Protection Degree and Migration in the Stochastic SIRS Model: A Queueing System Perspective

TL;DR

The paper addresses modeling epidemic spread under individual protection behavior and population mobility by formulating two Markov-based frameworks: (i) a per-individual SIRS Markov chain augmented with a Poisson-distributed protection degree $U\sim \text{Poisson}(\mu)$ and a contact-reduction factor $f=e^{-U}$, and (ii) an open Markov queueing network for population counts $S(t),I(t),R(t)$ with input $\lambda$, infection rate $\beta$, recovery $\gamma$, immunity loss $\alpha$, and revival $p$. The authors derive stationary expectations $E[S]=\frac{\gamma}{\beta k}$, $E[I]=\frac{\lambda}{\gamma(1-p)}$, $E[R]=\frac{\lambda}{\alpha(1-p)}$, and numerically characterize the limited distributions, validating with simulations and a second-wave Zhengzhou data set. The Zhengzhou case demonstrates practical applicability; Pearson correlation $\rho=0.800$, cosine similarity $0.987$, and CORT $0.656$, indicating strong agreement. Overall, the work provides a quantitative link between individual protection and mobility and epidemic outcomes, with implications for targeted interventions and policy.

Abstract

With the prevalence of COVID-19, the modeling of epidemic propagation and its analyses have played a significant role in controlling epidemics. However, individual behaviors, in particular the self-protection and migration, which have a strong influence on epidemic propagation, were always neglected in previous studies. In this paper, we mainly propose two models from the individual and population perspectives. In the first individual model, we introduce the individual protection degree that effectively suppresses the epidemic level as a stochastic variable to the SIRS model. In the alternative population model, an open Markov queueing network is constructed to investigate the individual number of each epidemic state, and we present an evolving population network via the migration of people. Besides, stochastic methods are applied to analyze both models. In various simulations, the infected probability, the number of individuals in each state and its limited distribution are demonstrated.

Figures (9)

• Figure 1: State transition of SIRS model: The arrows indicate the transitions from one state to another, where the transition probabilities are given. The loop arrow indicates remaining staying in the present state without making a transition of an individual.
• Figure 2: An open Markov queueing network of SIRS model considering a mobile population: Inside the dotted box is the system(an area), where there are three solid line boxes representing service centers(an epidemic state is a service center). Each service center has infinite servers. Arrows indicate the transitions of individuals.
• Figure 3: The infected probabilities with different protection degrees: We take the average value of the infected probability of individuals within the same-degree group. The protection degree with $\mu=1$, $\mu$=2, and without protection degree is respectively denoted by the blue circle, the green square and the red triangle plot. The protection degree lower the probability of being infected, and larger value of $\mu$ leads to smaller infected probability.
• Figure 4: The degree distribution of evolved network with different initial edges of each node: The edges of each node in the initial regular network are set to be 4, 8, 16 and the node joining the network with edges $m=4$. The stationary distributions are identical to each other though the initial edges is different.
• Figure 5: The number of individuals varying with time of the three states with different parameters: The figure respectively shows the influence of each parameter on the individual number. Sub-Fig. (a) is set as $\lambda$=3, $\beta$=0.001, $\gamma$=0.7, $\alpha$=0.8 and $p$=0.995. $\alpha$, $\lambda$, $\beta$, $\gamma$ and $p$ are changed respectively from (b)-(e).

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Theorem 1
• Proof
• Theorem 2
• Proof