A Compartmental Model for Epidemiology with Human Behavior and Stochastic Effects

TL;DR

This paper develops a deterministic and stochastic SIR framework that incorporates compliant and noncompliant behavioral subpopulations, where NPIs reduce transmission for the compliant group and noncompliance spreads socially. It derives a next-generation-based disease-free reproduction number and analyzes local stability of DFEs in the deterministic model, then extends the system with Gaussian white noise to examine stochastic effects on transmission and behavior. The authors prove global existence and positivity of the stochastic system and establish stochastic Lyapunov-based stability results for disease-free states, complemented by numerical simulations. Key findings show that stochastic perturbations generally strengthen extinction criteria and require more conservative planning than the deterministic model, with simulations illustrating scenarios from stable disease-free states to outbreak events under higher noise. The work lays groundwork for further stochastic analyses in behaviorally informed epidemiological models, including potential extensions to endemic states, spatial dynamics, and stochastic control of NPIs.

Abstract

We propose a compartmental model for epidemiology wherein the population is split into groups with either comply or refuse to comply with protocols designed to slow the spread of a disease. Parallel to the disease spread, we assume that noncompliance with protocols spreads as a social contagion. We begin by deriving the reproductive ratio for a deterministic version of the model, and use this to fully characterize the local stability of disease free equilibrium points. We then append the deterministic model with stochastic effects, specifically assuming that the transmission rate of the disease and the transmission rate of the social contagion are uncertain. We prove global existence and nonnegativity for our stochastic model. Then using suitably constructed stochastic Lyapunov functions, we analyze the behavior of the stochastic system with respect to certain disease free states. We demonstrate all of our results with numerical simulations.

Figures (7)

• Figure 1: A list of variables and parameters for \ref{['eq:SIRwithCompliance']}.
• Figure 2: The flow diagram for \ref{['eq:SIRwithCompliance']}. Arrows leaving a given population denote outward flow proportional to that population with the given rate. Colored arrows connote population transfer which is modified or introduced for our model (i.e., manners in which our model substantively differs from the basic SIR model).
• Figure 3: When the conditions of theorem \ref{['thm:E0expMSS']} (and thus theorem \ref{['eq:stabilityOfDFE']}(i)) are met, we see stability of the compliant disease free state $s = \tfrac{b}{\delta}, s^* = 0$ for both the deterministic and stochastic simulations.
• Figure 4: When the conditions of \ref{['eq:stabilityOfDFE']}(i) are met but the conditions of theorem \ref{['thm:E0expMSS']} are not met, we can only guarantee stability in the deterministic case. Because of this, we often observe significant outbreaks of the disease, as see in the right panel above. Here there is an immediate outbreak of the disease among the noncompliant population, and later an outbreak among the compliant population which is correlated with a random spike in noncompliance.
• Figure 5: In the case of $\xi = 1$ and $\nu = 0$, when the conditions of theorem \ref{['thm:WorstCase']} are met, the disease free equilibrium $s = 0, s^* = \tfrac{b}{\delta}$ is stable for both the stochastic and deterministic systems.

Theorems & Definitions (22)

• Definition 2.1: Stability of $U \subset U_{DF}$
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• Theorem 2.5
• proof
• Remark 2.6
• Theorem 3.1