcal{M}\). If R0(5. 31), i > 1, then (5.33) admits a unique positive solution [81]. The other endemic equilibria states satisfy
$$\displaystyle\begin{array}{rcl} S_{i}^{{\ast}}& \equiv & \frac{b} {b +\beta _{i}I_{i}^{{\ast}}-\alpha I_{i}^{{\ast}}}, {}\\ E_{i}^{{\ast}}& \equiv & \frac{g +\alpha +b + p_{i} -\alpha I_{i}^{{\ast}}} {a} I_{i}^{{\ast}}, {}\\ R_{i}^{{\ast}}& \equiv & 1 - S_{ i}^{{\ast}}- E_{ i}^{{\ast}}- I_{ i}^{{\ast}}, {}\\ \end{array}$$
for each \(i \in \mathcal{M}\). Stability of the disease-free solution can be established using the following lemma from [81].

Lemma 5.1

Let \(\varOmega \equiv \{ (x,y) \in \mathbb{R}_{+}^{2}: x + y \leq 1\}\) and h(x,y) ≡ (a − b)x + (c − b)y + b, where a,b,c > 0 are constants. Then it follows that
$$\displaystyle{ \max \{h(x,y): (x,y) \in \varOmega \}=\max \{ a,b,c\}. }$$

Theorem 5.7

If \(\sigma \in \mathcal{S}_{\mathrm{dwell}}\) and
$$\displaystyle{ p_{i} > \frac{\beta _{i}a} {b + a} - (\mu +g+\alpha ),\quad \forall i \in \mathcal{M}, }$$
(5.34)
then \(Q_{\mathrm{DFS}}^{(5.31)}\) is globally attractive in the meaningful domain. Moreover, the total number of infected individuals approaches zero (i.e., limt→∞Ic(t) = 0).

Proof

Define the mapping V (E, I) ≡ a E + (a + b)I [81], which satisfies V (0, 0) = 0 and V (E, I) > 0 for (E, I) ∈ Ω E I {(0, 0)} (where Ω E I is outlined in the proof of Theorem 5.5). The time-derivative of V trajectories of (5.31) is given by
$$\displaystyle\begin{array}{rcl} \dot{V }_{(5.27)}(t,S,E,I,R)& =& a(\beta _{\sigma }SI - aE - bE +\alpha EI) {}\\ & & \quad + (a + b)(aE - gI - bI -\alpha I - p_{\sigma }I +\alpha I^{2}), {}\\ & =& [\beta _{\sigma }aS - (a + b)(g +\alpha +b + p_{\sigma }) +\alpha aE +\alpha (a + b)I]I, {}\\ & \leq & [\beta _{\sigma }a(1 - E - I) - (a + b)(g +\alpha +b + p_{\sigma }) {}\\ & & \quad +\alpha aE +\alpha (a + b)I]I, {}\\ & =& [h_{\sigma }(E,I) - (a + b)(g +\alpha +b + p_{\sigma })]I, {}\\ \end{array}$$
where
$$\displaystyle{ h_{i}(E,I) \equiv (\alpha a -\beta _{i}a)E + (\alpha (a + b) -\beta _{i}a)I +\beta _{i}a,\quad \forall i \in \mathcal{M}. }$$
Applying Lemma 5.1 with the function h i and set Ω E I gives that
$$\displaystyle{ \dot{V }_{(5.27)} \leq [\max \{\alpha a,\beta _{\sigma }a,\alpha (a + b)\} - (a + b)(g +\alpha +b + p_{\sigma })]I. }$$
Then, since R0(5. 31), i < 1 for each \(i \in \mathcal{M}\) by Eq. (5.34), it follows that \(\dot{V }_{(5.27)} \leq 0\). Note that \(\dot{V } = 0\) if (E, I) = (c, 0) or if max{α a, β i a, α(a + b)} = (a + b)(g +α + b + p i ), which implies R0(5. 31), i = 1 (and is therefore not possible). It then follows by similar arguments to the proof of Theorem 5.5 that limt → I(t) = 0. Recall that I c  ≡ I N and
$$\displaystyle{ \dot{N}(t) = (b - d)N(t) -\alpha I_{c}(t) = (b - d -\alpha I(t))N(t). }$$
The case b < d is straightforward. The case b = d yields that \(\dot{N}(t) = -\alpha I(t)N(t) \leq 0\), from which it follows that N is bounded for all t since I → 0. The case b > d gives that N grows without bound since I → 0. Then, S c  ≡ S N and S → 1 implies that S c  → N. The fact that N ≡ S c + E c + I c + R c yields the result.
Equations (5.26), (5.29), and (5.34) define mode-dependent critical control rates for the SEIR model (5.24), SEIR model with vertical transmission (5.27), and SEIR model with disease-induced mortality (5.31), respectively:
$$\displaystyle\begin{array}{rcl} p_{i}^{(5.26),\mbox{ crit}}& \equiv & \frac{\beta _{i}a} {\mu +a} - (\mu +g),\quad \forall i \in \mathcal{M}, {}\\ p_{i}^{(5.29),\mbox{ crit}}& \equiv & \frac{(\beta _{i} +\mu q)a} {\mu (1-\rho ) + a} - (\mu +g),\quad \forall i \in \mathcal{M}, {}\\ p_{i}^{(5.34),\mbox{ crit}}& \equiv & \frac{\beta _{i}a} {b + a} - (\mu +g+\alpha ),\quad \forall i \in \mathcal{M}. {}\\ \end{array}$$
Observe that
$$\displaystyle{ p_{i}^{(5.34),\mbox{ crit}} \leq p_{ i}^{(5.26),\mbox{ crit}} \leq p_{ i}^{(5.29),\mbox{ crit}},\quad \forall i \in \mathcal{M}, }$$
as expected; the disease-induced mortality effectively reduces the average infectious period (making the disease easier to control and thus a decrease in the critical treatment rates) while the vertical transmission has the effect of increasing the basic reproduction number of each mode (hence an increase in the critical treatment rates).

5.4 Screening of Traveling Individuals

In this part, we return to ( 4.21) and consider restricting the travel of infected individuals as a control method for preventing epidemics. We consider the following control strategy:
  1. 1.

    Assume that θ(j) ∈ [0, 1] is the probability of successfully detecting an infected individual entering city \(j \in \mathcal{N} \equiv \{ 1,\ldots,n\}\) by travel.

     
  2. 2.

    Assume that susceptible individuals are not falsely identified as being infected. Denote infected individuals traveling to city j who are properly screened by Q(j).

     
  3. 3.

    Assume that individuals who are being screened do not die or give birth.

     
  4. 4.

    When the infected individuals are identified, assume that they will be isolated for treatment.

     
  5. 5.

    Assume that individuals in the screened classes recover at a switched rate q σ (j) > 0 in city j and enter the recovered population.

     
With the additional assumptions that the immigration rate is m(j) in city j and the functions f i (j) and h i (j) having standard incidence rate structures, the controlled version of ( 4.21) becomes the following switching system:
$$\displaystyle{ \begin{array}{rl} \dot{S}^{(j)}(t)& = m^{(j)} -\beta _{\sigma }^{(j)}\frac{S^{(j)}(t)I^{(j)}(t)} {N^{(j)}(t)} -\mu ^{(j)}S^{(j)}(t) +\sum _{l\in \mathcal{N}}\alpha ^{(l,j)}S^{(j)}(t) \\ &\quad -\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}\gamma _{\sigma }^{(j)}\frac{S^{(l)}(t)I^{(l)}(t)} {N^{(l)}(t)}, \\ \dot{I}^{(j)}(t)& =\beta _{ \sigma }^{(j)}\frac{S^{(j)}(t)I^{(j)}(t)} {N^{(j)}(t)} - g^{(j)}I^{(j)}(t) -\mu ^{(j)}I^{(j)}(t) +\alpha ^{(j,j)}I^{(j)}(t) \\ &\quad + (1 -\theta ^{(j)})\left [\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}I^{(l)}(t) +\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}\gamma _{\sigma }^{(j)}\frac{S^{(l)}(t)I^{(l)}(t)} {N^{(l)}(t)} \right ], \\ \dot{Q}^{(j)}(t)& = -q_{\sigma }^{(j)}Q^{(j)} +\theta ^{(j)}\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}I^{(l)}(t) \\ &\quad +\theta ^{(j)}\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}\gamma _{\sigma }^{(j)}\frac{S^{(l)}(t)I^{(l)}(t)} {N^{(l)}(t)}, \\ \dot{R}^{(j)}(t)& = g^{(j)}I^{(j)}(t) + q_{\sigma }^{(j)}Q^{(j)}(t) -\mu ^{(j)}R^{(j)}(t) +\sum _{l\in \mathcal{N}}\alpha ^{(l,j)}R^{(l)}(t), \\ &(S^{(j)}(0),I^{(j)}(0),Q^{(j)}(0),R^{(j)}(0)) = (S_{0}^{(j)},I_{0}^{(j)},Q_{0}^{(j)},R_{0}^{(j)}), \end{array} }$$
(5.35)
for all \(j \in \mathcal{N}\). The flow diagram for the model with screening is given in Fig. 5.14.
Fig. 5.14

