Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: A stability analysis

Abstract

In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov-LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single-group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint. KEYWORDSdelay differential equations, disease propagation, disease relapse, Lyapunov stability, multigroup model, nonlinear incidence 6254 ;40:6254-6275. ZHANG ET AL. 6255onset of symptoms, and an individual may be infective prior to the onset of symptoms. Prolonged latency between exposure and infectiousness is a characteristic of tuberculosis, for instance, latent compartments being incorporated even in the earliest models of tuberculosis transmission. 1 Also, several models have included both fast and slow pathways from susceptible to actively infected, with a proportion of exposed susceptibles progressing immediately to active infection or several sequential latent compartments to simulate the increased risk of progression to active disease in the years immediately following initial infection. 2,3 As noted in Lloyd, 4 an accurate description of the distribution of the latent and infectious periods is an important step towards the construction of an appropriate model. Assuming that the latency is exponentially distributed, the corresponding disease propagation model is represented by a system of nonlinear ODEs. This assumption, however, is equivalent to the fact that the chance of recovery within a given time interval is constant, regardless of the time since infection. This is sometimes unrealistic, as observed through the statistical studies of the transmission dynamics of measles in small communities (see, for instance, Lloyd 4 or Krylova and Earn 5 ), the chance of recovery being initially small, but increasing over time and corresponding to a distribution of the infectious period, which is less dispersed than an exponential and more centered around the mean. If, however, a general distribution is assumed for the latent period, then one obtains a delayed integro-differential system. 6,7 The most commonly used distributions of the latent period are the Gamma distribution, used, for instance, to model the spread of avian influenza (H7N7) in chicken 8 and the log-normal distribution, used, for instance, to model the spread of Ebola. 9 Since the host population can be divided geographically into communities, cities, and countries, or epidemiologically, to incorporate factors such as modes of transmission, contact patterns, and genetic susceptibility, a heterogeneous host population can subsequently be divided into several homogeneous groups, the within-group dynamics and the mixing patterns being then modeled separately. One o...

A multi‐group SEIRI epidemic model with logistic population growth under discrete Markov switching: Extinction, persistence, and positive recurrence

“…In a previous work, 16 Zhang et al have formulated the following general deterministic multi-group SEIRI model, in which all individuals into n distinct groups, each representing a homogeneous subpopulation.…”

“…For the stochastic system (2.6), the total population is divided into two distinct groups, that is, n = 2. According to literature, 16,44,45 we choose the initial value as (S 1 (0),…”

“…In a previous work, 16 Zhang et al have formulated the following general deterministic multi‐group SEIRI model, in which all individuals into $$$ n $$$ distinct groups, each representing a homogeneous subpopulation. …”

“…), multi-group epidemics models are more suitable to divide the host population into several homogeneous groups, which have attracted much attention. [11][12][13][14][15][16] For instance, Shu et al 13 investigated multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission. In a previous study, 14 Li et al investigated a class of multi-group epidemic models with distributed delays and established the basic reproduction number R 0 , which determines the global dynamics.…”

“…However, due to heterogeneity (sex, transmission, position, etc. ), multi‐group epidemics models are more suitable to divide the host population into several homogeneous groups, which have attracted much attention 11–16 . For instance, Shu et al 13 investigated multi‐group SEIR epidemic models with distributed and infinite delay and nonlinear transmission.…”