Global Stability Analysis of SEIR Model with Holling Type II Incidence Function

Abstract

A deterministic model for the transmission dynamics of a communicable disease is developed and rigorously analysed. The model, consisting of five mutually exclusive compartments representing the human dynamics, has a globally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number (ℛ₀), is less than unity; in such a case the endemic equilibrium does not exist. On the other hand, when the reproduction number is greater than unity, it is shown, using nonlinear Lyapunov function of Goh-Volterra type, in conjunction with the LaSalle's invariance principle, that the unique endemic equilibrium of the model is globally asymptotically stable under certain conditions. Furthermore, the disease is shown to be uniformly persistent whenever ℛ₀>1.

1. Introduction

Mathematical models have been widely used to gain insight into the spread and control of emerging and reemerging disease. The dynamics of these models is usually determined by a threshold quantity known as the basic reproduction number (denoted by ℛ₀), which is defined as the number of secondary cases generated by an infected individual in a completely susceptible population [1–5]. Characteristically, when ℛ₀ is less than unity, a small influx of infected individuals will not generate large outbreaks, and the disease dies out in time. On the other hand, when ℛ₀ exceeds unity, the disease will persist. A basic epidemic model supports at least two equilibria (a disease-free equilibria and endemic equilibria); ℛ₀ plays important role in the study of equilibria of a model. Several models found in the literature [2, 4–15] have been used to show that when ℛ₀ crosses the threshold, ℛ₀=1, a transcritical bifurcation takes place. That is, asymptotic local stability is transferred from the disease-free state to the new (emerging) endemic (positive) equilibria. In some cases, it can be shown that the transfer of asymptotic stability is independent of initial conditions; that is, it is global; see, for instance, [6, 8–12, 16]. Establishing global properties of a dynamical system using Lyapunov function is generally a nontrivial problem. This is owing to the fact that there are no systematic methods for constructing Lyapunov function for infectious disease models with standard incidence rate [17]. The most successful approach to the problem is the direct Lyapunov method which involves the use of quadratic function of the form 𝜔(𝑥₁,𝑥₂,…,𝑥_𝑛)=∑^𝑛_𝑖=1(𝑐_𝑖/2)(𝑥_𝑖−𝑥^∗_𝑖) or by using nonlinear Lyapunov function of Goh-Volterra type of the form 𝜔(𝑥₁,𝑥₂,…,𝑥_𝑛)=∑^𝑛_𝑖=1(𝑐_𝑖/2)(𝑥_𝑖−𝑥^∗_𝑖−𝑥^∗_𝑖ln(𝑥_𝑖/𝑥^∗_𝑖)). However, other methods used in establishing global properties of some epidemic models include Dulac’s criterion to eliminate the existence of the periodic solution and prove the global stability by the Poincaré Bendixson theorem [18] and those reported in Kamgang and Sallet [19] and Qiao et al. [20].

Let 𝑆(𝑡),𝐼(𝑡), and 𝑁(𝑡) denote the number of susceptible individuals, infectious individuals, and the total size of the population at time 𝑡, respectively. Further, let 𝛽(𝑁) be the average number of contacts that is sufficient to transmit infection (effective contact rate). Then, the force of infection, given by 𝛽(𝑁)𝐼/𝑁, represents the average number of contacts a susceptible individual makes with infectious individuals per unit time. If 𝛽(𝑁)=𝛽𝑁 (i.e., the contact rate depends on the total population, 𝑁), then the incidence function 𝑔₁(𝐼)=𝛽𝐼 is called mass action incidence. If 𝛽(𝑁)=𝛽 (a constant), then the incidence function 𝑔₂(𝐼)=𝛽𝐼/𝑁 is called standard incidence [4]. These two functions are widely used in modeling the transmission dynamics of the human diseases [1, 21]. Another widely used incidence function is the Holling type II incidence function, given by 𝑔₃(𝐼)=𝛽𝐼/(1+𝜔𝐼), with 𝜔>0, [22–28].

The nonlinear incidence function of type 𝑔₃(𝐼) was first introduced by Capasso and Serio [22], in their study of cholera epidemic. The main justification for using such a functional form of the incidence function stems from the fact that the number of effective contacts between infective individuals and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the preventive measures (and behavioral changes) taken by the susceptible individuals in response to the severity of the disease [23, 25–27].

A number of mathematical models have been developed in the literature to gain insights into the transmission dynamics of diseases with subpopulation (compartments) [4, 6–14, 29–31]. The choice of which compartment to include in a model depends on the characteristic of the particular disease being modelled and the purpose of the model [4].

