Stochastic Asymptotic Stability of SIR Model with Variable Diffusion Rates

Abstract

We introduce random fluctuations on contact and recovery rates in deterministic SIR model with disease deaths in nonparametric manner and obtain stochastic counterparts with general diffusion coefficients (functional contact and recovery rates).

$$\begin{aligned} \displaystyle dS&= \,\Big (-\beta S I +\mu (K-S)\Big )~dt - S I~ F_1\big (S,I,R\big ) ~dW_1\nonumber \\ dI&= \,\Big (\beta S I-\big (\alpha +\gamma +\mu \big )I\Big )~ dt + S I~ F_1\big (S,I,R\big ) ~dW_1-I~F_2\big (S,I,R\big ) ~dW_2 \\ dR&= \,\Big (\alpha I - \mu R\Big )~dt+ I~F_2\big (S,I,R\big ) ~dW_2. \nonumber \end{aligned}$$

(1)

The introduced stochastic model has functional diffusion coefficients which contains arbitrary local Lipschitz-continuous functions $F_i$’s defined on

$$\begin{aligned} {\mathbb {D}}=\{(S,I,R) \in {\mathbb {R}}^3: ~S\ge 0, ~I \ge 0, ~R \ge 0, ~S+I+R\le K\}. \end{aligned}$$

In this paper we prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria of the model and visualize our results with some simulations to confirm them.

Appendices

Appendix 1: Existence of a Unique Markovian, Continuous Time Solution

Consider the d-dimensional stochastic differential equation of the form

$$\begin{aligned} \displaystyle dX(t)=f\big (X(t),t\big )~dt+g\big (X(t),t\big )~dW(t) \end{aligned}$$

(23)

with an initial value $X(t_0)=X_0,~~ t_0 \le t \le T < \infty$ where $\displaystyle f:{\mathbb {R}}^{d} \times [t_0,T] \rightarrow {\mathbb {R}}^{d}$ and $\displaystyle g:{\mathbb {R}}^{d} \times [t_0,T] \rightarrow {\mathbb {R}}^{d \times m}$ are Borel measurable, $\displaystyle W=\{W(t)\}_{t\ge t_0}$ is an $\displaystyle {\mathbb {R}}^m$-valued Wiener process, and $X_0$ is an $\displaystyle {\mathbb {R}}^{d}$-valued random variable.

The infinitesimal generator $\mathbf {{\mathcal {L}}}$ associated with the SDE (23) is given by

$$\begin{aligned} \displaystyle {\mathcal {L}}=\frac{\partial }{\partial t}+ \sum _{i=1}^d f_i(x,t)\frac{\partial }{\partial x_i} +\frac{1}{2}\sum _{i,j=1}^m \Big (g(x,t)g^T(x,t)\Big )_{ij} ~\frac{\partial ^2}{\partial x_i \partial x_j}. \end{aligned}$$

(24)

Theorem 4

(${\mathbb {D}}$ -invariance) (Khas’minskiĭ [15] as appears in [11]) Let ${\mathbb {D}}$ and ${\mathbb {D}}_n$ be open sets in ${\mathbb {R}}^d$ with

$$\begin{aligned} \displaystyle {\mathbb {D}}_n~\subseteq ~ {\mathbb {D}}_{n+1}, ~~~~~~~ \mathbb {\bar{D}}_n~\subseteq ~ {\mathbb {D}}, ~~\mathrm{ and }\, {\mathbb {D}}~=\bigcup _n ~ {\mathbb {D}}_n \end{aligned}$$

and suppose $f$ and $g$ satisfy the existence and uniqueness conditions for solutions of (23) on each set $\{(t,x): t>t_0,\, x \in {\mathbb {D}}_n\}$. Suppose there is a nonnegative continuous function $V: {\mathbb {D}}\times [t_0,T]\rightarrow {\mathbb {R}}_+$ with continuous partial derivatives and satisfying ${\mathcal {L}}V \le c~V$ for some positive constant $c$ and $t>t_0,\, x \in {\mathbb {D}}$. If also,

