Global Analysis of an SEIRS Model with Saturating Contact Rate 1

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 80, Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Shulin Sun a, Cuihua Guo b, and Chengmin Li a a School of Mathematics and Computer Science Shanxi Normal University, Linfen Shanxi, , China b School of Mathematical Science Shanxi University, Taiyuan, Shanxi, , China sunsl 2004@yahoo.com.cn (Shulin Sun) gchzjq@yahoo.com.cn (Cuihua Guo) Shxshdlicm@163.com (Chengmin Li) Abstract The global properties of an SEIRS model with saturating contact rate are investigated systematically. We have proved that when the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Further, a unique endemic equilibrium exists for some conditions. Using a Krasnoselskii sub-linearity trick, it is shown that the endemic equilibrium is locally asymptotically stable for a special case. Finally, a non-linear Lyapunov function is used to show the globally asymptotic stability of the endemic equilibrium. Numerical simulations illustrate the main results in the paper. Keywords: SEIRS model; Saturating contact rate; Global stability; Lyapunov function 1 Introduction Mathematical model is a useful tool to analyze the spread and control of infections disease. These models always take the form of a non-linear ordinary differential equations(odes). However,global stability properties of these systems are generally a difficult problem. The global stability of SIR or SIRS 1 Foundation item: This work is supported by the National Natural Science Foundation of China (No , No ), the Youth Science and Technology Research Foundation of Shanxi (No ) and the Natural Science Foundation of Shanxi Normal University (No.ZR1003).

2 3992 Shulin Sun, Cuihua Guo and Chengmin Li models, in relation with the basic reproduction number, is known since eighties. Global stabilities for SEIRS and SEIS have long been conjectured. This was solved in 1995 by Li and Muldowney [1] using the Poincare-Bendixson properties of competitive systems in dimensions three combined with sophisticated use of compound matrices. But in this paper, the global dynamics of the model is resolved through the use of Lyapunov function. It should be pointed out that this kind of Lyapunov function has long history of applications to Lotka-Volterra models and was originally discovered by Volterra. The global stability of SEI model is studied by S.Bowong [2]. Sun et al [3] provided the global stability of an SEIR model with nonlinear incidence rate. Dessalegn [4] studied an SEI m RS model with standard incidence rate. Zhang and Ma [5] systematically analyze the global dynamics of SEIR model with saturating incidence rate. In this paper, we focus on the SEIRS model with saturating incidence rate. This study assumes that the recovered individuals eventually lose their infection-acquired immunity and become fully susceptible again. Finally, the global stability of the model is resolved through the use of Lyapunov functions. This paper is organized as follows. The model is formulated in Section 2. The global asymptotic stability of the disease-free equilibrium is established in Section 3. In Section 4, a sub-linearity trick [8] is used to establish the locally asymptotic stability, and a Lyapunov function is used to prove its global asymptotic stability for a special case. 2 The model and preliminary The total population size N(t) is divided into susceptible (S(t)), exposed (E(t)), infectious (I(t)) and recovered (R(t)). Let π be the recruitment of individuals (assumed susceptible) into the population. The natural death rate is denoted by. δ is the disease-induced death rate. The recovered individuals lose their infection-acquired immunity at a rate θ and become fully susceptible again. Let σ E be the constant rate at which the exposed individuals become infective, so that 1 σ E is the mean latent period. Let σ I be the constant rate for recovery, so that 1 σ I is the mean infective period. The susceptible individuals are infected at a rate of λ, where λ = βi 1+ωN (1) where β is the effective contact rate. Using the above definition and assumptions, we derive the following SEIRS