The classical SIR or SIRS model assumed that the disease incubation is negligible so that each susceptible individual (in the 𝑆 class) once infected becomes infectious (and move to 𝐼 class) and later recovers (to move to 𝑅 class) where they acquire permanent or temporary immunity [32]. However, more general models than SIR or SIRS models assumed that susceptible individuals, once infected, first go through the latent period (in the 𝐸 class) before becoming infectious; the resulting models are of SEIR or SEIRS type, depending on whether recovered individuals acquired permanent or temporary immunity [33, 34].

The model considered in this study is based on SEIR (𝑆: susceptible, 𝐸: exposed, 𝐼: infected, 𝑅: recovered) where recovered individuals acquire permanent immunity, so that they will not become infected again. This is owing to the fact that for many viral diseases such as measles, smallpox, rubella, HIV/AIDS, and influenza recovered individuals confer lifelong immunity [29, 33]. Huang and Takeuchi [29] study the classical SIR, SIS, SEIR, and SEI models with time delay and a general incidence function. Safi et. al. [13] consider the effect of periodic fluctuations on the transmission dynamics of a communicable disease using SEIRS model, subject to quarantine and isolation; the authors show that adding periodicity to the autonomous model does not alter the threshold dynamics of the model with respect to the control of the disease in the population. Zhang and Ma [35] considered the global dynamics of an SEIR model with saturating contact rate. Korobeinikov [36] established global asymptotic dynamics of SEIR and SIR models with several parallel infectious stages. Li et al. [33] analyzed the global dynamics of an SEIR model with vertical transmission and bilinear incidence. Li and Jin [34] consider global stability of an SEIR epidemic model with infectious force in latent, infected, and immune period. It should be stated, however, that the aforementioned three studies [33, 34, 36] considered mass action (bilinear) incidence to model the infection. This study focuses on the mathematical modeling of the transmission dynamics of an arbitrary disease with educated (counsel) and uneducated infectious stages.

This study has three important differences from those reported in [33, 34, 36]. The first is established essential qualitative features of an SEIR model with Holling type II incidence function. The second is that a public health education and counselling are offered to infected individuals (𝐼_𝑢). Recent studies further reinforced the widely held belief that one key strategy for preventing and controlling the spread of communicable disease such as HIV (especially in resource-poor nations) is to provide HIV-related public health education and counselling (such as sexual education and awareness of the risk and life-threatening consequences of HIV/AIDS) which would, hopefully, lead to reduction in risky sexual behavior and safer lifestyle within the community [7]. In addition to the above extensions, rigorous qualitative analysis will be provided for the resulting SEIR model. In particular, the paper gives special emphasis on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium.

The paper is organized as follows. The model is formulated and analysed for its basic dynamical features in Section 2. The stability analysis of the model is presented in Sections 3 and 4.

2. Model Formulation

The total population at time 𝑡, denoted by 𝑁(𝑡), is subdivided into five compartments of susceptible (𝑆(𝑡)), exposed (those who have been infected but are not yet infectious) (𝐸(𝑡)), uneducated infected individuals (𝐼_𝑢(𝑡)), educated infected individuals (𝐼_𝑒(𝑡)), and recovered (𝑅(𝑡)) individuals, so that 𝑁(𝑡)=𝑆(𝑡)+𝐸(𝑡)+𝐼_𝑢(𝑡)+𝐼_𝑒(𝑡)+𝑅(𝑡). (1)

The susceptible population is increased by the recruitment of individuals into the population (assumed susceptible), at a rate Π. Susceptible individuals may acquire infection, following effective contact with infected individuals (in the 𝐼_𝑢 or 𝐼_𝑒 class) at a rate 𝜆(𝑡), where 𝜆(𝑡)𝐼=𝛽^𝑢1+𝛼₁𝐼_𝑢+𝜂𝐼_𝑒1+𝛼₂𝐼_𝑒. (2)

In (2), 𝛽 is the effective contact rate (contact capable of leading to infection), while the modification parameter 0<𝜂<1 accounts for the assumed reduction in disease transmission by educated infected individuals in comparison to uneducated infected individuals in the 𝐼_𝑢 class. The population of susceptible individuals is further decreased by natural death (at a rate 𝜇). Thus, the rate of change of the susceptible population is given by 𝑑𝑆(𝑡𝑆(𝑡).𝑑𝑡=Π−𝜇+𝜆) (3)

The population of exposed individuals is generated by the infection of susceptible individuals (at the rate 𝜆(𝑡)). This population is decreased by development of disease symptoms (at a rate 𝜅) and natural death (at a rate 𝜇), so that 𝑑𝐸(𝑡)𝑆(𝑡)−𝐸(𝑡).𝑑𝑡=𝜆𝜅+𝜇 (4)