Flow of the multi-city model with screening (5.35) with n = 2. The red lines represent new infections (including from traveling individuals) and the blue lines represent the screening process. The population dynamics in each city have been omitted here

The meaningful physical domain for this system is
$$\displaystyle{ D_{(5.35)} \equiv \{ (S,I,Q,R) \in \mathbb{R}_{+}^{4n}:\sum _{ j\in \mathcal{N}}S^{(j)} + I^{(j)} + Q^{(j)} + R^{(j)} \leq N^{{\ast}}\}, }$$
where
$$\displaystyle{ N^{{\ast}} = \frac{\sum _{j=1}^{n}m^{(j)}} {\min \{\mu ^{(1)},\mu ^{(2)},\ldots,\mu ^{(n)}\}} > 0. }$$
For a given initial condition, the solution of (5.35) enters D(5. 35) (in finite or infinite time); D(5. 35) is positively invariant to (5.35) (see Proposition 2.1 in [167]).
Let us next consider the existence of a disease-free solution of (5.35) with I(j)(t) ≡ 0 for all \(j \in \mathcal{N}\): It is apparent that the screening class converges to zero, and thus the recovered class (since μ(j) > 0 for each \(j \in \mathcal{N}\)). The limiting system is given by
$$\displaystyle{ \begin{array}{rl} \dot{S}^{(j)}(t)& = m^{(j)} -\mu ^{(j)}S^{(j)}(t) +\sum _{l\in \mathcal{N}}\alpha ^{(l,j)}S^{(j)}(t),\quad \forall j \in \mathcal{N}. \end{array} }$$
(5.36)
Using the notation and methodology in [167], define the irreducible matrices A ≡ (α(l, j))1 ≤ l, j ≤ n and U ≡ diag{μ(1), , μ(n)} and the vector m = (m(1), , m(n)). Then (5.36) is in vector form as
$$\displaystyle{ \dot{S}(t) = m + (A - U)S(t), }$$
whose unique solution satisfies
$$\displaystyle{ S(t) \equiv (S_{0} + (A - U)^{-1}m)\exp ((A - U)t) - (A - U)^{-1}m, }$$
and
$$\displaystyle{ \lim _{t\rightarrow \infty }S(t) = -(A - U)^{-1}m \equiv S^{{\ast}}, }$$
where AU is nonsingular and m has nonnegative components (i.e., (AU)−1m has nonnegative entries). Hence, Eq. (5.35) admits the disease-free solution
$$\displaystyle{ Q_{\mathrm{DFS}}^{(5.35)} \equiv (S^{{\ast}},0,0,0). }$$
The basic reproduction number of (5.35) is, in general, the spectral radius of an integral operator. It is possible to provide a threshold theorem involving an approximation of the basic reproduction number, as follows.

Theorem 5.8

Let \(\alpha _{\min } \equiv \min \{\alpha ^{(l,j)}: l,j \in \mathcal{N},l\neq j\}\), \(\alpha _{\max } \equiv \max \{\alpha ^{(l,j)}: l,j \in \mathcal{N},l\neq j\}\), \(\theta _{\min } \equiv \min \{\theta ^{(j)}: j \in \mathcal{N}\}\), \(g_{\min } \equiv \min \{ g^{(j)}: j \in \mathcal{N}\}\) and \(\mu _{\min } \equiv \min \{\mu ^{(j)}: j \in \mathcal{N}\}\). For each \(i \in \mathcal{M}\), let \(\beta _{i} \equiv \max \{\beta _{i}^{(j)}: j \in \mathcal{N}\}\), and \(\gamma _{i} \equiv \max \{\gamma _{i}^{(j)}: j \in \mathcal{N}\}\). If \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and
$$\displaystyle{ \widehat{\mathrm{R}_{0}}^{(5.35)} \equiv \frac{\sum _{i=1}^{m}(\beta _{ i}S^{{\ast}} + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{ i}S^{{\ast}})\tau _{ i}} {\omega (g_{\min } +\mu _{\min } + (n - 1)\theta _{\min }\alpha _{\min })} < 1, }$$
(5.37)
then \(Q_{\mathrm{DFS}}^{(5.35)}\) is globally attractive in the meaningful domain; the disease is eradicated by the screening control.