3 Global analysis of an SEIRS model 3993 model with saturating incidence rate ds βsi = π + θr dt S 1+ωN de = βsi σ dt 1+ωN EE E di = σ dt EE σ I I I δi = σ I I θr R dr dt where π, θ,, β, σ E,σ I,δ,ω are assumed to be positive constant. Consider the biologically-feasible region D = {(S, E, I, R) R+ 4 : S + E + I+R π }. It follows from the system (2) that dn = π (S+E+I+R) δi dt π N. Thus, dn < 0 for N> π. Since dn π N, it can also be shown, dt dt using a standard comparison theorem, that N(t) N(0)e t + π (1 e t ). If N(0) π, thenn(t) π. Hence, the set D is positively-invariant. This result is summarized in below. Lemma 1. The region D is positively-invariant for the model (2). It is convenient to define the region (the stable manifold of the disease-free equilibrium (DFE)) D 0 = {(S, E, I, R) D : E = I = R =0}. Lemma 2[4]. Let the initial data S(0) > 0,E(0) > 0,I(0) > 0,R(0) > 0. Then, the solution (S(t),E(t),I(t),R(t)) of model (2) is postive for all t>0. The consequence of the above result is that it is sufficient to consider the dynamics of the flow generated by system (2) in the region D, where the model can be considered to be epidemiologically and mathematically well-posed. 3 Disease-free equilibrium(dfe) The model (2) has a DFE given by (2) ε 0 =(S 0,E 0,I 0,R 0 )=( π, 0, 0, 0). (3) The stability of ε 0 is studied using the next generation method [9]. The associated matrix F (of the new infection terms) and the M-matrix V (of the remaining transfer terms) are given as follows, respectively F = ( 0 βs 0 1+ωS ), V = ( k0 0 σ E k 1 where S 0 = π, k 0 = σ E +, k 1 = σ I + + δ. Clearly, F is non-negative, V is a non-singular M-matrix, V F has Z sign pattern. The associated basic reproduction number, denoted by R 0, is then given by R 0 = ρ(fv 1 ), where ρ is the spectral radius of FV 1. It follows that ), R 0 = βs 0 σ E. (4) 1+ωS 0 k 0 k 1

4 3994 Shulin Sun, Cuihua Guo and Chengmin Li The result below follows from Theorem 2 of [9]. Lemma 3. The DFE, ε 0, of the model (2)is locally-asymptotically stable (LAS) if R 0 < 1, and unstable if R 0 > 1. The quantity R 0 refers to the average number of secondary cases generated by a single infectious individual in a completely susceptible population. Lemma 3 shows that if R 0 < 1, a small influx of infectious individuals will not generate large outbreak of the disease. For disease elimination to be independent of the initial number of infectious individuals, a globally asymptotic stability has to be established for the DFE (for the case when R 0 < 1). This is done below. Theorem 1. The DFE, ε 0, of the model (2), is global asymptotically stable (GAS) in D if R 0 1. Proof. Consider the linear Lyapunov function V = σ E E + k 0 I. (5) Its time derivative along the solutions of system (2) satisfies V = σ E Ė + k 0I βsi = σe 1+ωN k 0k 1 I. (6) Now, using ( βn 1+ωN ) > 0, N π = S βs 0, we have βs 0 1+ωN 1+ωS 0. With this in mind, Eq.(6) becomes βs 0 βs 0 σ E V σ E I k 0 k 1 I = k 0 k 1 I( 1) = k 0 k 1 I(R 0 1). 1+ωS 0 1+ωS 0 k 0 k 1 Therefore, V 0 for R0 1 and V = 0 if and only if I = 0. Further, substituting I = 0 into the equation for Ṡ,Ė and Ṙ in (2) shows that S(t) π, E(t) 0 and R(t) 0as t. It follows that, by the LaSalle s Invariance Principle, every solution to the system (2) with initial conditions in D approaches ε 0 as t. Thus, since the region D is positively-invariant, the DFE, ε 0, is GAS in D if R 0 1. The epidemiological implication of the above is that the disease can be eliminated from the community if the threshold quantity R 0 1. The result of Theorem 1 is illustrated numerically by simulating the model (2)(Fig. 1). 4 Endemic equilibrium point (EEP) Let ε 1 =(S,E,I,R ) be the positive equilibrium of model (2). Then π + θr βs I S =0 1+ωN βs I (σ 1+ωN E + )E =0 σ E E (σ I + + δ)i =0 σ I I (θ + )R =0 (7)