The population of uneducated infected individuals is generated at the rate 𝜅. It is decreased by natural recovery (at a rate 𝛾₁), education (at a rate 𝜎), natural death (at the rate 𝜇), and disease-induced death (at a rate 𝛿₁). This gives 𝑑𝐼_𝑢(𝑡)−𝑑𝑡=𝜅𝐸𝜎+𝛾₁+𝜇+𝛿₁𝐼_𝑢(𝑡). (5)

The population of educated infected individuals is generated by the education of infected individuals (at the rate 𝜎). This population is decreased by recovery (at a rate 𝛾₂), natural death (at the rate 𝜇), and disease-induced death (at a rate 𝛿₂<𝛿₁). It is assumed that the disease-induced mortality rate of educated infected individuals is low in comparison with uneducated infected individuals. Hence, the rate of change of this population is given by 𝑑𝐼_𝑒𝑑𝑡=𝜎𝐼_𝑢(𝑡)−𝛾₂+𝜇+𝛿₂𝐼_𝑒(𝑡). (6)

Finally, the population of recovered individuals is generated by the recovery of uneducated and educated infected individuals (at the rates 𝛾₁ and 𝛾₂, resp.). It is decreased by natural death (at the rate 𝜇), so that 𝑑𝑅𝑑𝑙𝑡=𝛾₁𝐼_𝑢(𝑡)+𝛾₂𝐼_𝑒(𝑡)−𝜇𝑅(𝑡). (7)

Thus, the model for the transmission dynamics of an infectious disease in the presence of educated (counsel) infected individuals is given by the following nonlinear system of differential equations: 𝑑𝑆(𝑡𝑆(𝑡),𝑑𝑡=Π−𝜇+𝜆)𝑑𝐸(𝑡)𝑆(𝑡)−𝐸(𝑡),𝑑𝑡=𝜆𝜇+𝜅𝑑𝐼_𝑢(𝑡)−𝑑𝑡=𝜅𝐸𝜇+𝜎+𝛾₁+𝛿₁𝐼_𝑢(𝑡),𝑑𝐼_𝑒𝑑𝑡=𝜎𝐼_𝑢(𝑡)−𝜇+𝛾₂+𝛿₂𝐼_𝑒(𝑡),𝑑𝑅𝑑𝑙𝑡=𝛾₁𝐼_𝑢(𝑡)+𝛾₂𝐼_𝑒(𝑡)−𝜇𝑅(𝑡). (8) The model (8) extends some SEIR models in the literature such as those in [33, 34, 36] by (i)replacing the mass action incidence function with Holling type II incidence function, (ii)splitting the compartment of infected individuals into educated (counsel) and uneducated (noncounsel) infected individuals thereby allowing time-varying infection rate. The epidemiological implication of this assumption is that educated infected individuals transmit the disease at a reduced rate (0<𝜂<1) in comparison to the uneducated infected individuals (due to behavioral changes). For instance, In Zambia, the decline in HIV incidence since early 1990s is attributed to behavioral changes [37]. Public health education campaigns have also been successfully implemented in numerous countries and communities, such as Uganda, Thailand, Zambia, and the US gay community [38, 39]. Between 1991 and 1998, HIV prevalence dramatically declined in Uganda from 21 to 9.8% (with a corresponding reduction in nonregular sexual partners by 65% coupled with greater levels of awareness about HIV/AIDS [38]. The Ugandan programme fostered community mobilization towards change in risky behavior, without increasing stigma [40]. Further, unlike in the aforementioned modelling studies, detailed rigorous mathematical analysis of the model (8) will be provided.

2.1. Basic Properties

Since the model (8) monitors human populations, all its associated parameters are nonnegative. Further, the following nonnegativity result holds.

Theorem 1. The variables of the model (8) are nonnegative for all the time. In other words, solutions of the model system (8) with positive initial data will remain positive for all time 𝑡>0.