Proof

From Eq. (5.35) note that
$$\displaystyle{ \dot{S}^{(j)}(t) \leq m^{(j)} -\mu _{ j}S^{(j)}(t) +\sum _{ l\in \mathcal{N}}\alpha ^{(l,j)}S^{(j)}(t). }$$
Consider the comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}^{(j)}(t)& = m^{(j)} -\mu ^{(j)}x^{(j)}(t) +\sum _{l\in \mathcal{N}}\alpha ^{(l,j)}x^{(j)}(t), \\ x(0)& = (S_{0}^{(1)},\ldots,S_{0}^{(n)}) = S_{0}. \end{array} }$$
(5.38)
As remarked above, the solution of this system converges to − (AU)−1m = S. Choose
$$\displaystyle{ 0 <\epsilon = \frac{(1 -\widehat{\mathrm{ R}_{0}}^{(5.35)})(\omega (g_{\min } +\mu _{\min } + (n - 1)\theta _{\min }\alpha _{\min })} {2\sum _{i=1}^{m}(\beta _{i}S^{{\ast}} + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{i})\tau _{i}}. }$$
By comparison theorem, there exists a time t > 0 such that S(j)(t) ≤ x(j)(t) ≤ S +ε for t ≥ t and each \(j \in \mathcal{N}\) (i.e., by Theorem  1.10). Choose \(N \in \mathbb{N}\) as the smallest integer such that N ω > t. The following inequalities follow from (5.35) for each \(j \in \mathcal{N}\):
$$\displaystyle\begin{array}{rcl} \sum _{j\in \mathcal{N}}\dot{I}_{j}(t)& =& \sum _{j\in \mathcal{N}}[\beta _{\sigma }^{(j)}\frac{S^{(j)}(t)I^{(j)}(t)} {N^{(j)}(t)} - g^{(j)}I^{(j)}(t) -\mu ^{(j)}I^{(j)}(t) {}\\ & & \quad -\theta ^{(j)}\sum _{ l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}I^{(l)}(t) + (1 -\theta ^{(j)})\sum _{ l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}\gamma _{ \sigma }^{(j)}\frac{S^{(l)}(t)I^{(l)}(t)} {N^{(l)}(t)} ], {}\\ & \leq & \sum _{j\in \mathcal{N}}[\beta _{\sigma }\frac{S^{(j)}(t)I^{(j)}(t)} {N^{(j)}(t)} - g_{\min }I^{(j)}(t) -\mu _{\min }I^{(j)}(t) -\theta ^{(j)}\sum _{ l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}I^{(j)}(t) {}\\ & & \quad + (1 -\theta ^{(j)})\sum _{ l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}\gamma _{ \sigma }\frac{S^{(l)}(t)I^{(l)}(t)} {N^{(l)}(t)} ]. {}\\ \end{array}$$
Since
$$\displaystyle{ \sum _{j\in \mathcal{N}}\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}S^{(l)}I^{(l)} =\sum _{ j\in \mathcal{N}}\sum _{l\in \mathcal{N}\setminus \{j\}}\alpha ^{(l,j)}S^{(j)}I^{(j)} \leq \sum _{ j\in \mathcal{N}}(n - 1)\alpha _{\max }S^{(j)}I^{(j)}, }$$
then
$$\displaystyle\begin{array}{rcl} \sum _{j\in \mathcal{N}}\dot{I}_{j}(t)& \leq & \sum _{j\in \mathcal{N}}[\beta _{\sigma }(S^{{\ast}}+\epsilon ) - g_{\min } -\mu _{\min } \\ & &\quad - (n - 1)\theta _{\min }\alpha _{\min } + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{\sigma }(S^{{\ast}}+\epsilon )]I^{(j)}(t), \\ & \leq & \sum _{j\in \mathcal{N}}[\beta _{\sigma }S^{{\ast}} + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{\sigma }S^{{\ast}} \\ & &\quad +\epsilon (\beta _{\sigma } + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{\sigma }) - g_{\min } -\mu _{\min }- (n - 1)\theta _{\min }\alpha _{\min }]I^{(j)}(t), \\ & =& \lambda _{k,\epsilon }\sum _{j\in \mathcal{N}}I^{(j)}(t), {}\end{array}$$
(5.39)
where
$$\displaystyle\begin{array}{rcl} \lambda _{i,\epsilon }& \equiv & \beta _{i}S^{{\ast}} + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{ i}S^{{\ast}} \\ & &\quad +\epsilon (\beta _{i} + (n - 1)(1 -\theta _{\min })\alpha _{\max }\gamma _{i}) - g_{\min } -\mu _{\min }- (n - 1)\theta _{\min }\alpha _{\min }.{}\end{array}$$
(5.40)
Equation (5.39) implies
$$\displaystyle\begin{array}{rcl} \sum _{j\in \mathcal{N}}I^{(j)}((N + 1)\omega )& \leq & \sum _{ j\in \mathcal{N}}I^{(j)}(N\omega )\exp \left (\sum _{ i=1}^{m}\lambda _{ i,\epsilon }\tau _{i}\right ) {}\\ & =& \exp \left (\sum _{i=1}^{m}\lambda _{ i,\epsilon }\tau _{i}\right )\sum _{j\in \mathcal{N}}I^{(j)}(N\omega ), {}\\ \end{array}$$
from which it follows that \(\sum _{j\in \mathcal{N}}I^{(j)}((N + 1)\omega ) \leq \eta \sum _{j\in \mathcal{N}}I^{(j)}(N\omega ),\) where
$$\displaystyle{ \eta \equiv \exp \left (\sum _{i=1}^{m}\lambda _{ i,\epsilon }\tau _{i}\right ). }$$
Note that η ∈ (0, 1) since \(\widehat{\mathrm{R}_{0}}^{(5.35)} < 1\) and by the choice of ε above. Thus,
$$\displaystyle{ \sum _{j\in \mathcal{N}}I^{(j)}((N + 1)\omega ) \leq \eta \sum _{ j\in \mathcal{N}}I^{(j)}(N\omega ) <\sum _{ j\in \mathcal{N}}I^{(j)}(N\omega ). }$$
Similarly, it can be shown that
$$\displaystyle{ \sum _{j\in \mathcal{N}}I^{(j)}((N + h + 1)\omega ) \leq \eta \sum _{ j\in \mathcal{N}}I^{(j)}((N + h)\omega ),\quad \forall h \in \mathbb{N}, }$$
and so
$$\displaystyle\begin{array}{rcl} \sum _{j\in \mathcal{N}}I^{(j)}((N + h + 1)\omega )& \leq & \eta \sum _{ j\in \mathcal{N}}I^{(j)}((N + h)\omega ), {}\\ & \leq & \eta (\eta \sum _{j\in \mathcal{N}}I^{(j)}((N + h - 1)\omega ), {}\\ & \vdots & {}\\ & \leq & \eta ^{h+1}\sum _{ j\in \mathcal{N}}I^{(j)}(N\omega ). {}\\ \end{array}$$
Therefore the sequence \(\{\sum _{j\in \mathcal{N}}I^{(j)}((N + h)\omega )\}_{h=0}^{\infty }\) converges to zero as h → . Since I(j)(t) is bounded on t ∈ [0, N ω] for each j and since \(\sum _{j\in \mathcal{N}}I^{(j)}\) is bounded on each interval [t0 + (N + h)ω, (N + h + 1)ω] for \(h \in \mathbb{N} \cup \{ 0\}\), then it follows that I(j) converges to zero as h →  for each j. The limiting system is given by Eq. (5.36), which converges to the disease-free solution QDFS(5. 35).
Requiring that \(\widehat{\mathrm{R}_{0}}^{(5.35)} < 1\) in Eq. (5.37) defines a critical screening rate θcrit that guarantees disease eradication. More precisely,
$$\displaystyle{ \theta _{\mbox{ crit}} \equiv \frac{\sum _{i=1}^{m}(\beta _{i}S^{{\ast}}m - g_{\min } -\mu _{\min } + (n - 1)\alpha _{\max }\gamma _{i}S^{{\ast}})\tau _{i}} {\sum _{i=1}^{m}(\alpha _{\max }\gamma _{i})\tau _{i} +\omega \alpha _{\min }}. }$$
(5.41)

Example 5.7

Consider (5.35) with \(\mathcal{N} =\{ 1,2\}\) and \(\mathcal{M} =\{ 1,2\}\). Suppose that σ follows the seasonal switching rule outlined in ( 3.37) and the initial conditions are given by (S(1), I(1), Q(1), R(1), S(2), I(2), Q(2), R(2)) = (0. 5, 0. 1, 0, 0, 0. 4, 0, 0, 0) (i.e., the disease begins in city 1). The following model parameters are used: β1 = 4. 5, β2 = 0. 5, g(1) = 1. 5, g(2) = 1. 2, m(1) = 0. 1, m(2) = 0. 09, μ(1) = 0. 1, μ(2) = 0. 09, γ = 1, α(1, 1) = −0. 4, α(1, 2) = 0. 4, α(2, 2) = −0. 3, α(2, 1) = 0. 3. That is,
$$\displaystyle{ A = \left [\begin{array}{*{10}c} -0.4& 0.3\\ 0.4 &-0.3 \end{array} \right ],\quad U = \mbox{ diag}\{0.1,0.09\},\quad m = (0.1,0.09), }$$
giving that S = (0. 880, 1. 13). If θ(1) = θ(2) = 0 then \(\widehat{\mathrm{R}_{0}}^{(5.35)} = 1.47\) (see Fig. 5.15a). If θ(1) = 0. 95 and θ(2) = 0. 9 then \(\widehat{\mathrm{R}_{0}}^{(5.35)} = 0.987\) (see Fig. 5.15b); in this case, the critical screening rate is given by θcrit = 0. 871.
Fig. 5.15

Simulations of Example 5.7. (a ) θ(1) = θ(2) = 0. (b ) θ(1) = 0. 95, θ(2) = 0. 9

5.5 Switching Control for Vector-borne Diseases

In this part we return to the vector-borne model ( 4.35) for control strategy analysis. Switching cohort immunization is considered here: assume that a switching vaccination control is applied at a rate v σ  > 0 to the susceptible population (where immunization immediately moves an individual to the vaccinated class, V ). Assume also that a switching treatment control is applied at a rate p σ  > 0. Motivated by realistic difficulties and failures of a vaccine program, the probability that a vaccinated individual can still become infected through transmission is assumed to be nonzero (but reduced when compared to the susceptible individuals); let
$$\displaystyle{ \xi \beta _{\sigma }V (t)\int _{0}^{d}f(u)I(t - u)\mathrm{d}u, }$$
where ξ ∈ [0, 1], correspond to such a reduced transmission between vaccinated and infected (i.e., ξ is a measure of the vaccine efficacy). Applied to ( 4.35), the control model is given by
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu (1 - S(t)) -\beta _{\sigma }S(t)\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - v_{\sigma }S(t) +\theta V (t), \\ \dot{I}(t)& =\beta _{\sigma }(S(t) +\xi V (t))\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - (g +\mu +p_{\sigma })I(t), \\ \dot{R}(t)& = gI(t) + p_{\sigma }I(t) -\mu R(t), \\ \dot{V }(t)& = v_{\sigma }S(t) -\xi \beta _{\sigma }V (t)\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - (\theta +\mu )V (t), \\ (S(s),I(s),R(s),V (s))& = (S_{0},I_{0}(s),R_{0},V _{0}),\quad \forall s \in [-d,0].\end{array} }$$
(5.42)
The physical domain of interest for (5.42) is given by
$$\displaystyle{ D_{(5.42)} \equiv \{ (S,I,R,V ) \in \mathbb{R}_{+}^{4}: S + I + R + V = 1\}, }$$
and it is assumed that (S0, I0(0), R0, V0) ∈ D(5. 42). Equation (5.42) admits m disease-free equilibria due to the time-varying vaccination rates:
$$\displaystyle{ Q^{(5.42)} \equiv (S_{ i}^{{\ast}},I_{ i}^{{\ast}},R_{ i}^{{\ast}},V _{ i}^{{\ast}}) \equiv \left ( \frac{\mu (\theta +\mu )} {\mu +v_{i}(1-\theta )},0,0, \frac{v_{i}S_{i}^{{\ast}}} {\theta +\mu } \right ), }$$
for all \(i \in \mathcal{M}\). The movement of the population between compartments is illustrated in Fig. 5.16.
Fig. 5.16

