Stability analysis of HIV/AIDS epidemic model with nonlinear incidence and treatment
 Jianwen Jia^{1}Email author and
 Gailing Qin^{1}
https://doi.org/10.1186/s1366201711755
© The Author(s) 2017
Received: 25 December 2016
Accepted: 10 April 2017
Published: 12 May 2017
Abstract
An HIV/AIDS epidemic model with general nonlinear incidence rate and treatment is formulated. The basic reproductive number \(\Re_{0}\) is obtained by use of the method of the next generating matrix. By carrying out an analysis of the model, we study the stability of the diseasefree equilibrium and the unique endemic equilibrium by using the geometric approach for ordinary differential equations. Numerical simulations are given to show the effectiveness of the main results.
Keywords
1 Introduction
The human immunodeficiency virus (HIV) infection, which can lead to acquired immunodeficiency syndrome (AIDS), has become an important infectious disease in both the developed and the developing nations. It causes mortality of millions of people and expenditure of an enormous amount of money in health care and disease control.
The study of HIV/AIDS transmission dynamics has been of great interest to both applied mathematicians and biologists due to its universal threat to humanity. Mathematical models have been used extensively in the research of the epidemiology of HIV/AIDS, to help improve our understanding of the major contributing factors in a given epidemic [1–4]. Yusuf and Benyah [5] presented a deterministic model for controlling the spread of the disease, and the results show that the optimal way to mitigate the spread of the disease is for susceptible individuals to consistently practise safe sex as much as possible, while the ARV treatment should be initiated for patients as soon as they progress to the preAIDS stage of the disease. Huo et al. [6] considered a simple HIV/AIDS epidemic model with treatment, they incorporate the new compartment, that is, the treatment compartment T. Individuals in compartment T receive all kinds of treatments, these treatments do not completely eliminate HIV from the body. They study the effect of treatment on the transmission dynamics of the HIV/AIDS epidemic model.
In mathematical epidemiology, the disease incidence plays an important role in the study of the mathematical epidemiological model. The general form of incidence rate is written as \(\beta U(N)\frac {S}{N}I\). Both bilinear and standard incidences (\(\beta SI\) and \(\beta SI/N\) with N the total population) have been frequently used in classical epidemic models [6, 7]. However, several studies have suggested that the disease transmission process may have a nonlinear incidence rate [8, 9]. In addition, some general nonlinear incidence \(\beta g(I)S\) [10], \(Sg(I)\) [11], \(g(S,I)\) [12] and \(g(S,I,N)\) [13] are used in models. Contrasted to models with the bilinear or standard incidence, complex dynamic behaviors may occur when more general nonlinear incidences are used.
Muldowney [14] proposed a way to prove the asymptotical stability of periodic orbits through estimating the right derivative of the Lyapunov function, and the global asymptotical stability of the epidemic equilibrium was proved by using a PoincaréBendixson property and a general criterion for the orbital stability of periodic orbits concerned with higherdimensional nonlinear autonomous systems as well as the theory of competitive system of differential equations. This geometric method is also used in [15, 16] to resolve the global asymptotical stability of the epidemic equilibrium for an SEIR with bilinear and nonlinear incidence rates.
Motivated by the above work, in this paper, we consider an HIV/AIDS epidemic model with nonlinear incidence rate \(Sg(I)\) and treatment. Our paper is organized as follows. In Section 2 we formulate the complete mathematical model and define the basic reproductive number \(\Re_{0}\). Furthermore, the existence of equilibria of this model is given in Section 3. The stability analysis of the equilibria of the model is proposed in Section 4, which includes the stability analysis of the diseasefree equilibrium and the endemic equilibrium of the model. Some numerical simulations are given in Section 5. Finally, we summarize this work.
2 The model and the basic reproduction number
2.1 Formulation of the models
 (H_{1})::

\(g(0)=0\), \(g'(I)>0\), \(g''(I)\leq0\) for \(I\geq0\);
 (H_{2})::

\(\lim_{I\rightarrow0^{+}}\frac{g(I)}{I}=\beta\), \(0<\beta<\infty\).
