tability Analysis of a I Epidemic Model with tandard Incidence Cruz Vargas-De-León Received 19 April 2011; Accepted 19 Octuber 2011 leoncruz82@yahoo.com.mx Abstract In this paper, we study the global properties of classic I epidemic model with constant recruitment, disease-induced death and standard incidence term. We apply the Poincaré-Bendixson theorem, Dulac s criterion, and the method of Lyapunov function to establish conditions for global stability. For this system, three Dulac functions and two Lyapunov functions are constructed for the endemic steady state. Keywords. I epidemic model, tandard incidence term, Global stability, Direct Lyapunov method, Dulac s criterion, Poincaré-Bendixson theorem. 1 Introduction Continuous models, usually in the form of ordinary nonlinear differential equations, have formed a large part of the traditional mathematical epidemiology literature. In such models, the classical assumptions are that the total population is divided into any number of classes according to their epidemiological status, and that the transmission of the infection in the population is modelled by incidence terms. Many forms are possible for the incidence term in epidemic models, the most common are the simple mass action and standard incidence terms. In this paper, we study an epidemic model of I type, with constant recruitment, disease-induced death and standard incidence rate. This model is analyzed to determine the basic reproductive number, steady states, and stability. The question of global stability in mathematical epidemiology is a very interesting mathematical problem. To study the conditions of global stability for the steady states of two dimensional systems of nonlinear ordinary differential equation two wellknown method are used. One technique that is used, is Dulac s criterion to eliminate the existence of the periodic solution and prove the global stability by the Poincaré- Bendixson theorem (ee [1]. The other technique is the second Lyapunov method, 2000 Mathematics ubject Classifications: 34K20 and 92D30. Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, México Facultad de Estudios uperiores Zaragoza, Universidad Nacional Autónoma de México, México 1
the principle advantages of Lyapunov s direct method is the fact that it does not require the knowledge of solutions. However, the method requires an auxiliary function, called Lyapunov function, that is usually difficult to construct. The global stability is established by using a Lyapunov function and Laalle s invariance principle (ee [2]. The second Lyapunov method is a power technique for multidimensional systems. We study the global stability of I epidemic model using these two techniques. In recent years, the second Lyapunov method has been a popular technique to study global properties of epidemiological models. A Volterra-type Lyapunov function has been used in [3, 4] to prove global stability of the steady states of the classic I, IR and IR epidemic models with mass action incidence rate and constant population. In [5] using combinations of common quadratic, composite quadratic and Volterra-type functions to prove global stability of steady states of I, IR and IR epidemic models with constant recruitment, disease-induced death and mass action incidence rates. Generally speaking, within epidemic models with simple mass action incidence exist methods for constructing Lyapunov functions (ee [3, 4, 5], there are no systematic methods for constructing Lyapunov functions for epidemiological models with standard incidence rate (ee a recent article [6]. On the other hand, as is well known, the Lyapunov functions for a given system are not unique. In this paper, we construct two Lyapunov functions for the endemic steady state of a epidemic model type I with standard incidence rate. The paper is organized as follows. In ection 2, we prove that the I epidemic model is uniformly bound and determine the basic reproductive number and the steady states. The local and global stability of the disease-free steady state is established in ection 3. In ection 4, the local stability of the endemic steady state is established. ection 4 includes the global stability of endemic steady state using the Poincaré- Bendixson theorem, Dulac s criterion, and Lyapunov s direct method. Two Dulac functions and one Lyapunov function is constructed for endemic steady state in ection 5. In ection 6 includes the discussion and concluding remarks. 