IJEM International Journal of Engineering Mathematics 2314-6109 2356-7007 Hindawi Publishing Corporation 10.1155/2014/487589 487589 Research Article Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence Bernoussi Amine 1 http://orcid.org/0000-0001-7560-7315 Kaddar Abdelilah 2 Asserda Said 1 Zhong Shouming 1 Département de Mathématiques Faculté des Sciences Université Ibn Tofail Campus Universitaire BP 133, 14000 Kénitra Morocco univ-ibntofail.ac.ma 2 Département d’Economie et de Gestion, Faculté des Sciences Juridiques Economiques et Sociales Université Mohammed V-Souissi, Route Outa Hsaine BP 5295, 11100 Sala Al Jadida Morocco um5s.ac.ma 2014 7122014 2014 25 07 2014 14 11 2014 7 12 2014 2014 Copyright © 2014 Amine Bernoussi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

1. Introduction

Epidemic models have long been an important tool for understanding and controlling the spread of infectious diseases. Most of them are described by delayed differential equations. The introduction of time delay is often used to model the latent period, that is, the time from the acquisition of infection to the time when the host becomes infectious [1, 2].

Recently, considerable attention has been paid to model the relapse phenomenon, that is, the return of signs and symptoms of a disease after a remission. Hence, the recovered individual can return to the infectious class (see ). For the biological explanations of the relapse phenomenon, we cite two examples.

For malaria, Bignami  proposed that relapses derived from persistence of small numbers of parasite in the blood. Also, it has been observed that the proportion of patients who have successive relapses is relatively constant (see ).

For tuberculosis, relapse can be caused by incomplete treatment or by latent infection, being observed that HIV-positive patients are significantly more likely to relapse than HIV-negative patients, although it is often difficult to differentiate relapse from reinfection (see ).

In this paper, we propose the following epidemic model with time delay and relapse (delayed SIRI epidemic model) as follows (see [10, 11]): (1) d S d t = A - μ S - f S , I , d I d t = e - μ τ f S τ , I τ - μ + γ + α I + δ R , d R d t = γ I - μ + δ R . The initial condition for the above system is (2) S θ = φ 1 θ , I θ = φ 2 θ , R θ = φ 3 θ , i i θ - τ , 0 , with φ = ( φ 1 , φ 2 , φ 3 ) C + × C + × C + , such that φ i ( θ ) 0 ( - τ θ 0 , i = 1,2 , 3 ). Here C denotes the Banach space C ( [ - τ , 0 ] , R ) of continuous functions mapping the interval [ - τ , 0 ] into R , equipped with the supremum norm. The nonnegative cone of C is defined as C + = C ( [ - τ , 0 ] , R + ) .

Here ψ τ = ψ ( t - τ ) for any given function ψ , A = μ N , where N = S + I + R is the total number of population, S is the number of susceptible individuals, I is the number of infectious individuals, R is the number of recovered individuals, A is the recruitment rate of the population, μ is the natural death of the population, α is the death rate due to disease, f is the nonlinear incidence function, γ is the recovery rate of the infective individuals, δ is the rate that recovered individuals relapse and regained infectious class, and τ is the latent period.

In model (1) the incidence function f ( S , I ) is a locally Lipschitz continuous function on R + × R + satisfying f ( 0 , I ) = f ( S , 0 ) = 0 for S 0 , I 0 , and the following hold:

f is a strictly monotone increasing function of S 0 , for any fixed I > 0 , and f is a strictly monotone increasing function of I 0 , for any fixed S 0 .

ϕ ( S , I ) = f ( S , I ) / I is a bounded and monotone decreasing function of I > 0 , for any fixed S 0 , and K ( S ) = lim I 0 + ϕ ( S , I ) is a continuous and monotone increasing function on S 0 .

This incidence function includes different forms presented in literature (see, e.g., ).

System (1) always has a disease-free equilibrium P 0 = ( A / μ , 0,0 ) . On the other hand, under the hypothesis ( H 2 ) , if (3) R 0 K A / μ e - μ τ μ + γ + α - γ δ / μ + δ > 1 , then system (1) also admits a unique endemic equilibrium P * = ( S * , I * , R * ) , where S * , I * , and R * satisfy the following system (see ): (4) i i i i i i i i i i A - μ S - f S , I = 0 , e - μ τ f S , I - μ + γ + α I + δ R = 0 , i i i i i i i i i i i i γ I - μ + δ R = 0 . Hereafter, we replace μ + γ + α - γ δ / ( μ + δ ) by η .