$$\begin{aligned} \displaystyle \inf _{t>t_0, x \in {\mathbb {D}} \backslash {\mathbb {D}}_n } V(x,t) \rightarrow \infty ~~~~\mathrm{ as }\,n \rightarrow \infty \end{aligned}$$

then, for any $X_0$ independent of $\sigma (W)$ such that ${\mathbb {P}}(X_0 \in {\mathbb {D}})=1,$ there is a unique Markovian, continuous time solution $X$ of (23) with $X(0)=X_0$, and $X(t) \in {\mathbb {D}}$ for all $t>0$ (a.s.).

Appendix 2: Stability of Equilibria

Consider the d-dimensional SDE

$$\begin{aligned} \displaystyle dX(t)=f\big (X(t),t\big )~dt+g\big (X(t),t\big )~dW(t),~~t \ge t_0, ~~X(t_0)=x_0. \end{aligned}$$

(25)

Assume that $f$ and $g$ satisfy, in addition to the existence and uniqueness assumptions, $f(x^*,t)=0$ and $g(x^*,t)=0$ for equilibrium solution $x^*$ for $t \ge t_0$. Furthermore, let’s assume that $x_0$ be a non-random constant with probability 1.

Definition 1

The equilibrium solution $x^*$ of the SDE (25) is stochastically stable (stable in probability) if for every $\epsilon > 0$ and $s \ge t_0$

$$\begin{aligned} \displaystyle \lim _{x_0 \rightarrow x^*} {\mathbb {P}}\left( \sup _{t_0 \le s < \infty }\Vert X_{s,x_0}(t)\Vert \ge \epsilon \right) ~=~ 0 \end{aligned}$$

(26)

where $X_{s,x_0}(t)$ denotes the solution of (25) satisfying $X(s)=x_0$ at time $t \ge s$.

Definition 2

The equilibrium solution $x^*$ of the SDE (25) is said to be stochastically asymptotically stable if it is stochastically stable and

$$\begin{aligned} \displaystyle \lim _{x_0 \rightarrow x^*} {\mathbb {P}}\left( \lim _{t \rightarrow \infty } X_{s,x_0}(t) = x^* \right) ~=~ 1, \end{aligned}$$

(27)

Definition 3

The equilibrium solution $x^*$ of the SDE (25) is said to be globally stochastically asymptotically stable if it is stochastically stable and for every $x_0$ and every $s$

$$\begin{aligned} \displaystyle {\mathbb {P}}\left( \lim _{t \rightarrow \infty } X_{s,x_0}(t) = x^* \right) ~=~ 1. \end{aligned}$$

(28)

Theorem 5

(Arnold [3]) Assume that $f$ and $g$ satisfy the existence and uniqueness assumptions and they have continuous coefficients with respect to $t$.

• (i) Suppose that there exist a positive definite function $V~\in ~C^{2,1}\big (U_h \times [t_0,\infty )\big )$, where $U_h=\{x \in {\mathbb {R}}^d:\Vert x-x^*\Vert <h\}$ for $h>0$, such that

  $$\begin{aligned} \displaystyle \mathrm{ for \, all }~t \ge t_0, ~~~ x \in U_h: ~~~ {\mathcal {L}}V(x,t) \le 0. \end{aligned}$$

  (29)

  Then, the equilibrium solution $x^*$ of (25) is stochastically stable.

• (ii) If, in addition, $V$ is decrescent (there exists a positive definite function $V_1$ such that $V(x,t)\le V_1(x)$ for all $x \in U_h$) and ${\mathcal {L}}V(x,t)$ is negative definite, then the equilibrium solution $x^*$ is stochastically asymptotically stable.

• (iii) If the assumptions of part ii) hold for a radially unbounded function $V~\in ~C^{2,1}\big ({\mathbb {R}}^d \times [t_0,\infty )\big )$ defined everywhere then the equilibrium solution $x^*$ is globally stochastically asymptotically stable.