Proof. Let 𝑡₁=sup{𝑡>0∶𝑆>0,𝐸>0,𝐼_𝑢>0,𝐼_𝑒>0,𝑅>0∈[0,𝑡]}. Thus, 𝑡₁>0. It follows from the first equation of the system (8) that 𝑑𝑆(𝑡)𝑆(𝑡)(𝑡)𝑆(𝑡),𝑑𝑡=Π−𝜆−𝜇𝑆≥Π−𝜆+𝜇 (9) which can be rewritten as 𝑑𝑆(𝑡)𝑑𝑡exp𝜇𝑡+^𝑡₀𝜆(𝜏)𝑑𝜏≥Πexp𝜇𝑡+^𝑡₀𝜆(𝜏)𝑑𝜏. (10) Hence, 𝑆𝑡₁exp𝜇𝑡₁+^𝑡₀𝜆(𝜏)(0)≥𝑑𝜏−𝑆_𝑡10Πexp𝜇𝑦+^𝑦₀𝜆(𝜏)𝑑𝜏𝑑𝑦, (11) so that 𝑆(𝑡)≥𝑆(0)exp−𝜇𝑡₁−^𝑡1₀𝜆(𝜏)×𝑑𝜏exp−𝜇𝑡₁−^𝑡1₀𝜆(𝜏)×𝑑𝜏_𝑡10Πexp𝜇𝑦+^𝑦₀𝜆(𝜏)𝑑𝜏𝑑𝑦>0. (12) Similarly, it can be shown that 𝐸>0, 𝐼_𝑢>0, 𝐼_𝑒>0 and 𝑅>0, for all time 𝑡>0.

The previous result can also be established using the method in of [41, Appendix A].

We claim the following result.

Lemma 2. The closed set 𝒟=𝑆,𝐸,𝐼_𝑢,𝐼_𝑒,𝑅∈ℝ⁵₊∶𝑆+𝐸+𝐼_𝑢+𝐼_𝑒Π+𝑅≤𝜇 (13) is positively invariant.

Proof. Adding all the equations of the model (8) gives, 𝑑𝑁𝛿𝑑𝑡=Π−𝜇𝑁−₁𝐼_𝑢+𝛿₂𝐼_𝑒. (14) Since 𝑑𝑁/𝑑𝑡≤Π−𝜇𝑁, it follows that 𝑑𝑁/𝑑𝑡≤0 if 𝑁≥Π/𝜇. Thus, a standard comparison theorem [42] can be used to show that 𝑁≤𝑁(0)𝑒^−𝜇𝑡+(Π/𝜇)(1−𝑒^−𝜇𝑡). In particular, 𝑁(𝑡)≤Π/𝜇 if 𝑁(0)≤Π/𝜇. Thus, the region 𝒟 is positively invariant. Further, if 𝑁(0)>Π/𝜇, then either the solution enters 𝒟 infinite time or 𝑁(𝑡) approaches Π/𝜇 asymptotically. Hence, the region 𝒟 attracts all solutions in ℝ⁵₊.

Since the region 𝒟 is positively invariant, it is sufficient to consider the dynamics of the flow generated by the model (8) in 𝒟, where the usual existence, uniqueness, continuation results hold for the system [10].

3. Local Stability of Disease-Free Equilibrium (DFE)

The DFE of the model (8) is given by ℰ₀=𝑆^∗,𝐸^∗,𝐼^∗_𝑢,𝐼^∗_𝑒,𝑅^∗=Π𝜇,0,0,0,0. (15) The local stability of ℰ₀ will be explored using the next generation operator method [43, 44]. Using the notation in [44], the nonnegative matrix, 𝐹, of the new infection terms and the 𝑀-matrix, 𝑉, of the transition terms associated with the model (8) are given, respectively, by 0𝐹=𝛽Π𝜇𝜂𝛽Π𝜇000,000𝑉=𝜇+𝜅00−𝜅𝜇+𝜎+𝛾₁+𝛿₁00−𝜎𝜇+𝛾₂+𝛿₂. (16) It follows that the control reproduction number [4, 21], denoted by ℛ₀=𝜌(𝐹𝑉⁻¹), where 𝜌 is the spectral radius, is given by ℛ₀=𝛽Π𝜅𝑘₃+𝜂𝜎𝜇𝑘₁𝑘₂𝑘₃, (17) where 𝑘₁=𝜇+𝜅,𝑘₂=𝜇+𝜎+𝛾₁+𝛿₁,𝑘₃=𝜇+𝛾₂+𝛿₂. (18) Using in [44, Theorem 2], the following result is established.

Lemma 3. The DFE of the model (8), given by (15), is locally asymptotically stable (LAS) if ℛ₀<1 and unstable if ℛ₀>1.

The quantity ℛ₀ measures the average number of new infections generated by a single infected individual in a population. Lemma 3 implies that the disease can be eliminated from the community (when ℛ₀<1) if the initial sizes of the subpopulations of the model are in the basin of attraction of the DFE (ℰ₀). To ensure that disease elimination is independent of the initial sizes of sub-populations, it is necessary to show that the DFE is globally asymptotically stable (GAS) if ℛ₀<1. This is explored below.

