SIS and SIR Epidemic Models Under Virtual Dispersal

Abstract

We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number ${\mathcal {R}}_0$ is computed as a function of a patch residence-time matrix ${\mathbb {P}}$. Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: There is a unique globally stable endemic equilibrium when ${\mathcal {R}}_0>1$, while the disease-free equilibrium is globally stable when ${\mathcal {R}}_0\le 1$. Our further analysis indicates that the dispersal behavior described by the residence-time matrix ${\mathbb {P}}$ has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence-time matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single-outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease-prevalence-driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state-dependent ${\mathbb {P}}$ on disease dynamics.

Appendices

Appendix 1: Computation of ${\mathcal {R}}_0$

Proof

The general SIS model with residence time is described by the system (6)

$$\begin{aligned} \dot{I}={\text {diag}}(\bar{N}-I){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^tI-{\text {diag}}(d_I+\gamma _I)I. \end{aligned}$$

The right-hand member of the above system can be clearly decomposed as ${\mathcal {F}}+{\mathcal {V}}$ where

$$\begin{aligned} {\mathcal {F}}={\text {diag}}(\bar{N}-I){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^tI \quad {\text {and}}\quad {\mathcal {V}}=-{\text {diag}}(d_I+\gamma _I)I \end{aligned}$$

The Jacobian matrix at the DFE of ${\mathcal {F}}$ and ${\mathcal {V}}$ is given by:

$$\begin{aligned} F=D{\mathcal {F}}\bigg |_{DFE}={\text {diag}}(\bar{N}){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^t \quad {\text {and}}\quad V={\mathcal {V}}\bigg |_{DFE}=-{\text {diag}}(d_I+\gamma _I) \end{aligned}$$

The basic reproduction number ${\mathcal {R}}_0$ is given by the spectral radius of the next-generation matrix $-FV^{-1}$ (Diekmann et al. 1990; Driessche and Watmough 2002). Hence, we deduce that

$$\begin{aligned} {\mathcal {R}}_0=\rho (-{\text {diag}}(\bar{N}){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^t V^{-1} ) \end{aligned}$$

$\square$

Appendix 2: Proof of Theorem 2.1

The proof uses the method in Iggidr et al. (2012) which is based on Hirsch’s theorem (Hirsch 1984).

Theorem 5.1

(Hirsch 1984) Let $\dot{x}=F(x)$ be a cooperative differential equation for which ${\mathbb {R}}^n_+$ is invariant , the origin is an equilibrium, each DF(x) is irreducible, and that all orbits are bounded. Suppose that

$$\begin{aligned} x>y\implies DF(x)<DF(y)\quad {\text {for all}}\quad x,y. \end{aligned}$$

Then, all orbits in ${\mathbb {R}}^n_+$ tend to zero or there is a unique equilibrium $p^*$ in the interior of ${\mathbb {R}}^n_+$ and all orbits in ${\mathbb {R}}^n_+$ tend to $p^*$.

Proof of Theorem 2.1

Equation (6) can be written as:

$$\begin{aligned} \dot{I}=(F+V)I-{\text {diag}}(I){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^tI \end{aligned}$$

(14)

where $F=\text {diag}(\bar{N}){\mathbb {P}}\text {diag}({\mathcal {B}})\text {diag}(\tilde{N})^{-1}{\mathbb {P}}^t$ and $V=-\text {diag}(d_I+\gamma _I)$, as defined in “Appendix 1.” Let us denote by X(I) the semi-flow induced by (14). Hence,

$$\begin{aligned} DX(I)={\text {diag}}(\bar{N}-I) {\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^t+V-W(I_1,I_2) \end{aligned}$$