5 Global analysis of an SEIRS model I(t) t Fig. 1: Simulations of model (2), with π = 8,θ = 0.012,β = 0.15,ω = 1, = ,σ E =0.5,σ I =0.2and δ = (so that R 0 < 1), and various initial conditions. Adding the equations in the model (7) gives π N δi =0. (8) Using Eq.(8), one can express S,E,I and R in terms of N in the forms: S = π+ θσ I (π N ) (θ+)δ β(π N ) (1+ωN )δ + E = (σ I ++δ)(π N ) σ E δ I = π N δ R = σ I (π N ) (θ+)δ Substituting (9) in the second equation of (7) yields (9) π N F (N )=0, (10) δ where F (N )= βs σ E. Clearly, N = π is a zero point of (10), which corresponds to the diseasefree equilibrium ε 0. Since N [0,S 0 ], one has 1+ωN (σ E+)(σ I ++δ) F (0) = βσ Eπ(θδ + δ + θσ I ) (σ E + )(σ I + δ + )(θ + )(βπ + δ) σ E (θ + )(βπ + δ) < 0, F (S 0 )= βs 0 1+ωS 0 (σ E + )(σ I + + δ) σ E = (σ E + )(σ I + + δ) σ E (R 0 1). Obviously, F (S 0 ) > 0 for R 0 > 1. The existence follows from the intermediate value theorem.

6 3996 Shulin Sun, Cuihua Guo and Chengmin Li Suppose 1 ω <πand β ωδ > σ I. We have F (N βs )=( 1+ωN ) = δβ[π(θ + )(β ωδ) θσ I( + ωπ)] (θ + )[β(π N )+δ(1 + ωn )] > 0. 2 Now, F (N ) = 0 has only one positive root in the interval [0,S 0 ]. Thus, we have the following result: Theorem 2. Suppose that R 0 > 1, system (2) has a unique endemic equilibrium ε 1 =(S,E,I,R ) with coordinates satisfying (9) if 1 ω <πand β ωδ > σ I. (11) 4.1 Local asymptotic stability of EEP: Special case The locally asymptotic stability of the EEP is explored for a special case, where the disease-induced death rate δ is assumed to be negligible and is set to zero. Thus, under this setting (with δ = 0), the rate of change of the total population of the model (2) becomes dn = π N. Hence, N(t) π = N dt as t. Substituting N = N into (1) gives λ = βi. (12) 1+ωN Furthermore, it can be shown that the model (2) with δ = 0 and λ as given in (12), has the associated reproduction number (noting that S 0 = N = π ), given by R 0 = βs 0 1+ωS 0 σ E k 0 k 1, (13) where k 0 = σ E + and k 1 = σ I +. Additionally, it can also be shown that the model (2), under the above setting, has a unique endemic equilibrium, ε 1, of the form ε 1 =(S,E,I,R ), whenever R 0 > 1 and that 1 ω <π,β>σ I. It follows from the expression N = N S = N E I R. (14) Substituting(12), (14) into the system (2) gives the following reduced system Ė = λ(n E I R) k 0 E I = σ E E k 1 I (15) Ṙ = σ I I k θ R where k 0 = σ E +, k 1 = σ I +, k θ = θ +. we claim the following.