3.1. Global Stability of DFE

Theorem 4. The DFE of the model (8), given by (15), is GAS in 𝒟 whenever ℛ₀≤1.

Proof. Consider the following Lyapunov function: 𝜅𝑘ℱ=³+𝜂𝜎𝜂𝑘₁𝑘₂𝑘𝐸+³+𝜂𝜎𝜂𝑘₂𝐼_𝑢+𝐼_𝑒, (19) with Lyapunov derivative (where a dot represents differentiation with respect to time) given by ℱ̇=𝜅𝑘₃+𝜂𝜎𝜂𝑘₁𝑘₂𝐸̇+𝑘₃+𝜂𝜎𝜂𝑘₂𝐼̇_𝑢+𝐼̇_𝑒=𝜅𝑘₃+𝜂𝜎𝜂𝑘₁𝑘₂𝐼𝛽𝑆^𝑢1+𝛼₁𝐼_𝑢+𝜂𝐼_𝑒1+𝛼₂𝐼_𝑒−𝑘₁𝐸+𝑘₃+𝜂𝜎𝜂𝑘₂𝜅𝐸−𝑘₂𝐼_𝑢+𝜎𝐼_𝑢−𝑘₃𝐼_𝑒≤𝜅𝑘₃+𝜂𝜎𝜂𝑘₁𝑘₂𝛽Π𝜇𝐼_𝑢+𝜂𝐼_𝑒−𝑘₁𝐸+𝑘₃+𝜂𝜎𝜂𝑘₂𝜅𝐸−𝑘₂𝐼_𝑢+𝜎𝐼_𝑢−𝑘₃𝐼_𝑒=𝛽Π𝜅𝑘₃+𝜂𝜎𝜇𝜂𝑘₁𝑘₂𝐼_𝑢+𝜂𝐼_𝑒+𝑘𝜎−³+𝜂𝜎𝜂𝐼_𝑢−𝑘₃𝐼_𝑒=𝑘₃𝜂ℛ₀−1𝐼_𝑢+𝜂𝐼_𝑒. (20) Since all the parameters and variables of the model (8) are nonnegative (Theorem 1), it follows that ℱ̇≤0 for ℛ₀≤1 with ℱ̇=0 if and only if 𝐸=𝐼_𝑢=𝐼_𝑒=0. Hence, ℱ is a Lyapunov function on 𝒟. Therefore, the largest compact invariant subset of the set where ℱ̇=0 is the singleton {(𝐸,𝐼_𝑢,𝐼_𝑒)=(0,0,0)}. Thus, it follows, by the LaSalle's invariance principle [18, 45], that 𝐸,𝐼_𝑢,𝐼_𝑒⟶(0,0,0)as𝑡⟶∞. (21)
Since limsup_𝑡→∞𝐼_𝑢=0 and limsup_𝑡→∞𝐼_𝑒=0 (from (21)), it follows that, for sufficiently small 𝜛^∗>0, there exist constants 𝑀₁>0 and 𝑀₂>0 such that limsup_𝑡→∞𝐼_𝑢≤𝜛^∗ for all 𝑡>𝑀₁ and limsup_𝑡→∞𝐼_𝑒≤𝜛^∗ for all 𝑡>𝑀₂. Hence, it follows from the last equation of the model (8) that, for 𝑡>max{𝑀₁,𝑀₂}, 𝑅̇≤𝛾₁𝜛^∗+𝛾₂𝜛^∗−𝜇𝑅. (22) Thus, by comparison theorem [42], 𝑅^∞=limsup_𝑡→∞𝛾𝑅≤¹𝜛^∗+𝛾₂𝜛^∗𝜇, (23) so that, by letting 𝜛^∗→0, 𝑅^∞=limsup_𝑡→∞𝑅≤0. (24) Similarly (by using liminf_𝑡→∞𝐼_𝑢=0 and liminf_𝑡→∞𝐼_𝑒=0), it can be shown that 𝑅_∞=liminf_𝑡→∞𝑅≥0. (25) Thus, it follows from (24) and (25) that 𝑅_∞≥0≥𝑅^∞. (26) Hence, lim_𝑡→∞𝑅=0. (27) Similarly, it can be shown that lim_𝑡→∞𝑆(𝑡)=Π𝜇. (28) Thus, by combining (21), (27), and (28), it follows that every solution of the equations of the model (8), with initial conditions in 𝒟, approaches ℰ₀ as 𝑡→∞ (for ℛ₀<1).