Flow of the vector-borne model with treatment and vaccination (5.42). The red lines represent the horizontal transmission and the blue lines represent the treatment and vaccination strategies. Births/deaths are omitted here for illustrative purposes

In the absence of infection, the solution of (5.42) traverses between the disease-free equilibria as the vaccination rates vary with respect to time. This observation motivates studying convergence to a disease-free set: when I(t) ≡ 0, the number of individuals in the recovered class approaches zero exponentially and the reduced model is given by
$$\displaystyle{ \begin{array}{rl} \dot{S}(t)& =\mu (1 - S(t)) - v_{\sigma }S(t) +\theta V (t), \\ \dot{V }(t)& = v_{\sigma }S(t) - (\mu +\theta )V (t). \end{array} }$$
(5.43)
Define \(v_{\mbox{ min}} \equiv \min \{ v_{i}: i \in \mathcal{M}\}\) and \(v_{\mbox{ max}} \equiv \max \{ v_{i}: i \in \mathcal{M}\}\). Since S + V = 1 is invariant to (5.43),
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& \leq & \mu -(\mu +v_{\mbox{ min}})S(t) +\theta (1 - S(t)), {}\\ & =& (\mu +v_{\mbox{ min}}+\theta )\left (\frac{\overline{S}_{\mbox{ max}}} {S(t)} - 1\right )S(t), {}\\ \end{array}$$
so that \(\dot{S}(t) \leq 0\) if \(1 \geq S(t) \geq \overline{S}_{\mbox{ max}}\) where
$$\displaystyle{ \overline{S}_{\mbox{ max}} \equiv \frac{\mu +\theta } {\mu +v_{\mbox{ min}}+\theta }. }$$
Similarly, if \(0 < S(t) \leq \overline{S}_{\mbox{ min}} \equiv \mu (1+\theta )/(\mu +v_{\mbox{ max}}+\theta )\), then \(\dot{S}(t) \geq 0\) since
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& \geq & \mu -(\mu +v_{\mbox{ max}})S(t) +\theta (1 - S(t)), {}\\ & =& (\mu +v_{\mbox{ max}}+\theta )\left (\frac{\overline{S}_{\mbox{ min}}} {S(t)} - 1\right )S(t). {}\\ \end{array}$$
Further, \(\dot{V }(t) \leq 0\) whenever \(1 \geq V (t) \geq \overline{V }_{\mbox{ max}} \equiv v_{\mbox{ max}}/(\mu +v_{\mbox{ max}}+\theta )\) since
$$\displaystyle\begin{array}{rcl} \dot{V }(t)& \leq & v_{\mbox{ max}}(1 - V (t)) - (\mu +\theta )V (t), {}\\ & =& (\mu +v_{\mbox{ max}}+\theta )\left (\frac{\overline{V }_{\mbox{ max}}} {V (t)} - 1\right )V (t). {}\\ \end{array}$$
Finally, if \(0 < V (t) \leq \overline{V }_{\mbox{ min}} \equiv v_{\mbox{ min}}/(\mu +v_{\mbox{ min}}+\theta )\), then \(\dot{V }(t) \geq 0\) since
$$\displaystyle\begin{array}{rcl} \dot{V }(t)& \geq & v_{\mbox{ min}}(1 - V (t)) - (\mu +\theta )V (t), {}\\ & =& (\mu +v_{\mbox{ min}}+\theta )\left (\frac{\overline{V }_{\mbox{ min}}} {V (t)} - 1\right )V (t). {}\\ \end{array}$$
The solution of (5.43) converges to the set
$$\displaystyle{ \{(S,V ) \in \mathbb{R}_{+}^{2}: \overline{S}_{ \mbox{ min}} \leq S \leq \overline{S}_{\mbox{ max}},\overline{V }_{\mbox{ min}} \leq V \leq \overline{V }_{\mbox{ max}}\}, }$$
which can be shown as follows: Consider the comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}(t)& = \left \{\begin{array}{@{}l@{\quad }l@{}} (\mu +v_{\mbox{ min}}+\theta )\left (\frac{\overline{S}_{\mbox{ max}}} {x(t)} - 1\right )x(t),\quad &\mbox{ if }x(t)\neq 0, \\ \mu +\theta, \quad &\mbox{ if }x(t) = 0, \end{array} \right. \\ x(0)& = S_{0}.\end{array} }$$
(5.44)
The solution of (5.44) converges to \(\overline{S}_{\mbox{ max}}\). By the comparison theorem,, for any ε > 0 there exists t > 0 such that \(S(t) \leq x(t) \leq \overline{S}_{\mbox{ max}}+\epsilon\) for all t ≥ t. Similarly, the comparison system
$$\displaystyle{ \begin{array}{rl} \dot{x}(t)& = \left \{\begin{array}{@{}l@{\quad }l@{}} (\mu +v_{\mbox{ max}}+\theta )\left (\frac{\overline{S}_{\mbox{ min}}} {x(t)} - 1\right )x(t),\quad &\mbox{ if }x(t)\neq 0, \\ \mu -(\mu +v_{\mbox{ max}})x(t) +\theta (1 - x(t)), \quad &\mbox{ if }x(t) = 0, \end{array} \right. \\ x(0)& = S_{0}, \end{array} }$$
(5.45)
yields that \(\lim _{t\rightarrow \infty }S(t) \geq \overline{S}_{\mbox{ min}}\), with similar arguments with respect to V giving the desired result. Therefore, under the assumption that I(t) ≡ 0, the solution of (5.42) converges to the disease-free convex set
$$\displaystyle{ \varPsi _{\mbox{ cohort}} \equiv \{ (S,I,R,V ) \in \mathbb{R}_{+}^{4}: \overline{S}_{ \mbox{ min}} \leq S \leq \overline{S}_{\mbox{ max}},I = 0,R = 0,\overline{V }_{\mbox{ min}} \leq V \leq \overline{V }_{\mbox{ max}}\}. }$$
Define the following constants:
$$\displaystyle{ \lambda _{i} \equiv \beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + p_{i}),\quad \forall i \in \mathcal{M}. }$$
\(\mathcal{M}^{-}\equiv \{ i \in \mathcal{M}:\lambda _{i} < 0\}\) and \(\mathcal{M}^{+} \equiv \{ i \in \mathcal{M}:\lambda _{i} \geq 0\}\). The idea here is that the switched system is composed of a mixture of stable and unstable modes, where \(\mathcal{M}^{-}\) and \(\mathcal{M}^{+}\) denote the stable and unstable modes, respectively. To prove threshold conditions for disease eradication, we focus on the set Ψcohort. Before proceeding, we remind the reader of some switched systems notions: let \(\sigma \in \mathcal{S}_{\mbox{ dwell}}\), t2 > t1 ≥ 0, and let
$$\displaystyle\begin{array}{rcl} T_{i}(t^{1},t^{2})& \equiv & \vert \{t \in [t^{1},t^{2}]:\sigma (t) = i\}\vert, {}\\ T^{+}(t^{1},t^{2})& \equiv & \vert \{t \in [t^{1},t^{2}]:\sigma (t) \in \mathcal{M}^{+}\}\vert, {}\\ T^{-}(t^{1},t^{2})& \equiv & \vert \{t \in [t^{1},t^{2}]:\sigma (t) \in \mathcal{M}^{-}\}\vert, {}\\ N_{i}(t^{1},t^{2})& \equiv & \vert \{t_{ k} \in [t^{1},t^{2}):\sigma (t_{ k}) = i\}\vert, {}\\ N(t^{1},t^{2})& \equiv & \vert \{t_{ k} \in [t^{1},t^{2})\}\vert, {}\\ N^{-}(t^{1},t^{2})& \equiv & \vert \{t_{ k} \in [t^{1},t^{2}):\sigma (t_{ k}) \in \mathcal{M}^{-}\}\vert. {}\\ \end{array}$$
Roughly, these are the activation time in the ith mode, set \(\mathcal{M}^{+}\), set \(\mathcal{M}^{-}\), and the number of switches activating the ith mode, the total number of switches, and the number of switches activating modes in the set \(\mathcal{M}^{+}\), respectively. Note that \(\bigcup _{i=1}^{m}T_{i}(t_{0},t) = [t_{0},t]\). As an illustration, consider the switching rule in Fig. 5.17, which gives that
$$\displaystyle\begin{array}{rcl} & & T_{1}(0,5) = 2,\qquad T_{1}(0,4) = 1,\qquad T_{2}(3,3.5) = 0.5,\qquad T_{3}(0,5) = 2 {}\\ & & N_{1}(0,5) = 2,\ \ \quad N_{1}(0,4) = 1,\qquad N_{2}(3,3.5) = 1,\ \qquad N_{3}(0,5) = 1. {}\\ \end{array}$$
Fig. 5.17

