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Mathematical analysis of swine influenza epidemic model with optimal control

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Abstract

A deterministic model is designed and used to analyze the transmission dynamics and the impact of antiviral drugs in controlling the spread of the 2009 swine influenza pandemic. In particular, the model considers the administration of the antiviral both as a preventive as well as a therapeutic agent. Rigorous analysis of the model reveals that its disease-free equilibrium is globally asymptotically stable under a condition involving the threshold quantity-reproduction number \({\mathcal {R}}_c\). The disease persists uniformly if \({\mathcal {R}}_c>1\) and the model has a unique endemic equilibrium under certain condition. The model undergoes backward bifurcation if the antiviral drugs are completely efficient. Uncertainty and sensitivity analysis is presented to identify and study the impact of critical model parameters on the reproduction number. A time dependent optimal treatment strategy is designed using Pontryagin’s maximum principle to minimize the treatment cost and the infected population. Finally the reproduction number is estimated for the influenza outbreak and model provides a reasonable fit to the observed swine (H1N1) pandemic data in Manitoba, Canada, in 2009.

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Correspondence to Mudassar Imran.

Appendix

Appendix

Proof of Lemma 1

Proof

Adding the equations of the model (1) gives

$$\begin{aligned} \frac{dN}{dt}=\pi -\mu {N}-\delta (I_2+\theta _1H)\le \pi -\mu {N}. \end{aligned}$$
(12)

Since \(N(t)\ge 0\), it follows using a standard comparison theorem [29] that

$$\begin{aligned} N(t)\le {N(0)e^{-\mu t}+\frac{\pi }{\mu }\left( 1-e^{-\mu t}\right) }. \end{aligned}$$

Therefore, \(N(t)\le \pi /\mu \) if \(N(0)\le \pi /\mu \).

This proves the positive invariance of \({\mathcal {D}}\).

To prove that \({\mathcal {D}}\) is attracting, from (12), it is clear that \(\frac{dN}{dt}<0\), whenever \(N(t)>\pi /\mu \). Thus, either the solution enters \({\mathcal {D}}\) in finite time, or N(t) approaches \(\pi /\mu \), and the variables denoting the infected classes approach zero. Hence, \({\mathcal {D}}\) is attracting and all solutions in \(\mathbb R_+^9\) eventually enter \({\mathcal {D}}\). \(\square \)

DFE

\({\mathcal {R}}_{c}\) calculation

The matrices F (for the new infection terms) and V (of the transition terms) are given, respectively, by \(F={\begin{bmatrix} \beta \eta _1\varOmega&\beta \varOmega&\beta \eta _2\varOmega&\beta \eta _3\varOmega&\beta \eta _4\varOmega \\ 0&0&0&0&0 \\ 0&0&0&0&0 \\ 0&0&0&0&0 \\ 0&0&0&0&0 \\ \end{bmatrix}}\) \(V={\begin{bmatrix} K_1&0&0&0&0 \\ -\alpha {f}&K_2&0&0&0 \\ 0&-\gamma&K_3&0&0 \\ 0&0&-\psi&K_4&0 \\ 0&-\tau _1&-\tau _2&0&K_5\\ \end{bmatrix}}\), where, \(\varOmega =\displaystyle \frac{S_L^*+\theta _HS_H^*+\theta _PP^*}{N^*},\) \(K_1=\alpha +\mu ,\) \(K_2=\tau _1+\gamma +\mu ,\) \(K_3=\tau _2+\psi +\phi _2+\mu +\delta ,\) \(K_4=\psi _2+\phi _3+\mu +\theta _1\delta ,\) \(K_5=\phi _T+\mu .\)

Reproduction number \(R_c\) is given as

$$\begin{aligned} R_c=\rho (FV^{-1}) \end{aligned}$$

where, \(\rho \) denotes the spectral radius.

Proof of Theorem 1

Proof

Consider the model (1) with \(\theta _H=1\) and \(\theta _P=0\). Further, consider the Lyapunov function

$$\begin{aligned} {\mathcal {F}}=g_1L+g_2I_1+g_3I_2+g_4H+g_5T, \end{aligned}$$

where,

$$\begin{aligned}\begin{array}{lll} \begin{aligned} g_1&{}=\eta _1K_2K_3K_4K_5 + \alpha K_3K_4K_5 + \alpha \tau _1\eta _4K_3K_4 + \alpha \gamma \eta _2 K_4K_5 + \alpha \gamma \psi \eta _3K_5\\ &{}\quad + \alpha \gamma \tau _2\eta _4 K_4\\ g_2&{}=K_1(\gamma \tau _2\eta _4 K_5 + \gamma \psi \eta _3 K_5 + \gamma \eta _2 K_4K_5 + \tau _1\eta _4K_3K_4 + K_3K_4K_5) \\ g_3&{}=K_1K_2K_5(\eta _2K_4+\eta _3+ \tau _2 \eta _4)\\ g_4&{}=\eta _3K_1K_2K_3K_5\\ g_5&{}= \eta _4 K_1K_2K_3K_4 \end{aligned} \end{array} \end{aligned}$$