The previous result shows that the disease can be eliminated from the community if the associated reproduction number of the model is less than unity.

4. Existence and Stability for Endemic Equilibrium Point

In this section, the possible existence and stability of endemic (positive) equilibria of the model (8) (i.e., equilibria where at least one of the infected components of the model is nonzero) will be explored.

4.1. Existence of Endemic Equilibrium Point (EEP)

First of all, the persistence of the disease in the population will be investigated below. The model system (8) is said to be uniformly persistent if there exists a constant 𝑐 such that any solution (𝑆(𝑡),𝐸(𝑡),𝐼_𝑢(𝑡),𝐼_𝑒(𝑡),𝑅(𝑡)) satisfies ([31, 46]) liminf_𝑡→∞𝑆(𝑡)≥𝑐,liminf_𝑡→∞𝐸(𝑡)≥𝑐,liminf_𝑡→∞𝐼_𝑢(𝑡)≥𝑐,liminf_𝑡→∞𝐼_𝑒(𝑡)≥𝑐,liminf_𝑡→∞𝑅(𝑡)≥𝑐, (29) provided that (𝑆(0),𝐸(0),𝐼_𝑢(0),𝐼_𝑒(0),𝑅(0))∈𝒟^∘ (the interior of the region 𝒟).

Theorem 5. System (8) is uniformly persistent in 𝒟^∘ if and only if ℛ₀>1.

Proof. The theorem can be proved by using the approach used to prove Proposition 3.3 of [32], by applying a uniform persistence result in [46] and noting that the DFE of the model (8) is unstable whenever ℛ₀>1 (Lemma 3).

When ℛ₀>1, it follows (from Theorem (3)) that model (8) is uniformly persistent; by using in [47, Theorem 2.8.6] and in [42, Theorem D.3] it follows that the model (8) has at least one endemic equilibrium in 𝒟^∘. Thus the following result is established.

Lemma 6. The model (2) has at least one endemic equilibrium, given by ℰ₁=(𝑆^∗∗,𝐸^∗∗,𝐼_𝑢^∗∗,𝐼_𝑒^∗∗,𝑅^∗∗), whenever ℛ₀>1.

The uniqueness of the endemic equilibrium will be investigated at the end of the next subsection.

4.2. Global Stability of Endemic Equilibrium for Special Case

Here, the global asymptotic stability property of the endemic equilibrium of the model (8) is given for a special case when educated (counsel) infected individuals do not transmit infection (𝜂=0). The model (8), with 𝜂=0, then reduces to: 𝑑𝑆𝐼𝑑𝑡=Π−𝜇+𝛽𝑓_𝑢𝑆(𝑡),𝑑𝐸𝐼𝑑𝑡=𝛽𝑓_𝑢𝑆(𝑡)−𝜇+𝜅𝐸(𝑡),𝑑𝐼_𝑢(𝑡)−𝑑𝑡=𝜅𝐸𝜇+𝜎+𝛾₁+𝛿₁𝐼_𝑢(𝑡),𝑑𝐼_𝑒𝑑𝑡=𝜎𝐼_𝑢(𝑡)−𝜇+𝛾₂+𝛿₂𝐼_𝑒(𝑡)𝑑𝑅𝑑𝑡=𝛾₁𝐼_𝑢(𝑡)+𝛾₂𝐼_𝑒(𝑡)−𝜇𝑅(𝑡), (30) where 𝑓𝐼_𝑢=𝐼_𝑢1+𝛼₁𝐼_𝑢. (31) The reproduction number of the model (30) is given by ℛ_0𝑟=ℛ₀|_𝜂=0=𝛽Π𝜅𝜇𝑘₁𝑘₂. (32) We claim the following result.

Theorem 7. The endemic equilibrium of the reduced model, given by (30), is GAS in 𝒟^∘ if ℛ_0𝑟>1. Thus it is unique.