Example of a switching rule \(\sigma \in \mathcal{S}_{\mbox{ dwell}}\) with switch times t k  = 1, 3, 4 and \(\mathcal{M} =\{ 1,2,3\}\)

Some necessary Halanay-like results are needed for the disease eradication proofs and are reviewed here. In [174], Zhu used the following Halanay-like lemma to study switched system stability.

Lemma 5.2

Assume that β,α > 0 and the function \(u: [t_{0} - d,\infty ) \rightarrow \mathbb{R}_{+}\) satisfies the following delay differential inequality:
$$\displaystyle{ \dot{u}(t) \leq \beta \| u_{t}\|_{d} -\alpha u(t),\quad \forall t \geq t_{0}. }$$
If β −α ≥ 0, then
$$\displaystyle{ u(t) \leq \| u_{t_{0}}\|_{d}\exp ((\beta -\alpha )(t - t_{0})),\quad \forall t \geq t_{0}. }$$
If β −α < 0, then there exists a positive constant η satisfying η + βexp (ηd) −α < 0 such that
$$\displaystyle{ u(t) \leq \| u_{t_{0}}\|_{d}\exp (-\eta (t - t_{0})),\quad \forall t \geq t_{0}, }$$
where ∥utd sup−d≤s≤0u(t + s).

For completeness, an impulsive delayed version of a switching Halany-like result is presented here without proof (see Proposition 1 in [142]).

Proposition 5.1

For \(i \in \mathcal{M}\) , let a i ,b i ,g i ,h i ≥ 0 be constants and assume that a function \(u: [t_{0} - d,+\infty ) \rightarrow \mathbb{R}_{+}\) satisfies
$$\displaystyle{ \begin{array}{rlll} \dot{u}(t)& \leq b_{\sigma }\|u_{t}\|_{d} - a_{\sigma }u(t), &&t\neq t_{k},\quad t \geq t_{0}, \\ u(t)& \leq g_{\sigma }u(t^{-}) + h_{\sigma }\|u_{t}\|_{d},&&t = t_{k},\quad k \in \mathbb{N}, \end{array} }$$
(5.46)
for some \(\sigma \in \mathcal{S}_{\mathrm{dwell}}(d)\) (i.e., tk − tk−1 ≥ d for all \(k \in \mathbb{N}\)). Then, for t ≥ t0,
$$\displaystyle{ u(t) \leq \| u_{t_{0}}\|_{d}\left (\prod _{j=1}^{N(t_{0},t)}\delta _{ i_{j}}\right )\exp \left (\sum _{i\in \mathcal{M}^{+}}\lambda _{i}T_{i}(t_{0},t) -\sum _{i\in \mathcal{M}^{-}}\eta _{i}\tilde{T}_{i}(t_{0},t)\right ), }$$
(5.47)
where \(\tilde{T}_{i}(t_{0},t) \equiv T_{i}(t_{0},t) - N_{i}(t_{0},t)d\), \(\lambda _{i} \equiv b_{i}\max _{i\in \mathcal{M}}\{1/\delta _{i},1\} - a_{i}\), δi ≡ gi + hiexp (ξd), \(\xi \equiv \max \{\xi _{i}: i \in \mathcal{M}^{-}\}\), ξi > 0 is chosen for \(i \in \mathcal{M}^{-}\) so that ξi + biexp id) − ai < 0, \(\mathcal{M}^{+} \equiv \{ i \in \mathcal{M}:\lambda _{i} \geq 0\}\), \(\mathcal{M}^{-}\equiv \{ i \in \mathcal{M}:\lambda _{i} < 0\}\), and ηi > 0 is chosen for \(i \in \mathcal{M}^{-}\) so that \(\eta _{i} + b_{i}\max _{i\in \mathcal{M}}\{1/\delta _{i},1\}\exp (\eta _{i}d) - a_{i} < 0\).

Proposition 5.1 is placed into the following useful form for this section (set g i  = 1, h i  = 0, and δ i  = 1 for each \(i \in \mathcal{M}\)).

Proposition 5.2

For \(i \in \mathcal{M}\) , let β i ≥ 0 and α i ≥ 0. Assume that a function \(u: [t_{0} - d,\infty ) \rightarrow \mathbb{R}_{+}\) satisfies the following switching delay differential inequality:
$$\displaystyle{ \dot{u}(t) \leq \beta _{\sigma }\|u_{t}\|_{d} -\alpha _{\sigma }u(t), }$$
and \(\sigma \in \mathcal{S}_{\mathrm{dwell}}(d)\) . Let \(\mathcal{M}^{+} \equiv \{ i \in \mathcal{M}:\lambda _{i} \geq 0\}\) and \(\mathcal{M}^{-}\equiv \{ i \in \mathcal{M}:\lambda _{i} < 0\}\) where λ i ≡β i −α i for each \(i \in \mathcal{M}\) . For each \(i \in \mathcal{M}^{-}\) , choose η i > 0 such that
$$\displaystyle{ \eta _{i} +\beta _{i}\exp (\eta _{i}d) -\alpha _{i} < 0. }$$
Then,
$$\displaystyle{ u(t) \leq \| u_{t_{0}}\|_{d}\exp \left (\sum _{i\in \mathcal{M}^{+}}\lambda _{i}T_{i}(t_{0},t) -\sum _{i\in \mathcal{M}^{-}}\eta _{i}(T_{i}(t_{0},t) - N_{i}(t_{0},t)d)\right ),\quad \forall t \geq t_{0}. }$$
(5.48)

Note that if λ i  = β i α i  < 0 for \(i \in \mathcal{M}\) then it is always possible to choose η i  > 0 satisfying η i +β i exp(η i d) −α i  < 0. Letting F i (x) ≡ x +β i exp(x d) −α i , F(0) = β i α i  < 0 and F′(η i ) = 1 +β i dexp(η i d) > 0. By continuity of F i , there exists η i  > 0 such that F(η i ) = 0 and η i can be chosen as 0 < η i  < η i .

Proposition 5.3

Assume that β i ≥ 0 and α i ≥ 0 for \(i \in \mathcal{M}\) . Assume that a function \(u: [t_{0} - d,\infty ) \rightarrow \mathbb{R}_{+}\) satisfies the following switching delay differential inequality:
$$\displaystyle{ \dot{u}(t) \leq \beta _{\sigma }\|u_{t}\|_{d} -\alpha _{\sigma }u(t), }$$
and \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) . Let \(\mathcal{M}^{+} \equiv \{ i \in \mathcal{M}:\lambda _{i} \geq 0\}\) and \(\mathcal{M}^{-}\equiv \{ i \in \mathcal{M}:\lambda _{i} < 0\}\) where λ i ≡β i −α i for each \(i \in \mathcal{M}\) . For each \(i \in \mathcal{M}^{-}\) , choose η i > 0 such that
$$\displaystyle{ \eta _{i} +\beta _{i}\exp (\eta _{i}d) -\alpha _{i} < 0. }$$
Then, u is bounded on any compact interval and satisfies
$$\displaystyle{ u(t_{0} + j\omega ) \leq \| u_{t_{0}}\|_{d}\chi ^{j},\quad \forall j \in \mathbb{N}, }$$
(5.49)
where
$$\displaystyle{ \chi =\exp \left [\sum _{i\in \mathcal{M}^{+}}\lambda _{i}\tau _{i} -\sum _{i\in \mathcal{M}^{-}}\eta _{i}(\tau _{i} - d)\right ]. }$$