2 I Model with tandard Incidence ome infectious diseases do not confer immunity. uch infections do not have a recovered state and individuals become susceptible again after infection. This type of disease can be modelled by the I type. The population is divided into two classes, in the susceptible individuals of classes ( and infectious individuals of class (I. The system of differential equations for the I model is [7]: d di = Λ βi µ φi, = βi (α µ φi. (1 Where Λ is the rate of susceptible individuals recruited into the population (either by birth or immigration; the standard incidence is βi I, which is the average number of 2
infection transmissions by all infectious individuals I per day; µ is the natural death rate; φ is the recovery rate; and α is the disease related death rate. The parameters are positive constants and variables are non-negative. The differential equation of the total population of (1 is: Thus the total population size may vary in time. d ( = Λ µ( αi. (2 We shall show that the system (1 is uniformly bounded. Theorem 2.1. All solutions of system (1 are eventually confined in the compact subset Γ = { (, I R 2 : 0, I 0, Λ/µ }. Proof. Let ((t, I(t be any solution with positive initial conditions ( 0, I 0. Let V (, I = (ti(t. The time derivative along a solution of (1 is V = Λ µv αi Λ µv, it follows that V µv Λ. Applying the theory of differential inequalities [8], we obtain V (, I Λ µ (1 exp( µt V ( 0, I 0 exp( µt, and for t, we have lim sup V Λ. Hence all the solutions of (1 which initiate in t µ R 2 are eventually confined in the region Γ. This completes the proof. Clearly, the set Γ is positively invariant with respect to (1. Let Γ denote the interior of Γ. The system (1 has two steady states in the non-negative triangle R 2 : a diseasefree steady state E = (Λ/µ, 0, and a unique endemic steady state E = (, I with coordinates The parameter = Λ µ (α µ(r 0 1, I = (R 0 1Λ µ (α µ(r 0 1. R 0 = β α µ φ, is often called the basic reproductive number. It represents the average number of secondary infections that occur when one infectious individual is introduced into a completely susceptible population. Notice that the basic reproductive number is independent of the factor Λ/µ. 3
3 Global stability of disease-free steady state In the absence of the infectious disease, the model has a unique disease-free steady state E. To establish the local stability of E, we use the Jacobian of the model evaluated at E. tability of this steady state is then determined based on the eigenvalues of the corresponding Jacobian which are functions of the model parameters. The Jacobian matrix for system (1 is given by (µ J = βi2 (I 2 β2 (I φ 2 βi 2 β 2 (I 2 (I (α µ φ 2 The Jacobian of the model at E is: J(E = µ β φ, 0 β (α µ φ with the eigenvalues λ 1 = µ and λ 2 = β (α µ φ = (α µ φ(1 R 0. In the following Theorem established the local stability of the disease-free steady state. Theorem 3.1. The disease-free steady state E of (1 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1. Proof. ince λ 1 < 0, the disease-free steady state E is locally asymptotically stable if λ 2 < 0. Noting that λ 2 < 0 if and only if R 0 < 1. If R 0 > 1, then λ 2 is positive. Therefore E is unstable, the Theorem 2.1 is proven. We prove the global stability of the disease-free steady state E when the basic reproductive number is less than or equal to unity. Theorem 3.2. If R 0 1, then the disease-free steady state E of (1 is globally asymptotically stable in Γ. Proof. Define U : {(, I Γ : > 0} R by. U(, I = 1 2 I2. The time derivative of U computed along solutions of (1 is U (, I = I2 (β (α µ φ(, I 2 = (α µ φ ((1 R 0. ince all the model parameters are positive and variables are non-negative, it follows that U (, I 0 for R 0 1 with U (, I = 0 if and only if I = 0, or R 0 = 1 and 4