Proof. Consider the reduced model, given by (30). Let ℛ_0𝑟>1, so that the associated endemic equilibrium exists. Further, consider the following nonlinear Lyapunov function: ℱ=𝑆−𝑆^∗∗−𝑆^∗∗𝑆ln𝑆^∗∗+𝐸−𝐸^∗∗−𝐸^∗∗𝐸ln𝐸^∗∗+𝑘₁𝜅𝐼_𝑢−𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐼ln^𝑢𝐼_𝑢^∗∗ (33) with Lyapunov derivative ℱ̇=𝑆̇−𝑆^∗∗𝑆𝑆̇+𝐸̇−𝐸^∗∗𝐸𝐸̇+𝑘₁𝜅𝐼̇_𝑢−𝐼_𝑢^∗∗𝐼_𝑢𝐼̇_𝑢,=Π−𝛽𝑆𝑓𝐼_𝑢𝑆−𝜇𝑆−^∗∗𝑆Π−𝛽𝑆𝑓𝐼_𝑢𝐼−𝜇𝑆+𝛽𝑆𝑓_𝑢−𝑘₁𝐸𝐸−^∗∗𝐸𝛽𝑆𝑓𝐼_𝑢−𝑘₁𝐸+𝑘₁𝜅𝜅𝐸−𝑘₂𝐼_𝑢−𝐼_𝑢^∗∗𝐼_𝑢𝜅𝐸−𝑘₂𝐼_𝑢𝑆=Π1−^∗∗𝑆𝑆−𝜇𝑆1−^∗∗𝑆+𝛽𝑆^∗∗𝑓𝐼_𝑢−𝑘₁𝑘₂𝜅𝐼_𝑢−𝐸^∗∗𝛽𝑆𝑓𝐼_𝑢𝐸+𝑘₁𝐸^∗∗−𝑘₁𝐼_𝑢^∗∗𝐸𝐼_𝑢+𝑘₁𝑘₂𝐼_𝑢^∗∗𝜅. (34) It can be shown from (30) that, at endemic steady state, Π=𝜇+𝛽𝑓𝐼_𝑢^∗∗𝑆^∗∗,𝑘₁=𝛽𝑓𝐼_𝑢^∗∗𝑆^∗∗𝐸^∗∗,𝑘₂=𝜅𝐸^∗∗𝐼_𝑢^∗∗. (35) Using the relations (35) in (34) gives ℱ̇=𝜇𝑆^∗∗2−𝑆^∗∗𝑆−𝑆𝑆^∗∗−𝑆^∗∗2𝑓𝐼_𝑢^∗∗𝑆+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢−𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗𝐼_𝑢𝐼_𝑢^∗∗−𝐸^∗∗𝛽𝑆𝑓𝐼_𝑢𝐸+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝛽𝑆^∗∗𝐼_𝑢^∗∗𝑓𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗. (36) Adding and subtracting 𝛽𝑆^∗∗𝑓(𝐼_𝑢^∗∗) and 𝛽𝑆^∗∗𝑓²(𝐼_𝑢^∗∗)𝐼_𝑢/𝑓(𝐼_𝑢)𝐼_𝑢^∗∗ in (36) gives ℱ̇=𝜇𝑆^∗∗2−𝑆^∗∗𝑆−𝑆𝑆^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗×𝑆4−^∗∗𝑆−𝐸^∗∗𝑆𝑓𝐼_𝑢𝐸𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗−𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢−𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗𝐼_𝑢𝐼_𝑢^∗∗−𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗+𝛽𝑆^∗∗𝑓²𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗. (37) Rewriting the last four terms of (37) in terms of Π−𝜇𝑆^∗∗ (by using the relations in (35)) gives ℱ̇=𝜇𝑆^∗∗2−𝑆^∗∗𝑆−𝑆𝑆^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗×𝑆4−^∗∗𝑆−𝐸^∗∗𝑆𝑓𝐼_𝑢𝐸𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗−𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗+Π−𝜇𝑆^∗∗𝑓𝐼_𝑢𝑓𝐼_𝑢^∗∗−Π−𝜇𝑆^∗∗𝐼_𝑢𝐼_𝑢^∗∗−Π−𝜇𝑆^∗∗+Π−𝜇𝑆^∗∗𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗=𝜇𝑆^∗∗𝑆2−^∗∗𝑆−𝑆𝑆^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗×𝑆4−^∗∗𝑆−𝐸^∗∗𝑆𝑓𝐼_𝑢𝐸𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗−𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗+Π−𝜇𝑆^∗∗𝑓𝐼_𝑢𝑓𝐼_𝑢^∗∗−𝐼_𝑢𝐼_𝑢^∗∗𝑓𝐼−1+^𝑢∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗=𝜇𝑆^∗∗𝑆2−^∗∗𝑆−𝑆𝑆^∗∗+𝛽𝑆^∗∗𝑓𝐼_𝑢^∗∗×𝑆4−^∗∗𝑆−𝐸^∗∗𝑆𝑓𝐼_𝑢𝐸𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗−𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗+Π−𝜇𝑆^∗∗𝐼_𝑢𝑓𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢−𝑓𝐼_𝑢^∗∗𝐼_𝑢^∗∗×𝑓𝐼_𝑢𝑓𝐼_𝑢^∗∗−1. (38) Since 𝑓(𝐼_𝑢) is an increasing function and 𝑆^∗∗≤Π/𝜇, it follows that Π−𝜇𝑆^∗∗𝐼_𝑢𝑓𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢−𝑓𝐼_𝑢^∗∗𝐼_𝑢^∗∗𝑓𝐼_𝑢𝑓𝐼_𝑢^∗∗−1≤0, (39) finally, since the arithmetic mean exceeds the geometric mean, then 𝑆2−^∗∗𝑆−𝑆𝑆^∗∗𝑆≤0,4−^∗∗𝑆−𝐸^∗∗𝑆𝑓𝐼_𝑢𝐸𝑆^∗∗𝑓𝐼_𝑢^∗∗−𝐼_𝑢^∗∗𝐸𝐼_𝑢𝐸^∗∗−𝑓𝐼_𝑢^∗∗𝐼_𝑢𝑓𝐼_𝑢𝐼_𝑢^∗∗≤0. (40)(41)