Proof

The boundedness of u on any compact interval follows immediately from Theorem 5.2. From Eq. (5.48), for \(j \in \mathbb{N}\)
$$\displaystyle\begin{array}{rcl} & & u(t_{0} + j\omega ) {}\\ & & \quad \leq \| u_{t_{0}}\|_{d}\exp \left [\sum _{i\in \mathcal{M}^{+}}\lambda _{i}T_{i}(t_{0},t_{0} + j\omega ) -\sum _{i\in \mathcal{M}^{-}}\eta _{i}(T_{i}(t_{0},t_{0} + j\omega ) - N_{i}(t_{0},t_{0} + j\omega )d)\right ], {}\\ & & \quad =\| u_{t_{0}}\|_{d}\exp \left [\sum _{i\in \mathcal{M}^{+}}\lambda _{i}jT_{i}(t_{0},t_{0}+\omega ) -\sum _{i\in \mathcal{M}^{-}}\eta _{i}j(T_{i}(t_{0},t_{0}+\omega ) - N_{i}(t_{0},t_{0}+\omega )d)\right ], {}\\ & & \quad =\| u_{t_{0}}\|_{d}\exp \left [j\sum _{i\in \mathcal{M}^{+}}\lambda _{i}\tau _{i} - j\sum _{i\in \mathcal{M}^{-}}\eta _{i}(\tau _{i} - d)\right ], {}\\ & & \quad =\| u_{t_{0}}\|_{d}\chi ^{j}, {}\\ \end{array}$$
since \(\sigma \in \mathcal{S}_{\mbox{ periodic}}(\omega )\) implies that T i (t0, t0 + j ω) = j T i (t0, t0 +ω) and N i (t0, t0 + j ω) = j N i (t0, t0 +ω).

We are now in a position to prove some eradication results.

Theorem 5.9

For each \(i \in \mathcal{M}^{-}\), let ηi > 0 satisfy
$$\displaystyle{ \eta _{i} +\beta _{i}(\overline{S}_{\mathrm{max}} +\xi \overline{V }_{\mathrm{max}})\exp (\eta _{i}d) - (\mu +g + p_{i}) < 0. }$$
Let \(\lambda ^{+} \equiv \max \{\lambda _{i}: i \in \mathcal{M}^{+}\}\) and \(\lambda ^{-} \equiv \min \{\eta _{i}: i \in \mathcal{M}^{-}\}\). Let \(\sigma \in \mathcal{S}_{\mathrm{dwell}}(d)\) such that there exists M > 0 and \(\widetilde{t} > 0\) satisfying
$$\displaystyle{ \sup _{t\geq \widetilde{t}} \frac{t -\widetilde{ t}} {T^{-}(\widetilde{t},t) - N^{-}(\widetilde{t},t)d} \leq M. }$$
(5.50)
If there exists q ≥ 0 such that
$$\displaystyle{ T^{+}(\widetilde{t},t) \leq q(T^{-}(\widetilde{t},t) - N^{-}(\widetilde{t},t)d)), }$$
(5.51)
$$\displaystyle{ q\lambda ^{+} <\lambda ^{-}, }$$
(5.52)
then the solution of (5.42) satisfies \(\lim _{t\rightarrow \infty }(S(t),I(t),R(t),V (t)) \in \varPsi _{\mathrm{cohort}}\); the solution converges to the disease-free set and is therefore eradicated.

