Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates
Introduction
Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations, such as measles, mumps, gonorrhea, HIV/AIDS, West-Nile virus and vector borne diseases such as Malaria. Heterogeneity in host population can be the result of many factors. Groups can be geographical such as communities, cities, and countries, or epidemiological, to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Much research has been done on multigroup models, for example, see [1], [2], [3], [4], [5], [6] and references therein. It is well known that global dynamics of multigroup models with higher dimensions, especially the global stability of the endemic equilibrium, is a very challenging problem. The question of uniqueness and global stability of the endemic equilibrium, when the basic reproduction number R0 is greater than 1, has largely been open. Recently, the paper [7] proposed a graphtheoretic approach to the method of global Lyapunov functions and used it to establish the global stability of a unique endemic equilibrium of a multi-group SIR model with varying subpopulation sizes. Their result completely solved the open problem of the uniqueness and global stability of endemic equilibrium for this class of multi-group models. By using the results or ideas of [7], the papers [8], [9], [10], [11], [12], [13] investigated uniqueness and global stability of the endemic equilibrium for several class of multigroup models, with the basic reproduction number R0 is greater than 1, and some open problems were resolved.
In this paper, we consider a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. It covers many models in the literature, for example, the ones in [3], [4], [8], [10], [12], [14], [15]. The population is divided into n distinct groups (n ⩾ 1). For 1 ⩽ k ⩽ n, the kth group is further partitioned into four compartments: the susceptible, exposed, infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), Ik(t) and Rk(t), respectively. The new multigroup epidemic model with group mixing and nonlinear incidence rates as follows:where φk(Sk) denotes the net growth of the susceptible class in the kth group, the nonlinear term βkjf(Sk, Ij) represents the cross infection from group j to group k; The matrix B = (βij)n×n is the irreducible contact matrix, where βij ⩾ 0; γkgk(Ek) accounts for the progression of individuals in group k from the exposed class into the infectious class; μkgk(Ek), αkψk(Ik) and qk(Rk) denote the removal of the exposed, infectious and recovered classes in the kth group, respectively, which include the mortality of individuals in the above-mentioned classes; pkψk(Ik) denotes the production of the recovered individuals from infectious ones in the kth group. All constants μk, γk, αk and pk are assumed to be positive.
In Section 2, we first obtain that the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, then the disease free equilibrium is globally asymptotically stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally asymptotically stable and thus the disease persists in the population. And in Section 3, some numerical simulations are showed to illustrate the effectiveness of the proposed result.
Section snippets
Main results
Since the variables Rk do not appear in the first three equations of (1), we can work on the reduced system as follows:For the functions gk, ψk and φk in (2), we assume that
- (G1)
gk, ψk are local Lipschitz on [0, ∞) with gk(0) = ψk(0) = 0, gk, ψk are continuous, positive, on (0, ∞), the function is non-increasing on (0,∞), and for positive constant δk > 0;
- (G2)
φk are local Lipschitz on [0, ∞) with φk(0) > 0,
Numerical example
Consider the system (2) when k = 2, one has a two-group model as follows:whereandIf βij are chosen asthen we have R0 = 0.5 < 1, hence P0 = (2, 0, 0, 3, 0, 0) is the unique
Acknowledgement
This project is partially supported by a William and Mary Charles Center Biomathematics Summer Scholarship, NSF Grants DMS-1022648, DMS-0703532, EF-0436318 and William and Mary HHMI travel award.
References (20)
- et al.
A deterministic model for gonorrhea in a nonhomogeneous population
Math. Biosci.
(1976) - et al.
Global dynamics of a SEIR model with varying total population size
Math. Biosci.
(1999) - et al.
Global analysis of an epidemic model with nonmonotone incidence rate
Math. Biosci.
(2007) - et al.
Global stability of multi-group epidemic models with distributed delays
J. Math. Anal. Appl.
(2010) - et al.
Global-stability problem for coupled systems of differential equations on networks
J. Differ. Eqn.
(2010) Complete global stability for an SIR epidemic model with delay – distributed or discrete
Nonlinear Anal. Real World Appl.
(2010)- et al.
Global stability of epidemiological models with group mixing and nonlinear incidence rates
Nonlinear Anal. Real World Appl.
(2010) - et al.
Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population
Nonlinear Anal. Real World Appl.
(2010) - et al.
A Lyapunov functional for a stage structured predator–prey model with nonlinear predation rate
Nonlinear Anal. Real World Appl.
(2010) - et al.
The effects of pulse vaccination on SEIR model with two time delays
Appl. Math. Comput.
(2008)
Cited by (75)
Study of SEIR epidemic model and scenario analysis of COVID-19 pandemic
2021, Ecological Genetics and GenomicsCitation Excerpt :In epidemiology there are different models to predict and explain the dynamics of an epidemic. The data of the Covid-19 outbreak can also be studied through various mathematical models such as SIR, SEIR (Susceptible, Exposed, Infected and Recovery), SIQR (Susceptible, Infectious, Quarantined and Recovered) and so on [7–12]. Tang, Wang, Li and Bragazzi [13] presented a compartmental deterministic model that would integrate the clinical development of the disease, the epidemiological status of the patient and the measures for intervention.
A multi-group SEIRA model for the spread of COVID-19 among heterogeneous populations
2020, Chaos, Solitons and FractalsCitation Excerpt :Barriers can be risen by the geo-spatial configuration of a country and differential connectivity between its regions, or even by the way the different communities or social classes interact in a city. Even though work has been done in multi-group SIR models [11] and the stability of their endemic equilibrium [10,21], the particularities of the COVID-19 pandemics and human interaction networks require to be modeled to be correctly represented. In this work, we present a general multi-group SEIRA model for representing the spread of novel COVID-19 through populations with heterogeneous characteristics, such as a heavily centralized organization with poor connections between the provinces, substantial social inequality among the population, or age/behavioral groups.
Optimal control strategies for a two-group epidemic model with vaccination-resource constraints
2020, Applied Mathematics and ComputationInput-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay
2019, Applied Mathematics and ComputationRazumikhin method conjoined with graph theory to input-to-state stability of coupled retarded systems on networks
2017, NeurocomputingCitation Excerpt :As we know, the stability of CSNs is the premise of the applications of CSNs in practice. Consequently, it is significant to study the stability of CSNs and there exist a great number of literatures on this topic, see [8–10] and the references therein. In addition, in the implementation of practical coupled systems, time delays are inevitably encountered and they can turn a stable system into an unstable one [11].