In  (1990), Tudor developed and analyzed qualitatively one of the first SIRI epidemic models for the spread of a herpes-type infection in either human or animal populations. This model consists of a system of nonlinear ordinary differential equations with a bilinear incidence rate (i.e., f ( S , I ) = β S I ) and a constant total population (i.e., N = S + I + R = constant ).

In  (1997), Moreira and Wang extended a Tudor-model to include nonlinear incidence functions. By using an elementary analysis of Liénard’s equation and Lyapunov’s direct method, they derived sufficient conditions for the global asymptotic stability of the disease-free and endemic equilibria.

In  (2000), Castillo-Garsow et al. considered an SIRI model for drug use in a population of adolescents. The authors assumed that N = constant and f ( S , I ) = β S I / N ; they estimated the parameters of the model and they determined a rough approximation of the basic reproductive number. Based on these parameters, they performed some simulations that clearly showed the endemic character of tobacco use among adolescents.

In  (2004), Blower developed a compartmental model for genital herpes, assuming standard incidence rate (i.e., f ( S , I ) = β S I / N ) and constant recruitment rate.

In  (2006), Korobeinikov proved that the endemic equilibrium of the model (1) with δ = 0 and τ = 0 is globally asymptotically stable.

In  (2007), van den Driessche and Zou proposed an integrodifferential equation to model a general relapse phenomenon in infectious diseases. The resulting model, in particular case, is a delay differential equation with a constant population and standard incidence. The basic reproduction number for this model is identified and some global results are obtained by employing the Lyapunov-Razumikhin technique.

In  (2007), Van Den Driessche and co-authors formulated a delay differential SIRI model (System (1) with f ( S , I ) = β S I ). For this system, the endemic equilibrium is locally asymptotically stable if R 0 > 1 , and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level.

In  (2011), Liu et al. proposed a mathematical model for a disease with a general exposed distribution, the possibility of relapse and nonlinear incidence rate ( f ( S , I ) = β S ( I / N ) ). By the method of Lyapunov functionals, they showed that the disease dies out if R 0 = 1 and that the disease becomes endemic if R 0 > 1 . They also analyzed, as a special case of this model, the system (1) with f S , I = g S · I ; the result confirms that the endemic equilibrium is globally asymptotically stable.

In  (2012), Abta et al. considered a global asymptotic stability of a delayed SIR model (system (1) with f ( S , I ) = β S I / ( 1 + α 1 S + α 2 I ) and δ = 0 ).

In  (2013), Vargas-De-Leon presented the global stability conditions of an ordinary SIRI model with bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptible individuals into the community and that the disease produces nonnegligible death in the infectious class. The author presented the construction of Lyapunov functions using suitable combinations of known functions, common quadratic and Volterra-type, and a composite Volterra-type function.

In  (2013), Georgescu and Zhang analyzed the dynamics of an ordinary SIRI model under the assumption that the incidence of infection is given by f ( S , I ) = C ( S ) g ( I ) . They obtained by means of Lyapunov’s second method sufficient conditions for the local stability of equilibria and they showed that global stability can be attained under suitable monotonicity conditions.

In  (2013), Shuai and van den Driessche presented two systematic methods for the construction of Lyapunov functions for general infectious disease models (Ordinary SEIRI, SIS, etc.). Specifically, a matrix-theoretic method using the Perron eigenvector is applied to prove the global stability of the disease-free equilibrium, while a graph-theoretic method based on Kirchhoff’s matrix tree theorem and two new combinatorial identities are used to prove the global stability of the endemic equilibrium.

In  (2014), Xu investigated a delayed SIRI model (system (1) with f ( S , I ) = β S I ). The author established the global stability of a disease-free equilibrium and an endemic equilibrium by means of suitable Lyapunov functionals and LaSalle’s invariance principle.

In this paper we extend the global stability results presented in  to a delayed SIRI epidemic model (system (1)) with a general nonlinear incidence function. It is shown that global stability can be attained under suitable monotonicity conditions and it is established that the basic reproduction number R 0 is a threshold parameter for the stability of a delayed SIRI model. The rest of the paper is organized as follows. In Section 2, the global stability of disease-free and endemic equilibria are established. In Section 3, numerical simulations and concluding remarks are provided. In the appendix, some results on the global stability are stated.