Further, since all the model parameters are nonnegative, it follows that ℱ̇≤0 for ℛ_0𝑟>1 with ℱ̇=0 if and only if 𝑆=𝑆^∗∗, 𝐸=𝐸^∗∗,𝐼_𝑢=𝐼_𝑢^∗∗. Hence, ℱ is a Lyapunov function on 𝒟^∘. Therefore, the largest compact invariant subset of the set where ℱ̇=0 is the singleton {(𝑆,𝐸,𝐼_𝑢)=(𝑆^∗∗,𝐸^∗∗,𝐼_𝑢^∗∗)}. Thus, it follows, by the laSalle's invariance Principle [18, 45], that 𝑆(𝑡)→𝑆^∗∗,𝐸(𝑡)→𝐸^∗∗ and 𝐼_𝑢(𝑡)→𝐼_𝑢^∗∗ as 𝑡→∞. Since limsup_𝑡→∞𝐼_𝑢=𝐼_𝑢^∗∗, it follows that, for sufficiently small 𝜖>0, there exist constants 𝑇₁>0 such that limsup_𝑡→∞𝐼_𝑢≤𝐼_𝑢^∗∗+𝜖 for all 𝑡>𝑇₁. It follows from the fourth equation of (30) that, for 𝑡>𝑇₁, 𝐼̇_𝑒(𝑡)≤𝜎𝐼_𝑢^∗∗+𝜖−𝑘₃𝐼_𝑒. (42) Thus, by comparison theorem [42], 𝐼^∞_𝑒=limsup_𝑡→∞≤𝜎𝐼_𝑢^∗∗+𝜖𝑘₃. (43) Hence, by letting 𝜖→0, we have 𝐼^∞_𝑒=limsup_𝑡→∞≤𝜎𝐼_𝑢^∗∗𝑘₃. (44) Similarly, by using liminf_𝑡→∞𝐼_𝑢(𝑡)=𝐼_𝑢^∗∗, it can be shown that 𝐼_𝑒∞=liminf_𝑡→∞≥𝜎𝐼_𝑢^∗∗𝑘₃. (45) Thus, it follows from (44) and (45) that 𝐼_𝑒∞≥𝜎𝐼_𝑢^∗∗𝑘₃≥𝐼^∞_𝑒. (46) Hence, lim_𝑡→∞𝐼_𝑒(𝑡)=𝜎𝐼_𝑢^∗∗/𝑘₃=𝐼_𝑒^∗∗. In a similar way, it can be shown that lim_𝑡→∞𝑅(𝑡)=𝑅^∗∗. Thus, every solution to the equations of the reduced model, with initial condition in 𝒟^∘, approaches the endemic equilibrium of the reduced system (30) as 𝑡→∞ for ℛ_0𝑟>1. This shows that the endemic equilibrium is unique.

5. Conclusions

In this paper, an SEIR epidemic model with a nonlinear (Holling type II) incidence rate is designed and analysed. Some of the theoretical and epidemiological findings of the study are as follows. (i)The model (8) has a locally stable disease-free equilibrium whenever the associated reproduction number is less than unity. (ii)The DFE of the model (8) is shown to be globally asymptotically stable when ℛ₀<1. (iii)The model (8) is uniformly persistent in 𝒟^∘ if and only if ℛ₀>1. (iv)The endemic equilibrium of the reduced model (8), with 𝜂=0, is shown to be globally asymptotically stable, when it exists.

Acknowledgments

S. Garba acknowledges, with thanks, the support on part of the University of Pretoria Research and Development Project (RDP). The authors are grateful to the two anonymous reviewers and the handling editor for their constructive comments, which have enhanced the paper.