2.2 The basic reproduction number
3 The existence of the equilibria
Theorem 3.1
The system (2.2) always has a diseasefree equilibrium \(E_{0}=(S_{0}, I_{0}, T_{0}, A_{0}, R_{0})=(\frac{\Lambda}{\mu_{1}+d}, 0, 0, 0, \frac {\mu_{1}\Lambda}{d(\mu_{1}+d)})\). If \(\Re_{0}>1\), the assumptions (H_{1}) and (H_{2}) are satisfied, then, besides \(E_{0}\), system (2.2) has a unique endemic equilibrium \(E^{*}=(S^{*}, I^{*}, T^{*}, A^{*}, R^{*})\).
Proof
It is easy to verify that system (2.2) always has a diseasefree equilibrium \(E_{0}\).
Correspondingly, model (2.2) has a unique endemic equilibrium \(E^{*}(S^{*}, I^{*}, T^{*}, A^{*}, R^{*})\), where \(T^{*}=\frac{k_{2}}{n} I^{*}\), \(A^{*}=\frac{k_{1}+\frac{\alpha_{2} k_{2}}{n}}{d+\delta _{1}} I^{*}\), \(R^{*}=\frac{\mu_{1}}{d} S^{*}\).
The proof of Theorem 3.1 is completed. □
4 Analysis of stability
4.1 Stability of the diseasefree equilibrium
Theorem 4.1
The diseasefree equilibrium \(E_{0}\) is globally asymptotically stable if \(0<\Re_{0}<1\), and unstable if \(\Re_{0}>1\).
Proof
So we only need to consider the sign of \(\lambda_{4}\) and \(\lambda_{5}\). Since \(\lambda_{4}+\lambda_{5}=b\), \(\lambda_{4} \lambda_{5}=c\), and when \(0<\Re_{0}<1\), i.e. \(\frac{\beta n\Lambda}{\mu _{1}+d}< mn\alpha_{1} k_{2}< mn\), we have \(b>0\), \(c>0\), hence \(\lambda_{4}<0\), \(\lambda_{5}<0\). So all roots of (4.1) have negative real parts, i.e. the equilibrium \(E_{0}\) is locally asymptotically stable in Ω when \(0<\Re_{0}<1\).
From the above, we know that if \(0<\Re_{0}<1\), the equilibrium \(E_{0}\) is locally asymptotically stable and by Theorem 3.1 there are no endemic equilibrium in Ω. By [19], any solution of (2.2) starting in Ω must approach either an equilibrium or a closed orbit in Ω. By [20], if the solution path approaches a closed orbit, then this closed orbit must enclose an equilibrium. Nevertheless, the only equilibrium existing in Ω is \(E_{0}\) when \(0<\Re_{0}<1\) and it is located in the boundary of Ω, therefore there is no closed orbit in Ω. Hence any solution of system (2.2) with initial condition in Ω must approach the point \(E_{0}\) as time tends to infinity. Therefore, the diseasefree equilibrium \(E_{0}\) is globally asymptotically stable in Ω when \(0<\Re_{0}<1\).
When \(\Re_{0}>1\), we have \(c<0\), so the equation \(h(\lambda)=0\) has a positive root. Therefore, the equilibrium \(E_{0}\) is unstable.
The proof of Theorem 4.1 is completed. □
4.2 Stability of the endemic equilibrium
Theorem 4.2
If \(\Re_{0}>1\), then the endemic equilibrium \(E^{*}\) of the system (2.2) is locally asymptotically stable.
Proof
Obviously, equation (4.2) has real roots \(\lambda_{1}=d<0\), \(\lambda _{2}=(d+\delta_{1})<0\), and other roots of (4.2) are given by the roots of \(h(\lambda)=\lambda ^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3}=0\).
By the RouthHurwitz criteria, we see that all roots of the equation \(h(\lambda)=0\) have negative real parts, i.e. the epidemic equilibrium \(E^{*}\) of system (2.2) is locally asymptotically stable in Ω.