2. Global Stability Analysis of Delayed SIRI Model

In this section, we discuss the global stability of a disease-free equilibrium P 0 and an endemic equilibrium P * of system (1). Since ( d / d t ) ( S + I + R ) A - μ ( S + I + R ) , we have limsup ( S + I + R ) A / μ . Hence we discuss system (1) in the closed set (5) Ω = φ 1 , φ 2 , φ 3 C + × C + × C + : φ 1 + φ 2 + φ 3 A μ . It is easy to show that Ω is positively invariant with respect to system (1). Next we consider the global asymptotic stability of the disease-free equilibrium P 0 and the endemic equilibrium P * of (1) by Lyapunov functionals, respectively.

Proposition 1.

If R 0 1 , then the disease-free equilibrium P 0 is globally asymptotically stable.

Proof.

Define a Lyapunov functional (6) V 0 t = e - μ τ A / μ S τ 1 - K A / μ K u d u + I + η t - τ t I u d u + δ μ + δ R . We will show that d V 0 ( t ) / d t 0 for all t 0 . We have (7) d V 0 t d t = e - μ τ 1 - K A / μ K S τ S ˙ τ + e - μ τ f S τ , I τ d τ W - μ + γ + α I + δ R + η I - I τ W + γ δ μ + δ I - δ R = e - μ τ 1 - K A / μ K S τ A - μ S τ W + K A / μ K S τ e - μ τ f S τ , I τ - η I τ = μ e - μ τ 1 - K A / μ K S τ A μ - S τ W + η I τ ϕ S τ , I τ η K A / μ K S τ e - μ τ - 1 . By the hypothesis ( H 1 ) , we obtain that (8) 1 - K A / μ K S τ A μ - S τ 0 , where equality holds if and only if S = A / μ .

Furthermore, It follows from the hypothesis ( H 2 ) that (9) ϕ S τ , I τ η K A / μ e - μ τ K S τ K S τ η K A / μ e - μ τ K S τ K A / μ e - μ τ η R 0 .

Therefore, R 0 1 ensures that d V 0 ( t ) / d t 0 for all t 0 , where d V 0 ( t ) / d t = 0 holds if ( S , I , R ) = ( A / μ , 0,0 ) . Hence, it follows from system (1) that { P 0 } is the largest invariant set in S , I , R d V 0 ( t ) / d t = 0 . From the Lyapunov-LaSalle asymptotic stability, we obtain that P 0 is globally asymptotically stable. This completes the proof.

Proposition 2.

If R 0 > 1 , then the endemic equilibrium P 1 * is globally asymptotically stable.

Proof.