Proof

From the switched model of a vector-borne disease (5.42),
$$\displaystyle\begin{array}{rcl} \dot{S}(t)& =& \mu -\beta _{\sigma }S(t)\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - v_{\sigma }S(t) +\theta V (t), {}\\ & \leq & \mu (1 - S(t)) - v_{\sigma }S(t) +\theta V (t), {}\\ & \leq & \mu (1 - S(t)) - v_{\mathrm{min}}S(t) +\theta V (t), {}\\ & \leq & \mu +\theta - (\mu +\theta + v_{\mathrm{min}})S(t), {}\\ \end{array}$$
since V (t) = 1 − S(t) − I(t) − R(t) ≤ 1 − S(t). Similarly,
$$\displaystyle\begin{array}{rcl} \dot{V }(t)& =& v_{\sigma }S(t) -\xi \beta _{\sigma }V (t)\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - (\mu +\theta )V (t), {}\\ & \leq & v_{\sigma }S(t) - (\mu +\theta )V (t), {}\\ & \leq & v_{\mbox{ max}}S(t) - (\mu +\theta )V (t), {}\\ & \leq & v_{\mbox{ max}}(1 - V (t)) - (\mu +\theta )V (t), {}\\ & \leq & v_{\mbox{ max}} - (v_{\mbox{ max}} +\mu +\theta )V (t). {}\\ \end{array}$$
For any ε > 0, there exists a time t > 0 for which \(S(t) \leq \overline{S}_{\mbox{ max}}+\epsilon\) and \(V (t) \leq \overline{V }_{\mbox{ max}}+\epsilon\) for all t ≥ t. Let l be the smallest positive integer such that \(t_{l} >\max \{\widetilde{ t},t^{{\ast}}\}\). Then,
$$\displaystyle\begin{array}{rcl} \dot{I}(t)& =& \beta _{\sigma }(S(t) +\xi V (t))\int _{0}^{d}f(u)I(t - u)\mathrm{d}u - (\mu +g + p_{\sigma })I(t), {}\\ & \leq & \beta _{\mbox{ max}}(1+\xi )\sup _{t-d\leq s\leq t}I(s) - (\mu +g + p_{\mbox{ min}})I(t),\quad \forall t \in [0,t_{l}). {}\\ \end{array}$$
By inspection, I(t) ≤ ∥ I0 ∥  d exp(η t) for all t ∈ [0, t l ) where η > 0 satisfies
$$\displaystyle{ \eta +\beta _{\mbox{ max}}(1+\xi )\exp (\eta d) - (\mu +g + p_{\mbox{ min}}) > 0 }$$
by Lemma 5.2. In general,
$$\displaystyle{ \dot{I}(t) \leq \beta _{\sigma }[(\overline{S}_{\mbox{ max}}+\epsilon ) +\xi (\overline{V }_{\mbox{ max}}+\epsilon )]\sup _{t-d\leq s\leq t}I(s) - (\mu +g + p_{\sigma })I(t), }$$
(5.53)
for all t ∈ [tk−1, t k ) and k − 1 ≥ l, where \(I_{t_{l}} \in \mathrm{ PC}([-d,0], \mathbb{R}_{+})\). Define the constants
$$\displaystyle{ \lambda _{i,\epsilon } \equiv \beta _{i}[(\overline{S}_{\mbox{ max}}+\epsilon ) +\xi (\overline{V }_{\mbox{ max}}+\epsilon )] - (\mu +g + p_{i}),\quad \forall i \in \mathcal{M}. }$$
For each \(i \in \mathcal{M}^{-}\), let ηi, ε > 0 satisfy
$$\displaystyle{ \eta _{i,\epsilon } +\beta _{i}[(\overline{S}_{\mbox{ max}}+\epsilon ) +\xi (\overline{V }_{\mbox{ max}}+\epsilon )]\exp (\eta _{i,\epsilon }d) - (\mu +g + p_{i}) < 0. }$$
Proposition 5.2 thus implies that
$$\displaystyle{ I(t) \leq I_{0}^{{\ast}}\exp \left [\sum _{ i\in \mathcal{M}^{+}}\lambda _{i,\epsilon }T_{i}(t_{l},t) -\sum _{i\in \mathcal{M}^{-}}\eta _{i,\epsilon }(T_{i}(t_{l},t) - N_{i}(t_{l},t)d)\right ], }$$
(5.54)
for all t ∈ [tk−1, t k ), k − 1 ≥ l, where I0 ≡ ∥ I0 ∥  d exp(η t l ).
Define \(\lambda _{\epsilon }^{+} \equiv \max \{\lambda _{i,\epsilon }: i \in \mathcal{M}^{+}\}\) and \(\lambda _{\epsilon }^{-} \equiv \{\eta _{i,\epsilon }: i \in \mathcal{M}^{-}\}\). Then, by definition,
$$\displaystyle{ \beta _{i}[(\overline{S}_{\mbox{ max}}+\epsilon ) +\xi (\overline{V }_{\mbox{ max}}+\epsilon )]\exp (\eta _{i,\epsilon }d) - (\mu +g + p_{i}) < -\eta _{i,\epsilon } \leq -\lambda _{\epsilon }^{-},\quad \forall i \in \mathcal{M}, }$$
which can be rewritten as
$$\displaystyle{ \beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\exp (\eta _{i,\epsilon }d) - (\mu +g + p_{i}) + G_{i}\epsilon < -\eta _{i,\epsilon } \leq -\lambda _{\epsilon }^{-},\quad \forall i \in \mathcal{M}, }$$
where
$$\displaystyle{ G_{i} \equiv \beta _{i}(1+\xi )\exp (\eta _{i,\epsilon }d),\quad \forall i \in \mathcal{M}. }$$
Also,
$$\displaystyle{ \beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\exp (\eta _{i}d) - (\mu +g + p_{i}) < -\eta _{i} \leq -\lambda ^{-},\quad \forall i \in \mathcal{M}. }$$
Therefore, there exists a constant F1 such that −λ ε + ≤ −λ + F1ε. Letting \(\nu \in \mathop{\mathrm{arg\,max}}\limits \{\lambda _{i}: i \in \mathcal{M}^{+}\}\),
$$\displaystyle{ q\lambda _{\epsilon }^{+} \leq q\lambda ^{+} + F_{ 2}\epsilon, }$$
where
$$\displaystyle{ F_{2} \equiv q\beta _{\nu }(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + c_{\nu }). }$$
Hence,
$$\displaystyle{ q\lambda _{\epsilon }^{+} -\lambda _{\epsilon }^{-}\leq q\lambda ^{+} -\lambda ^{-} + (F_{ 1} + F_{2})\epsilon. }$$
Since q λ+λ < 0, there exists a positive constant δ such that q λ ε +λ ε  ≤ −0. 5δ. Choose
$$\displaystyle{ \epsilon = \frac{\delta (F_{1} + F_{2})} {2}, }$$
then q λ ε +λ ε  ≤ −0. 5δ.
It thus follows from Eqs. (5.50), (5.51), and (5.54) that
$$\displaystyle\begin{array}{rcl} I(t)& \leq & I_{0}^{{\ast}}\exp \left [\lambda _{\epsilon }^{+}\sum _{ i\in \mathcal{M}^{+}}T_{i}(t_{l},t) -\lambda _{\epsilon }^{-}\sum _{ i\in \mathcal{M}^{-}}(T_{i}(t_{l},t) - N_{i}(t_{l},t)d)\right ], {}\\ & =& I_{0}^{{\ast}}\exp [\lambda _{\epsilon }^{+}T^{+}(t_{ l},t) -\lambda _{\epsilon }^{-}(T^{-}(t_{ l},t) - N^{-}(t_{ l},t)d)], {}\\ & \leq & I_{0}^{{\ast}}\exp [q\lambda _{\epsilon }^{+}(T^{-}(t_{ l},t) - N^{-}(t_{ l},t)d) -\lambda _{\epsilon }^{-}(T^{-}(t_{ l},t) - N^{-}(t_{ l},t)d)], {}\\ & =& I_{0}^{{\ast}}\exp [(q\lambda _{\epsilon }^{+} -\lambda _{\epsilon }^{-})(T^{-}(t_{ l},t) - N^{-}(t_{ l},t)d)], {}\\ & \leq & I_{0}^{{\ast}}\exp \left [(q\lambda _{\epsilon }^{+} -\lambda _{\epsilon }^{-})\frac{(t - t_{l})} {M} \right ],\quad \forall t \geq t_{l}. {}\\ \end{array}$$
Equation (5.51) guarantees that T(t l , t) − N(t l , t)d ≥ 0. Therefore,
$$\displaystyle{ I(t) \leq I_{0}^{{\ast}}\exp [-0.5\delta (t - t_{ l})],\quad \forall t \geq t_{l}. }$$
It follows that limt → R(t) = 0 and (5.42) reduces to system (5.43), from which the result follows.
Intuitively, Eqs. (5.51) and (5.52) describe the time spent in the unstable mode \(\mathcal{M}^{+}\), with corresponding worst-case growth rate λ+, versus the time spent in the stable modes \(\mathcal{M}^{-}\), with corresponding most conservative decay rate λ. The constant q characterizes said relationships. If Eq. (5.52) holds, then
$$\displaystyle\begin{array}{rcl} & & q\max _{i\in \mathcal{M}^{+}}\{\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + p_{i})]\} {}\\ & & \quad +\min _{i\in \mathcal{M}^{-}}\{\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\exp (\eta _{i}d) - (\mu +g + p_{i})\} < 0. {}\\ \end{array}$$
Let \(\nu \in \mathop{\mathrm{arg\,max}}\limits \{\lambda _{i}: i \in \mathcal{M}^{+}\}\) and \(\zeta \in \mathop{\mathrm{arg\,min}}\limits \{\eta _{i}: i \in \mathcal{M}^{+}\}\). Then
$$\displaystyle{ \lambda ^{+} =\beta _{\nu }(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + p_{\nu }) }$$
and
$$\displaystyle{ \lambda ^{-} =\beta _{\zeta }(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\exp (\eta _{\zeta }d) - (\mu +g + p_{\zeta }). }$$
Hence, (5.52) implies that
$$\displaystyle{ \overline{\mathrm{R}_{0}}^{(5.42)} \equiv q\frac{(\beta _{\nu } +\beta _{\zeta }\exp (\eta _{\zeta }d))(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})} {2\mu + 2g + p_{\nu } + p_{\zeta }} < 1, }$$
(5.55)
an approximation of the disease’s basic reproduction number. In fact, (5.52) implies (5.55); (5.52) is a stricter requirement on the model parameters. Controlled eradication under periodic variations can be established as follows.

Theorem 5.10

For each \(i \in \mathcal{M}^{-}\), let ηi > 0 satisfy
$$\displaystyle{ \eta _{i} +\beta _{i}(\overline{S}_{\mathrm{max}} +\xi \overline{V }_{\mathrm{max}})\exp (\eta _{i}d) - (\mu +g + p_{i}) < 0. }$$
If \(\sigma \in \mathcal{S}_{\mathrm{periodic}}(\omega )\) and
$$\displaystyle{ \varLambda _{\mathrm{cohort}} \equiv \sum _{i\in \mathcal{M}^{+}}\lambda _{i}\tau _{i} -\sum _{i\in \mathcal{M}^{-}}\eta _{i}(\tau _{i} - d) < 0, }$$
(5.56)
then the solution of (5.42) satisfies \(\lim _{t\rightarrow \infty }(S(t),I(t),R(t),V (t)) \in \varPsi _{\mathrm{cohort}}\); the solution converges to the disease-free set and is therefore eradicated.