(15)

where $W(I_1,I_2)={\text {diag}}({\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^t I)$. Since ${\mathbb {P}}$ is irreducible and $I\le \bar{N}$, DX(I) is clearly Metzler irreducible matrix. That means, the flow is strongly monotone. Plus, DX(I) is clearly decreasing with respect of I. Hence, by Hirsch’s theorem all trajectories either go to zero or go to an equilibrium point $\bar{I}\gg 0$. From the relation (15), we have $DX(0)=F+V$ where F and V are the one defined previously in “Appendix 1.” However, since F a nonnegative matrix and V is Metzler, we have the following equivalence

$$\begin{aligned} \alpha (F+V)<0\iff \rho (-FV^{-1})<1 \end{aligned}$$

where $\alpha (F+V)$ is the stability modulus, i.e., the largest real part of eigenvalues, of $F+V$ and $\rho (-FV^{-1})$ the spectral radius of $-FV^{-1}$. Hence, the DFE is globally asymptotically stable if ${\mathcal {R}}_0=\rho (-FV^{-1})<1$. And if ${\mathcal {R}}_0>1$, i.e., $\alpha (F+V)>0$, the DFE is unstable (Driessche and Watmough 2002). Since, we have proved that DX(I) is a Metzler matrix, to prove the local stability of the endemic equilibrium $\bar{I}\gg 0$, we only need to prove that it exists $w\gg 0$ such that $DX(\bar{I})w<0$ (Berman and Plemmons 1994). The endemic equilibrium $\bar{I}\gg 0$ satisfies the equation

$$\begin{aligned} (F+V)\bar{I}-{\text {diag}}(\bar{I}){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^t\bar{I}=0 \end{aligned}$$

Hence,

$$\begin{aligned} DX(\bar{I})\bar{I}=-W(\bar{I})\bar{I}<0 \end{aligned}$$

Hence, with $w=\bar{I}$, we deduce that $\bar{I}$ is locally stable. With the attractivity of $\bar{I}$ guaranteed Hirsh’s theorem, we conclude that the endemic equilibrium $\bar{I}\gg 0$ is globally asymptotically stable if ${\mathcal {R}}_0>1$.

Finally, if ${\mathcal {R}}_0=1$, we have $\alpha (F+V)=0$. It exists $c\gg 0$ such that $(F+V)^tc=0$. By considering the Lyapunov function $V=\left\langle c |I\right\rangle$. This function is definite positive and its derivation along the trajectories if (14) is

$$\begin{aligned} \dot{V}= & {} \left\langle c |\dot{I}\right\rangle \nonumber \\= & {} \left\langle c |(F+V)I-{\text {diag}}(I){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^tI\right\rangle \nonumber \\= & {} -\left\langle c |{\text {diag}}(I){\mathbb {P}}{\text {diag}}({\mathcal {B}}){\text {diag}}(\tilde{N})^{-1}{\mathbb {P}}^tI\right\rangle \nonumber \\\le & {} 0 \end{aligned}$$

(16)

Plus $\dot{V}=0$ only at the DFE. Hence, the DFE is GAS if ${\mathcal {R}}_0=1$. This completes the proof of the Theorem 2.1. $\square$

Appendix 3: Proof of Theorem 2.2

Proof

Since System (6) has an attracting compact $\Omega$, then according to Theorem (2.1), we can expect that $\lim _{t\rightarrow \infty } I_i(t)<\frac{b_i}{d_i}$; thus, for time large enough, we can have $\frac{b_i}{d_i}-I_i>0$, therefore we have

$$\begin{aligned} \dot{I}_i>I_i\left( \frac{b_i}{d_i}-I_i\right) \left( \sum _{j=1}^{n}\frac{\beta _jp_{ij}^2}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}\right) -(d_i+\gamma _i )I_i \end{aligned}$$

which indicates that when ${\mathcal {R}}_0^i({\mathbb {P}})>1$

$$\begin{aligned} \frac{\dot{I}_i}{I_i}\big \vert _{I_i=0}= & {} \frac{b_i}{d_i}\left( \sum _{j=1}^{n}\frac{\beta _jp_{ij}^2}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}\right) -(d_i+\gamma _i )>0. \end{aligned}$$

Then apply the average Lyapunov Theorem (Hutson 1984), we can conclude that $\liminf _{t\rightarrow \infty } I_i(t)>0$; i.e., the disease in the residence Patch i is persistent if ${\mathcal {R}}_0^i({\mathbb {P}})>1$ .

