Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

doi:10.1016/j.amc.2011.05.056

Applied Mathematics and Computation

Volume 218, Issue 2, 15 September 2011, Pages 280-286

https://doi.org/10.1016/j.amc.2011.05.056 Get rights and content

Abstract

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.

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 S_k(t), E_k(t), I_k(t) and R_k(t), respectively. The new multigroup epidemic model with group mixing and nonlinear incidence rates as follows: $\{\begin{matrix} S_{k}^{'} = φ_{k} (S_{k}) - \sum_{j = 1}^{n} β_{kj} f_{kj} (S_{k}, I_{j}), \\ E_{k}^{'} = \sum_{j = 1}^{n} β_{kj} f_{kj} (S_{k}, I_{j}) - μ_{k} g_{k} (E_{k}), \\ I_{k}^{'} = γ_{k} g_{k} (E_{k}) - α_{k} ψ_{k} (I_{k}), \\ R_{k}^{'} = p_{k} ψ_{k} (I_{k}) - q_{k} (R_{k}), \end{matrix}$ where φ_k(S_k) denotes the net growth of the susceptible class in the kth group, the nonlinear term β_kjf(S_k, I_j) represents the cross infection from group j to group k; The matrix B = (β_ij)_n×n is the irreducible contact matrix, where β_ij ⩾ 0; γ_kg_k(E_k) accounts for the progression of individuals in group k from the exposed class into the infectious class; μ_kg_k(E_k), α_kψ_k(I_k) and q_k(R_k) 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; p_kψ_k(I_k) denotes the production of the recovered individuals from infectious ones in the kth group. All constants μ_k, γ_k, α_k and p_k 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 R_k do not appear in the first three equations of (1), we can work on the reduced system as follows: $\{\begin{matrix} S_{k}^{'} = φ_{k} (S_{k}) - \sum_{j = 1}^{n} β_{kj} f_{kj} (S_{k}, I_{j}), \\ E_{k}^{'} = \sum_{j = 1}^{n} β_{kj} f_{kj} (S_{k}, I_{j}) - μ_{k} g_{k} (E_{k}), \\ I_{k}^{'} = γ_{k} g_{k} (E_{k}) - α_{k} ψ_{k} (I_{k}) . \end{matrix}$ For the functions g_k, ψ_k and φ_k in (2), we assume that

(G₁)
g_k, ψ_k are local Lipschitz on [0, ∞) with g_k(0) = ψ_k(0) = 0, g_k, ψ_k are continuous, positive, on (0, ∞), the function $\frac{u}{ψ_{k} (u)}$ is non-increasing on (0,∞), and $\lim_{u \to 0^{+}} \frac{u}{ψ_{k} (u)} = δ_{k}$ for positive constant δ_k > 0;
(G₂)
φ_k are local Lipschitz on [0, ∞) with φ_k(0) > 0,

Numerical example

Consider the system (2) when k = 2, one has a two-group model as follows: $\{\begin{matrix} S_{1}^{'} = φ_{1} (S_{1}) - [β_{11} \frac{S_{1} I_{1}}{1 + I_{1}^{2}} + β_{12} \frac{S_{1} I_{2}}{1 + I_{2}^{2}}], \\ E_{1}^{'} = [β_{11} \frac{S_{1} I_{1}}{1 + I_{1}^{2}} + β_{12} \frac{S_{1} I_{2}}{1 + I_{2}^{2}}] - μ_{1} g_{1} (E_{1}), \\ I_{1}^{'} = γ_{1} g_{1} (E_{1}) - α_{1} ψ_{1} (I_{1}), \\ S_{2}^{'} = φ_{2} (S_{2}) - [β_{21} \frac{S_{2} I_{1}}{1 + I_{1}^{2}} + β_{22} \frac{S_{2} I_{2}}{1 + I_{2}^{2}}], \\ E_{2}^{'} = [β_{21} \frac{S_{2} I_{1}}{1 + I_{1}^{2}} + β_{22} \frac{S_{2} I_{2}}{1 + I_{2}^{2}}] - μ_{2} g_{2} (E_{2}), \\ I_{2}^{'} = γ_{2} g_{2} (E_{2}) - α_{2} ψ_{2} (I_{2}), \end{matrix}$ where $\begin{matrix} f_{kj} (S_{k}, I_{j}) = \frac{S_{k} I_{j}}{1 + I_{j}^{2}}, k, j = 1, 2, \\ φ_{1} (u) = 2 - u, φ_{2} (u) = 3 - u, g_{1} (u) = g_{2} (u) = ψ_{1} (u) = ψ_{2} (u) = u \end{matrix}$ and $γ_{1} = \frac{1}{2}, α_{1} = 2, μ_{1} = \frac{1}{4}, γ_{2} = 1, α_{2} = \frac{1}{3}, μ_{2} = 3 .$ If β_ij are chosen as $β_{11} = \frac{5}{24}, β_{12} = \frac{1}{24}, β_{21} = \frac{1}{36}, β_{22} = \frac{5}{36},$ then 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.

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Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

Abstract

Introduction

Section snippets

Main results

Numerical example

Acknowledgement

Math. Biosci.

Math. Biosci.

Math. Biosci.

J. Math. Anal. Appl.

J. Differ. Eqn.

Nonlinear Anal. Real World Appl.

Nonlinear Anal. Real World Appl.

Nonlinear Anal. Real World Appl.

Nonlinear Anal. Real World Appl.

Appl. Math. Comput.