The proof of Theorem 4.2 is completed. □
Next, we turn to showing the global stability of the equilibrium \(E^{*}\).
In order to study the global asymptotic stability of the endemic equilibrium \(\overline{E}^{*}\) of the system (4.3), by use of the geometrical approach developed by Li and Muldowney [15], we obtain the simple sufficient condition that \(\overline{E}^{*}\) is globally asymptotically stable when \(\Re_{0}>1\).
Proposition 4.1
The system (4.3) is uniformly persistent if and only if \(\Re_{0}>1\).
Proof
Combining the local stability analysis for the equilibrium in Theorem 4.1 and Theorem 4.3 in [22], we know that system (4.3) is uniformly persistent if and only if \(\Re_{0}>1\).
The proof of Proposition 4.1 is completed. □
Theorem 4.3
Assume that \(\Re_{0}>1\), then the endemic equilibrium \(\overline{E}^{*}\) of system (4.3) is globally asymptotically stable when \(\mu_{1}< k_{1}+k_{2}\).
Proof
The proof of Theorem 4.3 is completed. □
Theorem 4.4
If \(\mu_{1}< k_{1}+k_{2}\), then the epidemic equilibrium \(E^{*}=(S^{*}, I^{*}, T^{*}, A^{*}, R^{*})\) of system (2.2) is globally asymptotically stable when \(\Re_{0}>1\).
Proof
From an analysis of the above, we can know that the endemic equilibrium \(E^{*}(S^{*},I^{*},T^{*}, A^{*},R^{*})\) is globally attractive in Ω. Combined with the local stability of \(E^{*}\), the endemic equilibrium \(E^{*}\) is globally asymptotically stable in Ω.
The proof of Theorem 4.4 is completed. □
5 Numerical simulations
In this section, some numerical results of system (2.1) are presented for supporting the analytic results obtained above. We choose \(g(I)=\frac{\beta I}{1+\alpha I}\), it is not difficult to verify that assumptions (H_{1}) and (H_{2}) are satisfied.
 (1)When \(\alpha=0.85\), \(\mu_{1}=0.8\), \(k_{1}=0.05\), \(k_{2}=0.25\). By directly computing, we have \(\Re_{0}=0.5297<1\). According to Theorem 4.1 the diseasefree equilibrium \(E_{0}=(2.4562, 0, 0, 0, 37.5948)\) is globally asymptotically stable (see Figure 1). It shows that the disease eventually tends to go extinct.
 (2)When \(\alpha=0.1\), \(\mu_{1}=0.03\), \(k_{1}=0.15\), \(k_{2}=0.35\). By directly computing, we have \(\Re_{0}=1.7787>1\), \(E^{*}=(7.328, 2.363, 1.592, 3.352, 11.22)\) and \(\mu_{1}=0.03< k_{1}+k_{2}=0.5\). So the conditions of Theorem 4.4 are satisfied, then the endemic equilibrium \(E^{*}\) is globally asymptotically stable (see Figure 2). This shows the disease is persistent.
6 Conclusion
In this paper, a simple model HIV/AIDS epidemic model in which we consider a nonlinear incidence rate with a general form is introduced. The global dynamics of our model is determined by the basic reproduction number \(\Re_{0}\). When \(\Re_{0}\) is less than unity, the diseasefree equilibrium is globally asymptotically stable. When \(\Re_{0}\) is bigger than 1, the unique endemic equilibrium is globally asymptotically stable under the condition \(\mu _{1}< k_{1}+k_{2}\). Our results suggest that appreciable change in the susceptible individuals’ sexual habits faster reduces both incidence and prevalence the disease. Numerical simulations are given to support our analytic results.
Declarations
Acknowledgements
We greatly appreciate the editor and the anonymous referees careful reading and valuable comments, their critical comments and helpful suggestions greatly improve the presentation of this paper. This work is supported by Natural Science Foundation of Shanxi province (20130110022).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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