Proof

Beginning from Eq. (5.53) in the proof of Theorem 5.9, choose the smallest positive integer B such that \(B\omega >\max \{\widetilde{ t},t^{{\ast}}\}\). By Proposition 5.3, I((B + j)ω) ≤ ∥ I B ω  ∥  d δ j for each \(j \in \mathbb{N}\), where
$$\displaystyle{ \delta \equiv \exp \left [\sum _{i\in \mathcal{M}^{+}}\lambda _{i,\epsilon }\tau _{i} -\sum _{i\in \mathcal{M}^{-}}\eta _{i,\epsilon }(\tau _{i} - d)\right ] }$$
and ∥ I B ω  ∥  d  ≤ K for some constant K > 0. It follows from (5.56) and the arguments in the proof of Theorem 5.9 that ε > 0 can be chosen sufficiently small to guarantee that 0 < δ < 1. Thus, limt → I(t) = 0, from which limt → R(t) = 0 follows. Equation (5.42) reduces to (5.43) and the result holds.
Equation (5.56) implies that
$$\displaystyle\begin{array}{rcl} & & \sum _{i\in \mathcal{M}^{+}}[\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + p_{i})]\tau _{i} {}\\ & & \quad +\sum _{i\in \mathcal{M}^{-}}[\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\exp (\eta _{i}d) - (\mu +g + p_{i})](\tau _{i} - d), {}\\ & & \qquad <\sum _{i\in \mathcal{M}^{+}}[\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}}) - (\mu +g + p_{i})]\tau _{i} +\sum _{i\in \mathcal{M}^{-}}(-\eta _{i})(\tau _{i} - d), {}\\ & & \qquad =\sum _{i\in \mathcal{M}^{+}}\lambda _{i}\tau _{i} -\sum _{i\in \mathcal{M}^{-}}\eta _{i}(\tau _{i} - d), {}\\ & & \qquad < 0. {}\\ \end{array}$$
That is, (5.56) implies that \(\overline{\mathrm{R}_{0}}^{(5.42)} < 1\) where
$$\displaystyle{ \overline{\mathrm{R}_{0}}^{(5.42)} \equiv \frac{\sum _{i\in \mathcal{M}^{+}}\beta _{i}(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})\tau _{i} +\sum _{i\in \mathcal{M}^{-}}\beta _{i}\exp (\eta _{i}d)(\overline{S}_{\mbox{ max}} +\xi \overline{V }_{\mbox{ max}})(\tau _{i} - d)} {\sum _{i\in \mathcal{M}^{+}}(\mu +g + p_{i})\tau _{i} +\sum _{i\in \mathcal{M}^{-}}(\mu +g + p_{i})(\tau _{i} - d)}. }$$
(5.57)
\(\overline{\mathrm{R}_{0}}^{(5.42)}\) may be viewed as an approximate basic reproduction number and it should be noted that the theorem condition is stricter than requiring \(\overline{\mathrm{R}_{0}}^{(5.42)} < 1\). A comparison of these switching control strategies (i.e., switching vaccination and switching treatment) is reserved for Sect.  6.2.1
The results of this section are illustrated with simulation. Consider the initial conditions (S0, I0, R0, V0) = (0. 9, 0. 1, 0, 0), baseline model parameters in Table 5.1, and dwell-time satisfying periodic switching rule
$$\displaystyle{ \sigma = \left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &\mbox{ if }t \in [k,k + \frac{3} {12}),k \in \mathbb{N} \cup \{ 0\}, \\ 2,\quad &\mbox{ if }t \in [k + \frac{3} {12},k + 1), \end{array} \right. }$$
(5.58)
which is motivated from seasonal variations in the model parameters. The period of the switching rule is ω = 1 with τ1 = 3∕12 (modeling a winter season or rainy season, depending on climate) and τ2 = 9∕12 (summer seasons or dry season). As in [104], let
$$\displaystyle{ f(u) \equiv \frac{\exp (-u)} {1 -\exp (-d)}. }$$
Let v1 = 3, v = 2, p = 2, p = 0. 5. For the susceptible cohort immunization program, the model parameters give \(\overline{S}_{\mbox{ max}} = 0.3548\), \(\overline{V }_{\mbox{ max}} = 0.7317\), and λ1 = 0. 8948 (i.e., \(\mathcal{M}^{+} =\{ 1\}\)). Letting η2 = 1 (i.e., \(\mathcal{M}^{-} =\{ 2\}\)) implies that
$$\displaystyle{ \varLambda _{\mbox{ cohort}} = -0.4263, }$$
and convergence to the disease-free set Ψcohort by Theorem 5.10 (Fig. 5.18).
Fig. 5.18

Simulations of the switching control scheme (5.42) with parameters in Table 5.1

Table 5.1

Epidemiological parameters

Parameter

Description

Value

 

β σ

Average number of contacts per unit time

[8, 1.6]

 

μ

Natural birth/death rate

1

 

g

Recovery rate

1.5

 

d

Upper bound on the incubation time

0.1

 

The parameter values given in brackets represent the switching value associated with σ = 1 and σ = 2, respectively

5.6 Discussions

Cohort immunization (i.e., time-constant vaccination) has been implemented by a number of countries, as discussed earlier. As mentioned, the predominant strategy for measles immunization follow a recommendation of doses at 15 months of age and approximately 6 years of age in many parts of the Western world [139]. For background studies in the literature on epidemic models with such a control program, the reader is referred to [4, 69, 102, 138] and the references therein. The control strategies considered in Sect. 5.1 assume immediate movement from susceptible classes to the recovered class. This ignores the time involved to obtain immunity by completing a vaccination program. Motivated by this, consider the usual vaccination schedule for hepatitis B where individuals are given three vaccinations separated by 1 month and 6 months [101]. The authors further note that approximately 30–50% of individuals will gain anti-HB antibodies after the first dose, 80–90% after the second dose, and virtually all 1 month after the final dose. Based on their work on hepatitis B and measles in [101], the model (5.13) was analyzed. The application of vaccination and treatment schemes to switched SIR models in Sects. 5.1 and 5.2 are largely based on, and extend, the works in [94, 96].

Hepatitis B, Chagas’ disease, HIV/AIDS, and tuberculosis are examples of diseases displaying latency periods [103, 140] and therefore appropriately modeled by the SEIR model (5.22), which has been extensively studied in the literature (e.g., see [69, 72, 80, 81, 103, 134]). The SEIR model with vertical transmission (5.27) was analyzed with switching because of the number of infectious diseases with latency period that are transmitted by both horizontal and vertical modes (e.g., rubella, herpes simplex, hepatitis B, Chagas’ disease [140]). The SEIR model with disease-induced deaths (i.e., Eq. (5.31)) is an appropriate modeling choice for disease like AIDS [140]. A summary of the critical control rates guaranteeing eradication in the various theorems provided in Sects. 5.15.2, and 5.3 is given in Table 5.2.
Table 5.2

Critical control rates of the epidemic models with periodic switching

Control strategy

Disease model

Critical control rate

 

Newborn vaccinations

(5.4)

ρcrit ≡ 1 −ω(μ + g)∕(i = 1 m β i τ i )

 

Susceptible vaccinations

(5.8)

\(v_{\mbox{ crit}} \equiv \mu \left (\sum _{i=1}^{m}\beta _{i}\tau _{i}/(\omega (\mu +g)) - 1\right )\)

 

Vaccinations with progressive immunity

(5.13)

\(v_{\mbox{ crit}} \equiv \frac{\mu \left (\sum _{i=1}^{m}\beta _{ i}\tau _{i}/(\omega (\mu +g))-1\right )} {1-\mu \sum _{i=1}^{m}\beta _{i}^{V }\tau _{i}/(\omega (\mu +g))}\)

 

Treatment of infected

(5.17)

paverage-crit ≡ i = 1 m β i τ i ω − (μ + g)

 

SIR with general FOI and treatments

(5.20)

p i crit ≡ h i ′(0) − (μ + g)

 

SEIR with treatments

(5.24)

p i crit ≡ β i a∕(μ + a) − (μ + g)

 

Vertical SEIR with treatments

(5.27)

\(p_{i}^{\mbox{ crit}} \equiv \frac{(\beta _{i}+\mu q)a} {\mu (1-\rho )+a} - (\mu +g)\)

 

Disease-induced mortality SEIR with treatments

(5.31)

p i crit ≡ β i a∕(b + a) − (μ + g +α)

 

Note that \(p_{\mbox{ average-crit}} \equiv \frac{\sum _{i=1}^{m}p_{i}^{\mbox{ crit}}\tau _{i}} {\omega }\). The critical rates indexed by i are mode-dependent

In the mathematical epidemic modeling literature, a time-constant entry/exit screening strategy was studied by Liu and Takeuchi [100] consisting of a two-city SIS model with transport-related infections and a screening process. Entry and exit screening were performed during the spread of SARS in 2003; temperature screening using thermal scanning and questionnaires were given to assess symptoms for possible exposure at mass transit centers [100]. More recently, global travel has been a major factor in the spread of the H1N1 strain of influenza in 2009, Ebola virus in 2015, and Zika virus in 2016. Motivated by this and the time-invariant entry screening models investigated in [100, 147], screening strategies were considered in Sect. 5.4. The formulation and analysis of the screening strategy for a switched multi-city model in Sect. 5.4. In Sect. 5.5, Halanay-like switching results were used to prove convergence to disease-free sets (and thus disease eradication). Halanay-like inequalities have been generalized to include switching (for example, [164]), time-varying parameters (for example, [120, 172]), and impulsive effects (for example, [160, 163]). The works [142, 143] form the basis for the derivations and results found in Sect. 5.5. Other switching control strategies (e.g., reduced contact rates) are detailed later in this monograph, while other possibilities (e.g., purposeful shifts in population behavior) are theoretically unlocked by the findings here.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Xinzhi Liu
    • 1
  • Peter Stechlinski
    • 1
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

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