The Lyapunov derivative is given by (where a dot represents differentiation with respect to t)

$$\begin{aligned} {\mathcal {\dot{F}}}&=g_1\dot{L}+g_2\dot{I}_1+g_3\dot{I}_2+g_4\dot{H}_1+g_5\dot{T},\\&=g_1[\lambda (S_L+S_H+\theta _P P)-(\alpha +\mu ){L}] +g_2[\alpha {L}-(\tau _1+\gamma +\mu )I_{1}]\\&\quad +g_3[\gamma I_1-(\tau _2+\psi +\phi _2+\mu +\delta )I_{2}] +g_4[\psi I_2-(\psi _2+\phi _3+\mu +\theta _1\delta )H]\\&\quad +g_5[\tau _1{I}_1+\tau _2{I}_2-(\phi _T+\mu )T], \end{aligned}$$

so that,

$$\begin{aligned} {\mathcal {\dot{F}}}&=g_1\lambda [S_L(t)+\theta _H S_H(t) + \theta _P P]-K_1K_2K_3K_4K_5(\eta _1L+I_{1}+\eta _{2}{I}_{2}+\eta _{3}{H}+\eta _{4}{T}),\\&\le g_1\lambda \theta _H[S_L(t)+S_H(t) + P(t)]-K_1K_2K_3K_4K_5\frac{\lambda N}{\beta },\, \mathrm{since}\,\ {\theta _P < 1 < \theta _H}\\&\le {g}_1\lambda \theta _H{N}-K_1K_2K_3K_4K_5\frac{\lambda N}{\beta },\, \mathrm{since}\,\ (S_L(t)+S_H(t) + P(t))\le {N(t)} \ \mathrm{in} \ {\mathcal {D}} . \end{aligned}$$

It can be shown that \(g_1=\frac{{{\mathcal {R}}_c}}{\beta \varOmega }K_1K_2K_3K_4K_5.\) Hence,

$$\begin{aligned} \dot{{\mathcal {F}}}&\le \frac{{{\mathcal {R}}_c}}{\beta R^*}K_1K_2K_3K_4K_5\lambda {N}-K_1K_2K_3K_4K_5\frac{\lambda {N}}{\beta },\\&=K_1K_2K_3K_4K_5\frac{\lambda {N}}{\beta }\bigg (\frac{{{\mathcal {R}}_c}}{R^*}-1\bigg ). \end{aligned}$$

Thus, \(\dot{{\mathcal {F}}}\le 0\) if \({{\mathcal {R}}_c}\le R^*\) with \(\dot{{\mathcal {F}}}=0\) if and only if \(L=I_1=I_2=H=T=0\). Further, the largest compact invariant set in \(\{(S_L,S_H,P,L,I_1,I_2,H,T,R)\in {\mathcal {D}}: \dot{{\mathcal {F}}}=0\}\) is the singleton \(\{{\mathcal {E}}_0\}\). It follows from the LaSalle invariance principle (Chapter 2, Theorem 6.4 of [30]) that every solution to the equations in (1) with \(\theta _H=1\) and \(\theta _P=0\) and with initial conditions in \({\mathcal {D}}\) converges to DFE, \({\mathcal {E}}_0\), as \(t\rightarrow \infty \). That is, \([L(t),A(t),I_1(t),I_2(t),H(t),H_2(t),T(t)]\rightarrow (0,0,0,0,0,0,0)\) as \(t\rightarrow \infty \). Substituting \(L=A=I_1=I_2=H=H_2=T=0\) into the first three equations of the model (1) gives \(S_L(t)\rightarrow {S_L^*},\,S_H(t)\rightarrow {S_H^*}\) and \(P(t)\rightarrow {P^*}\) as \(t\rightarrow \infty \). Thus, \([S_L(t),S_H(t),P(t),L(t),I_1(t),I_2(t),H(t),T(t),R(t)] \rightarrow (S_L^*,S_H^*,P^*,0,0,0,0,0,0)\) as \(t\rightarrow \infty \) for \(\tilde{{\mathcal {R}}_c}\le R^*,\) so that the DFE, \({\mathcal {E}}_0\), is GAS in \({\mathcal {D}}\) if \({{\mathcal {R}}_c}\le R^*\). \(\square \)

Proof of Theorem 2

Proof

Let X = \(\{(S_L, S_H ,P, L, I_1, I_2, H, T, R) \in \mathbb {R}_+^9: L=I_1=I_2=H=T=0\}\). Thus X is the set of all disease free states of (1) and it can be easily verified that X is positively invariant. Let M = \(\mathcal {D}\cap X\). Since both \(\mathcal {D}\) and X are positively invariant, M is also positively invariant. Also note that \({\mathcal {E}}_0 \in M\) and \({\mathcal {E}}_0\) attracts all the solutions in X. So, \(\varOmega (M)\) = \(\{{\mathcal {E}}_0\}\).