If $p_{ij}>0$ and $p_{kj}=0$ for all $k=1,\ldots ,n, \text{ and } k\ne i$, this implies that if there is a portion of the residence Patch i population flowing into the residence Patch j, then there is no other residence Patch k where $k\ne j$, i.e.,

$$\begin{aligned} \beta _jp_{ij}\sum _{k=1,k\ne i}^{n}p_{kj}I_k=0 \end{aligned}$$

which also implies that

$$\begin{aligned} \left( \frac{b_i}{d_i}-I_i\right) \sum _{j=1}^{n}\frac{\beta _jp_{ij}\sum _{k=1,k\ne i}^{n}p_{kj}I_k}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}} =0. \end{aligned}$$

then we can conclude that Model (6) can have an equilibrium since under these conditions,

$$\begin{aligned} \frac{b_i}{d_i}\sum _{j=1}^{n}\frac{\beta _jp_{ij}\sum _{k=1,k\ne i}^{n}p_{kj}I_k}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}=\frac{b_i}{d_i}\frac{\beta _i \sum _{k=1,k\ne i}^{n}p_{ki}I_k}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}} =0. \end{aligned}$$

Therefore, if the conditions $p_{kj}=0$ for all $k=1,\ldots ,n, \text{ and } k\ne j$ whenever $p_{ij}>0$ hold, then we have

$$\begin{aligned} \dot{I}_i\vert _{I_i=0}= & {} \left[ I_i\left( \frac{b_i}{d_i}-I_i\right) \left( \sum _{j=1}^{n}\frac{\beta _jp_{ij}^2}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}\right) \right. \nonumber \\&\qquad \left. +\left( \frac{b_i}{d_i}-I_i\right) \sum _{j=1}^{n}\frac{\beta _jp_{ij}\sum _{k=1,k\ne i}^{n}p_{kj}I_k}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}} -(d_i+\gamma _i )I_i \right] \bigg \vert _{I_i=0} \\= & {} \frac{b_i}{d_i}\sum _{j=1}^{n}\frac{\beta _jp_{ij}\sum _{k=1,k\ne i}^{n}p_{kj}I_k}{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}=0. \end{aligned}$$

Therefore, $I_i=0$ is the invariant manifold for Model (6).

On the other hand, when these conditions hold, then we have

$$\begin{aligned} {\mathcal {R}}_0^i({\mathbb {P}})=R_0^i \times \sum _{j=1}^n\left( \frac{\beta _j}{\beta _i}\right) p_{ij}\left( \frac{\left( p_{ij} \frac{b_i}{d_i}\right) }{\sum _{k=1}^{n}p_{kj}\frac{b_k}{d_k}}\right) =R_0^i \times \sum _{j=1}^n \left( \frac{\beta _j}{\beta _i}\right) p_{ij}. \end{aligned}$$

Therefore, if ${\mathcal {R}}_0^i({\mathbb {P}})=R_0^i \times \sum _{j=1}^n \left( \frac{\beta _j}{\beta _i}\right) p_{ij}<1$, then we have the following inequality:

$$\begin{aligned} \frac{\dot{I}_i}{I_i}= & {} I_i\left( \frac{b_i}{d_i}-I_i\right) \left( \sum _{j=1}^{n}\frac{\beta _jp_{ij}^2}{\sum _{k=1}^{n}p_{ki}\frac{b_k}{d_k}}\right) -(d_i+\gamma _i )I_i\\\le & {} I_i \left[ \frac{b_i}{d_i}\left( \sum _{j=1}^{n}\frac{\beta _jp_{ij}^2}{\sum _{k=1}^{n}p_{ki}\frac{b_k}{d_k}}\right) -(d_i+\gamma _i )\right] \\= & {} I_i \left[ \sum _{j=1}^{n}\beta _jp_{ij}-(d_i+\gamma _i )\right] <0. \end{aligned}$$

Therefore, we have $\lim _{t\rightarrow \infty } I_i(t)=0$; i.e., there is no endemic in the residence Patch i. $\square$