I = 0. Hence, U is a Lyapunov function on Γ. Thus, I 0 as t. Using I = 0 in the first equation of (1 shows that Λ/µ as t. Therefore, it follows from the Laalle s Invariance Principle [2], that every solution of the equations in the model (1, with initial conditions in Γ, approaches E as t. This completes the proof. 4 tability of endemic steady state In this section, we want to introduce the two methods to analyze the global stability of the endemic steady state of the system (1: (i Dulac s criterion and the Poincaré- Bendixson theorem, and (ii Direct Lyapunov method. At first, we now consider the local stability of E. Theorem 4.1. The endemic steady state E of (1 is locally asymptotically stable if R 0 > 1. Proof. The Jacobian matrix for system (1 evaluated at the endemic steady state E is (µ β(i 2 J(E ( I β( 2 2 ( I φ 2 =, β(i 2 β( 2 ( I 2 ( I (α µ φ 2 that can be rewritten as (µ β(i 2 J(E = ( I 2 β I ( I (α µ 2, β(i 2 ( I 2 β I ( I 2 when we take into account the identity (3: φ µ α = β I, (3 which is obtained by the endemic steady state. The trace of J(E is tr(j(e = (µ βi I < 0. Thus tr(j(e < 0. Also, using the identity obtained by the endemic steady state we obtain Λ = µ( I αi, (4 5
det(j(e = (µ (α µi βi ( I 2 = βλi ( I 2 > 0. Thus det(j(e > 0. Here, the eigenvalues of the Jacobian matrix J(E have negative real parts. This means that E is asymptotically stable whenever it exists. This establishes Theorem 4.1. 4.1 Dulac s criteria Now, we prove the global stability of endemic steady state E whenever it exists, using the Dulac criterion and the Poincaré-Bendixson theorem. Theorem 4.2. If R 0 > 1, then the unique endemic steady state E of (1 is globally asymptotically stable in the interior of Γ. Proof. We first rule out periodic orbits in Γ using Dulac s criteria (ee [1]. Denote the right-hand side of (1 by (P (, I, Q(, I and we construct Dulac function Then we have Φ(, I =, > 0, I > 0. βi (ΦP (ΦQ I (Λ φi = β 2 µ βi (α µ φ, β for all > 0, I > 0. Thus, system (1 does not have a limit cycle in Γ. From Theorem 4.1, if R 0 > 1 holds then E is locally asymptotically stable. A simple application of the classical Poincaré-Bendixson theorem and Theorem 2.1, it suffices to show that the unique endemic steady state E is globally asymptotically stable in the region Γ. This proves Theorem 4.2. 4.2 Method of Lyapunov Functions Finally in this section, we use the method of Lyapunov to give another proof of Theorem 4.2. Proof. Define L : {(, I Γ :, I > 0} R by [ ( ] L(, I = ( (I I ( I ln I (α 2µ( I ( ln I β I I I 1. (5 (6 6
Then L is C 1 on the interior of Γ, E is the global minimum of L on Γ, and L(, I = 0. Computing the derivative of (4.2 along the solutions of system (1, we obtain L (, I = [( (I I ] d( = [( (I I ] (α 2µ( I β (α 2µ( I β (Λ µ( αi (I I I 2 ( βi (φ µ αi. (I I di I 2, Using (3 and (4, we have L (, I = [( (I I ] ( µ( (α µ(i I (α 2µ( I (I ( I I I. Notice that, Thus, I = I( (I I ( I ( L (, I = [( (I I ] (α 2µ( I (I I = µ ( 2 ( I ( (I I 2.. (7 ( µ( (α µ(i I ( I( (I I, I ( α µ (α 2µ I Clearly, L (, I < 0 always holds except at the endemic steady state, E. Furthermore, L(, I as 0 or, and L(, I as I 0 or I. Therefore, we may conclude that function (4.2 is a Lyapunov function for system (1 and that, by the Lyapunov asymptotic stability theorem [9], the endemic steady state E is globally asymptotically stable in the interior of Γ, when it exists. This proves Theorem 4.2. Remark 1. In order to determine L (, I, the equation for d used directly. given in (1 is not 7
5 Another auxiliary functions As is well known, that there may exist more of a Dulac function for a given system and that the Lyapunov functions are never unique. In this section, we present the constructions of novel Dulac functions and Lyapunov functions to study global properties of the model (1. We proved the Theorem 4.2 based on two good choices of a Dulac function. Taking the Dulac function D(, I = β 1 1 I 1 for system (1 and noting that (DP (DQ I = (Λ φi β 2, I we can see that system (1 has no periodic orbits in the interior of the first quadrant. Hence the endemic steady state (, I is globally asymptotically stable in the interior of the first quadrant. We find other Dulac function, B(, I = β 1 I 1. Then it follows that (BP (BQ I the argument is similar to that given above. = µ βi 1, Remark 2. Dulac functions D(, I and B(, I are commonly used in two-dimensional population models. We proved the Theorem 4.2 using another Lyapunov functions. The second function is given by φ W (, I = 2(α 2µ [( (I I ] 2 βi [ µ( I ( (I I ( I ln ( ln ( (α 2µ I µ ] ( ( I ( I I I ln I I Then the time derivative of W computed along solution of the system (1 is. W φ (, I = (α 2µ [( (I I ] (Λ µ( αi βi [( (I I ] µ( I (Λ µ( αi ( Λ ( βi µ φ I ( ( (α 2µ β (I I (φ µ α. I µ 8