By setting \(x(t) = (L(t),I_1(t),I_2(t),H(t),T(t)) ^ T\), equations for the infected components of (1) can be written as

$$\begin{aligned} x'(t)&= Y(x)x(t) \end{aligned}$$
(13)

where Y(x) = \(\left[ \left( \frac{S_L +\theta _H S_H + \theta _P P}{N \varOmega }\right) F - V\right] \) and it is clear that \(Y({\mathcal {E}}_0) = F - V\). Also it is easy to check that \(Y({\mathcal {E}}_0)\) is irreducible. We will apply the Lemma A.4 in [1] to show that M is a uniform weak repeller. Since \({\mathcal {E}}_0\) is a steady state solution, we can consider it to be a periodic orbit of period \(T = 1\). P(tx), the fundamental matrix of the solutions for (7) is \(e^{tY}\). Since the spectral radius of \(Y({\mathcal {E}}_0) = {{\mathcal {R}}_c}- 1 > 0\), the spectral radius of \(e^{Y({\mathcal {E}}_0)} > 1\). So condition 2 of Lemma A.4 is satisfied. Taking \(x={\mathcal {E}}_0\), we get \(P(T,{\mathcal {E}}_0) = e^{Y({\mathcal {E}}_0)}\) which is a primitive matrix, because \(Y({\mathcal {E}}_0)\) is irreducible, as mentioned in Theorem A.12(i) [40]. This satisfies the condition 1 of Lemma A.4. Thus, M is a uniform weak repeller and disease is weakly persistent. By definition, \(M = \partial \mathcal {D}\). M is trivially closed and bounded relative to \(\mathcal {D}\) and hence, compact. Therefore by Theorem 1.3 of [42], we have that M is a uniform strong repeller and disease is uniformly persistent. \(\square \)

Endemic equilibrium

The coefficients are defined as follows

$$\begin{aligned} A_1&=\pi \theta _H \theta _P\\ A_2&=\pi (1-p)[\theta _P(\sigma _H + \mu ) + \theta _H \mu ] + \pi \theta _H p (\theta _P (\sigma _L+\mu )+\mu ) + \pi \theta _P A_1\\ A_3&= \pi \mu (1-p) (\sigma _H + \mu ) + \pi \theta _H p \mu (\sigma _L + \mu ) + \pi \theta _P B\\ A_4&=\pi (1-p)[\theta _P(\sigma _H+\mu ) + \mu \theta _H] + \pi p [\mu + \theta _P (\sigma _L +\mu )] + \pi A_1\\ A_5&= \pi \mu (1-p)(\sigma _H + \mu ) + \pi p \mu (\sigma _L +\mu ) + \pi A_2\\ A_6&= \pi (1-p)\theta _H \theta _P + \pi p \theta _P\\ A_7&= \sigma _L \theta _H (1-p) + \sigma _H p\\ A_{8}&= \frac{1}{K_1} + \frac{\alpha }{K_1K_2}+ \frac{\alpha \gamma }{K_1K_2 K_3} + \frac{\alpha \gamma \psi }{K_1K_2K_3K_4}+ \frac{\alpha \tau _1 }{K_1K_2K_5} + \frac{\alpha \gamma \tau _2}{K_1K_2K_3K_5} \\&\quad +\frac{\alpha \gamma \phi _{I_2}}{\mu K_1K_2K_3} + \frac{\alpha \gamma \psi \phi _H}{\mu K_1K_2K_3K_4} + \bigg (\frac{\phi _T}{K_1K_5}\bigg )\frac{\alpha \tau _1 }{K_1K_2K_5} + \bigg (\frac{\phi _T}{K_1K_5}\bigg )\frac{\alpha \gamma \tau _2}{K_1K_2K_3K_5}.\\ \end{aligned}$$

Sensitivity analysis

The estimated parameters are presented in Table 3.

Table 3 Mean values of the model parameters with their assigned distributions

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Cite this article

Imran, M., Malik, T., Ansari, A.R. et al. Mathematical analysis of swine influenza epidemic model with optimal control. Japan J. Indust. Appl. Math. 33, 269–296 (2016). https://doi.org/10.1007/s13160-016-0210-3

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Keywords

  • Influenza
  • Reproduction number
  • Backward bifurcation
  • Uncertainty and sensitivity analysis
  • Optimal control
  • Statistical inference

Mathematics Subject Classification

  • 92B08
  • 49J15
  • 34C23