A Natural Stochastic SIS Model, Analysis of Moments and Comparison of Different Perturbation Techniques

TL;DR

This work introduces a natural stochastic SIS model by replacing the transmission rate $\beta$ with a nonnegative diffusion, ensuring the nonnegativity of transmission and enabling rigorous extinction/persistence analysis through an ergodic-mean criterion. It establishes boundedness of the infected population, derives extinction and persistence thresholds via $R_0^S=\mathbb{E}[Y_\infty]/\gamma$, and provides diffusion examples (notably CIR) to illustrate the dynamics. A novel average-analysis framework based on the Feynman-Kac formula and perturbation theory is developed to approximate expectations and moments of functions of the infected population, yielding first-order correction terms that improve over the mean-field approximation. The paper further compares perturbations with the same mean (e.g., Gray’s Brownian perturbation vs. CIR) and demonstrates that they can drive different average dynamics, offering a principled way to choose perturbations in stochastic epidemic modeling. Overall, the approach delivers a practical, analytically tractable toolkit for studying stochastic SIS models with nonnegative diffusion perturbations and for assessing perturbation sensitivity in long-term disease dynamics.

Abstract

In this study, a new and natural way of constructing a stochastic Susceptible-Infected-Susceptible (SIS) model is proposed. This approach is natural in the sense that the disease transmission rate, $β$, is substituted with a generic, almost surely non-negative one-dimensional diffusion. The condition $β\geq 0$ is essential in the deterministic model but generally overlooked in stochastic counterparts (see [12, 16]). Under different conditions on the parameters, the dynamics of the infected population such as boundedness, extinction, and persistence are identified. The new stochastic model agrees with its deterministic version, where the basic reproduction number $R^D_0$ determines the limiting dynamics: extinction when $R_0^D < 1$ and persistence when $R_0^D > 1$. A novel analytic technique is also provided to approximate the expectation of any well-behaved function of the infected population, including its moments, using an increasing power of correction terms. This is useful since the average dynamics of stochastic SIS models are not tractable due to their nonlinearity. Finally, using the first-order correction terms, two different perturbations with the same expectations: (1.4) performed in [12] and the Cox-Ingersoll-Ross (CIR) perturbation proposed here are compared in terms of their expected effect on the infected population dynamics. This comparison provides insight into how different small perturbations affect the overall dynamics of the model.

Figures (5)

• Figure 1: The results of two simulations of the model \ref{['perturbed SIS model']} with parameters: $\beta=y=b=0.89$, $\gamma = 0.92$, $\sigma=0.1$, $a=0.05$ and $x=0.8$. Note that $R^S_0 < 1$, $\mathbb{E}[Y_t]=0.89,\:\forall t>0$, due to remark \ref{['CIR model fixed expectation remark']} and $2ab/\sigma^2 = 8.9>1$ ensuring positivity of the perturbation.
• Figure 2: The figure showing the results of two simulations with parameters: $\beta=y=b=0.5$, $\gamma = 0.4$, $\sigma=0.1$, $a=0.05$ and $x=0.8$. Note that $R^S_0 = 1.5>1$, $\mathbb{E}[Y_t]=0.5,\:\forall t>0$, due to remark \ref{['CIR model fixed expectation remark']} and $2ab/\sigma^2 = 5 >1$ ensuring positivity of the perturbation. The deterministic limit or $\mathtt{I}^*=f^{-1}(-5/4)=0.2$.
• Figure 3: The plot showing the average of simulation sample paths in blue, deterministic or mean-field solution $u^{(1)}_0(t)$ in orange and $u^{(1)}_0(t)$ plus the first correction term ($u^{(1)}_1(t)$) in green. The parameters are (a): $c=0.1$, $x=0.3$, $y=b=0.45$, $\gamma=0.5$, $\sigma=0.063$, $a=0.02$ and (b): $c=0.1$, $x=0.3$, $y=b=0.2$, $\gamma=0.1$, $\sigma=0.032$, $a=0.02$. The number of averaged simulations is $1500$.
• Figure 4: The figure showing the variance of the 1500 simulated sample paths in blue and the first correction term to the variance \ref{['eqn variance correction terms']} in orange. The used parameters are: $c=0.1$, $x=0.3$, $y=b=0.5$, $\gamma=0.3$, $\sigma=0.063$, $a=0.02$, hence the infection persists and the perturbation is always positive.
• Figure 5: The plot showing the deterministic solution $u^{(1)}_0(t,x,y)=g_0(t,x)$ plotted in blue dashed lines, the leading correction term for the Gray perturbation \ref{['gray perturbed SIS model']} on the expectation in green solid line and leading correction term for the CIR perturbation \ref{['CIR process eqn']} in orange solid line. The used parameter setup is the same as the figure \ref{['figure average and correction terms']}, (a): $c=0.1$, $x=0.3$, $y=b=0.45$, $\gamma=0.5$, $\sigma=0.063$, $a=0.02$ and (b): $c=0.1$, $x=0.3$, $y=b=0.2$, $\gamma=0.1$, $\sigma=0.032$, $a=0.02$.

Theorems & Definitions (16)

• Proposition 2.1.1
• proof
• Theorem 2.2.1
• proof
• Remark 2.2.1
• Theorem 2.3.1
• proof
• Remark 2.3.1
• Remark 2.3.2
• Remark 3.2.1