Using (3, (4 and the following identity obtained of endemic steady state we get µ = Λ βi I φ I, (8 W φ (, I = (α 2µ [( (I I ] ( µ( (α µ(i I Furthermore, we have Using (9 (11, we get βi [( (I I ] µ( I ( µ( (α µ(i I ( ( 1 ( Λ 1 ( I β I I ( I φ I ( ( (α 2µ β (I I I µ I. I I I I = (I I I (, (9 I = I ( (I I ( I, (10 ( I = I ( (I I ( I (. (11 W φ (, I = (α 2µ [( (I I ] ( µ( (α µ(i I βi [( (I I ] µ( I ( µ( (α µ(i I ( ( Λ ( ( I ( (I I β ( I ( ( (I I φ( I ( ( ( (α 2µ I β (I I ( (I I I µ ( I. ( Cancelling identical terms with opposite signs, yields ( W µφ (Λ φi (, I = (α 2µ ( ( β (α µ I I µ φ(α µ (α 2µ (I I 2. ( 2 (α 2µ (I I 2 I µ 9
Hence W (, I is the negative definite. imilarly by the Lyapunov asymptotic stability theorem [9], then implies that E is globally asymptotically stable in the interior of Γ. Remark 3. We note that a Lyapunov function that directly use the equation d (1, difficult the construction of a suitable Lyapunov function. of 6 Discussion In this paper, we have studied the I epidemic model with standard incidence rate. Our analysis establishes that the global stability of the I epidemic model is completely determined by the basic reproductive number. If the basic reproductive number is less than one, will only be a disease-free steady state, which is globally asymptotically stable in the feasible region; and the disease will die out from the population irrespective to the initial conditions. The proof is based on the construction of a power Lyapunov function. If the basic reproductive number is greater than one, a unique endemic steady state exists, which is globally stable in the interior of the feasible region, and the disease is present in the population and will become endemic. The proof is based on the construction of a Dulac function and applying Dulac s criterion and the Poincaré- Bendixson theorem, and a second proof is based on the second Lyapunov method. We constructed three Dulac functions and two global Lyapunov functions to study the global stability of the endemic steady state. The Lyapunov functions are obtained by the suitable combination of known functions in the literature and of novel functions. These constructions are an elegant example of the non-uniqueness of Dulac functions and Lyapunov functions in epidemiological models. Acknowledgements We would like to thank the anonymous referees for their valuable comments and suggestions. References [1] J.K. Hale, Ordinary Differential Equations, John Wiley&ons, New York, 1969. [2] J. La alle,. Lefschetz, tability by Liapunov s Direct Method with Applications, Academic Press, New York, 1961. [3] E. Beretta, V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Appl., Part A, 12 (1986 677-694. [4] A. Korobeinikov, G. C. Wake, Lyapunov functions and global stability for IR, IR, and I epidemiological models, Appl. Math. Lett. 15 No. 8 (2002 955-960. 10
[5] C. Vargas-De-León, Constructions of Lyapunov Functions for Classic I, IR and IR Epidemic models with Variable Population ize. Foro-Red- Mat: Revista Electrónica de Contenido Matemático 26 (2009. http://www.redmat.unam.mx/foro/volumenes/vol026. [6] C. Vargas-De-León, On the Global tability of I, IR and IR Epidemic Models with tandard Incidence. Chaos, olitons and Fractals, (2011, DOI: 10.1016/j.chaos.2011.09.002. [7] J. Zhou, H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994 809 834. [8] G. Birkhoff and G.C. Rota, Ordinary Differential Equations, Ginn Boston, 1962. [9] A.M. Lyapunov, The General Problem of the tability of Motion, Taylor and Francis, London, 1992. Please cite this article as: C. Vargas-De-León, tability Analysis of a I Epidemic Model with tandard Incidence. Foro-Red-Mat: Revista Electrónica de Contenido Matemático Vol 28, Num. 4, 1 11 (2011. http://www.red-mat.unam.mx/foro/volumenes/vol028. 11