Global stability of a network-based SIRS epidemic model with nonmonotone incidence rate

Abstract

This paper studies the dynamics of a network-based SIRS epidemic model with vaccination and a nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger. Using the analytical method, epidemic threshold R 0 is obtained. When R 0 is less than one, we prove the disease-free equilibrium is globally asymptotically stable and the disease dies out, while R 0 is greater than one, there exists a unique endemic equilibrium. By constructing a suitable Lyapunov function, we also prove the endemic equilibrium is globally asymptotically stable if the inhibitory factor α is sufficiently large. Numerical experiments are also given to support the theoretical results. It is shown both theoretically and numerically a larger α can accelerate the extinction of the disease and reduce the level of disease.

Some aspects of the Markovian SIRS epidemic on networks and its mean‐field approximation

“…Then, since ∀ i, r ∈ G h and ∀ m ∈ {1, … , n}, ∑ z∈G m (a iz − a rz ) = 0, we have that (20) is equal to zero. Finally, from (16) and (19), we come to have…”

“…Moreover, we adopt an individual (node)-based approach, see also other studies 9,14 for SIRS-type node-based models. As opposite, a large part of the literature consider models in which the structure of the network is simplified by using a degree-based mean-field (DBMF) approach, [15][16][17][18] where all nodes with the same degree are assumed to be statistically equivalent. Thus, these kinds of models only reflect the evolution in time of the fraction of nodes with a certain degree k in each compartment, while neglecting the states of each single individual.…”

Inverse problem for adaptive SIR model: Application to COVID-19 in Latin America

“…The SIR model is a highly nonlinear dynamical system, particularly so in the case of COVID-19, due to its diverse characteristics. The rates of transmission and removal in the case of the COVID-19 depend on the evolution of the epidemic disease over time, see ( Acedo, Morano, Santonja, & Villanueva, 2016 ; Liu, Wei, & Zhang, 2019 ). Every new epidemics must be studied and mathematical models constructed to answer important questions on handling the disease ( Coronel, Huancas, & Sepúlveda, 2019 ).…”

Global dynamics of a network-based SIQS epidemic model with nonmonotone incidence rate

“…Liu et al. [31] investigated the network-based SIRS infectious disease model with vaccination and nonmonotone incidence rate and demonstrated global stability of endemic equilibrium.…”