To prove global stability of the endemic equilibrium, we define a Lyapunov functional V ( t ) = V 1 ( t ) + V 2 ( t ) + V 3 ( t ) + V + , with (10) V 1 t = S - S * - S * S f S * , I * f u , I * d u , V 2 t = e μ τ I - I * - I * ln I I * , V 3 t = δ e μ τ μ + δ R - R * - R * ln R R * , V + = t - τ t f S , I - f S * , I * - f S * , I * f S u , I u f S * , I * W W W W W · ln f S u , I u f S * , I * d u . The time derivative of the function V ( t ) along the positive solution of system (1) is (11) d V t d t = 1 - f S * , I * f S , I * A - μ S - f S , I W + e μ τ 1 - I * I W · e - μ τ f S τ , I τ + δ R - μ + α + γ I W + δ e μ τ μ + δ 1 - R * R γ I - μ + δ R W + f S , I - f S τ , I τ + f S * , I * W · ln f S τ , I τ f S , I . Using the relation A = μ S * + f ( S * , I * ) , simple calculations give (12) d V t d t = 1 - f S * , I * f S , I * - μ S - S * + f S * , I * + f S * , I * f S , I * f S , I - e μ τ μ + α + γ I - I * I f S τ , I τ - δ e μ τ I * I R + e μ τ μ + α + γ I * + δ e μ τ μ + δ γ I - γ I R * R + μ + δ R * + f S * , I * ln f S τ , I τ f S , I . Here by using (13) e μ τ μ + α + γ I * - δ e μ τ μ + δ γ I * = f S * , I * , μ + δ R * = γ I * , ln f S τ , I τ f S , I W = ln f S * , I * f S , I * + ln I * f S τ , I τ I f S * , I * + ln I f S , I * I * f S , I , straightforward calculations give (14) d V t d t = - μ 1 - f S * , I * f S , I * S - S * - f S * , I * f S * , I * f S , I * - 1 - ln f S * , I * f S , I * - f S * , I * I * f S τ , I τ I f S * , I * - 1 - ln I * f S τ , I τ I f S * , I * + f S * , I * f S , I f S , I * - I I * + I f S , I * I * f S , I - 1 - f S * , I * I f S , I * I * f S , I - 1 - ln I f S , I * I * f S , I + δ γ I * e μ τ μ + δ 2 - I * R I R * - I R * I * R = - μ 1 - f S * , I * f S , I * S - S * - f S * , I * f S * , I * f S , I * - 1 - ln f S * , I * f S , I * - f S * , I * I * f S τ , I τ I f S , I * - 1 - ln I * f S τ , I τ I f S , I * + I I * f S * , I * 1 - f S , I * f S , I ϕ S , I ϕ S , I * - 1 - f S * , I * I f S , I * I * f S , I - 1 - ln I f S , I * I * f S , I - δ γ e μ τ μ + δ I R * R I * R I R * - 1 2 . It follows from ( H 1 ) and ( H 2 ) that (15) - μ 1 - f S * , I * f S , I * S - S * 0 , I I * f S * , I * 1 - f S , I * f S , I ϕ S , I ϕ S , I * - 1 W 0 . Furthermore, since the function g ( x ) = 1 - x + ln ( x ) is always nonpositive for any x > 0 , and g ( x ) = 0 if and only if x = 1 , then d V ( t ) / d t 0 , for all t 0 , where the equality holds only at the equilibrium point ( S * , I * , R * ) . Hence, the functional V satisfies all the conditions of Theorem A.2. This proves that P 1 * is globally asymptotically stable.

3. Numerical Simulations and Concluding Remarks

In this section, we give a numerical simulation supporting the theoretical analysis given in Section 2. Let (16) f S , I = β S I 1 + α 1 S + α 2 I . We take the parameters of the system (1) as follows: (17) A = 10 , α 1 = 0.9 , α 2 = 0.9 , μ = 0.01 , γ = 0.02 , α = 0.005 , β = 0.1 , δ = 0.001 , τ = 10 . By Proposition 2, the endemic equilibrium P * is globally asymptotically stable; see Figure 1.

Solutions ( S , I , R ) of the SIRI epidemic model (1) are globally asymptotically stable and converge to the endemic equilibrium P * .

In this paper, we presented a mathematical analysis and numerical simulations for an SIRI epidemiological model applied to the evolution of the spread of disease with relapse in a given population. We denote R 0 the basic reproduction number. It is defined as the average number of contagious persons infected by a typical infectious in a population of susceptible. We prove in this paper that the basic reproduction number, R 0 , depends on the incubation period and we show that the disease-free equilibrium P 0 is globally asymptotically stable if R 0 1 and that a unique endemic equilibrium P * is globally asymptotically stable if R 0 > 1 .

Appendix The Lyapunov-LaSalle Theorem

In the following, we present the method of Lyapunov functionals in the context of a delay differential equations: (A.1) d x d t = f x t , where f : C R n is completely continuous and solutions of (A.1) are unique and continuously dependent on the initial data. We denote by x ( ϕ ) the solution of (A.1) through ( 0 , ϕ ) . For a continuous functional V : C R , we define (A.2) V ˙ = limsup h 0 + 1 h V x h ϕ - V ϕ , the derivative of V along a solution of (A.1). To state the Lyapunov-LaSalle type theorem for (A.1), we need the following definition.

Definition A.1 (see [<xref ref-type="bibr" rid="B26">32</xref>, page 30]).

We say V : C R is a Lyapunov functional on a set G in C for (A.1) if it is continuous on G ¯ (the closure of G ) and V ˙ 0 on G . We also define E = { ϕ G ¯ : V ˙ ( ϕ ) = 0 } , and M is the largest set in E which is invariant with respect to (A.1).

The following result is the Lyapunov-LaSalle type theorem for (A.1).

Theorem A.2 (see [<xref ref-type="bibr" rid="B26">32</xref>, page 30]).

If V is a Lyapunov functional on G and x t ( ϕ ) is a bounded solution of (A.1) that stays in G , then ω -limit set ω ( ϕ ) M ; that is, x t ( ϕ ) M as t + .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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