7 Global analysis of an SEIRS model 3997 Theorem 3. The unique endemic equilibrium of the reduced model (15), ε 1 =(E,I,R ), is LAS if R 0 > 1 and <π, β>σ 1 ω I. Proof. The proof of the above theorem is based on the krasnoselskii sublinearity, as given in Hethcote and Thieme [8] (see also [6,7]). The method entails showing that the linearized system of the system (where a prime denotes the differentiation with respect to t) given by (where x is an equilibrium solution of (16)) x = f(x) (16) has no solution of the form Z (t) =Df( x)z Z(t) =Z 0 e αt (17) with Z 0 C 3 \{0}, Z 0 = (Z 0,Z 1,Z 2 ),Z i,α C and Re(α) 0, where C denotes the complex number. Linearizing the system (15) at the endemic equilibrium ε 1, gives where λ = given by which satisfies where Ė =( λ k 0 )E +( βs λ )I λ R 1+ωN I = σ E E k 1 I Ṙ = σ I I k θ R (18) βi. Hence, the EEP, ε 1+ωN 1 =(E,I,R ), of system (15) is H = E = I = σ E k1 E R = σ I k θ I 0 σ E βs I k 0 (1+ωN ) ε 1 = Hε 1, (19) βs k 0 0 (1+ωN ) k1 0 0 σ 0 I k θ 0 Furthermore, substituting a solution of the form (17) into the linearized system (18) at equilibrium gives. αz 0 =( λ k 0 )Z 0 +( βs 1+ωN λ )Z 1 λ Z 2 αz 1 = σ E Z 0 k 1 Z 1 αz 2 = σ I Z 1 k θ Z 2. (20)

8 3998 Shulin Sun, Cuihua Guo and Chengmin Li System (20) can be simplified to give (1 + λ +α βs k 0 )Z 0 = Z k 0 (1+ωN ) 1 λ k 0 (Z 1 + Z 2 ) (1 + α k 1 )Z 1 = σ E k1 Z 0. (21) (1 + α k θ )Z 2 = σ I k θ Z 1 Solving for Z 1 and Z 2 in terms of Z 0, gives { Z1 = σ E Z 2 = α+k 1 Z 0 σ E σ I Z (α+k 1 )(α+k θ ) 0. (22) Substituting (22) into the first equation in (21), and simplifying, gives βs [1 + F 1 (α)]z 0 = Z k 0 (1+ωN ) 1 =(HZ) 1 [1 + F 2 (α)]z 1 = σ E k1 Z 0 =(HZ) 2. (23) [1 + F 3 (α)]z 2 = σ I k θ Z 1 =(HZ) 3 System (23) has the general form where [1 + F i (α)]z i 1 =(HZ) i,i=1, 2, 3, (24) F 1 (α) = λ +α k 0 + λ σ E k 0 ( 1 α+k 1 + σ I (α+k 1 )(α+k θ ) ) F 2 (α) = α k 1. (25) F 3 (α) = α k θ This notation (HZ) i (i =1, 2, 3) denotes the ith coordinate of the vector HZ, and Z i 1 (i =1, 2, 3) is as given in (22). We have that all entries of the matrix H are non-negative, satisfying ε 1 = Hε 1, with all coordinates of ε 1 positive. Hence, if Z is any solution of (24), then it is possible to find a minimal positive real number s, depending on Z, such that (see [6,7]) Z sε 1 (26) where Z =( Z 0, Z 1, Z 2 ) and. is a norm in C. Then aim here is to show that if Re(α) < 0, then the linearized system (18) has a solution of the form (17). Now, we need to show that Re(α) 0 is not satisfied which will then be sufficient to conclude that Re(α) < 0. Hence, we have two general cases for Re(α), namely α = 0 and α 0. Case (i): α =0. For α = 0, then system given in (20) becomes a homogeneous linear system of the form 0 = GZ i, i =0, 1, 2;

9 Global analysis of an SEIRS model 3999 where G = λ k 0 βs 1+ωN λ λ σ E k σ I k θ. βs det(g) = k 0 k 1 k θ k 1 k θ λ (1 + σ E + σ Eσ I )+σ E k θ. (27) k 1 k 1 k θ 1+ωN Solving the equation in (15) at the endemic steady-state, gives { k0 E = λ S I = σ E k 0 k 1 λ S (28) Furthermore, σ E βs =1. (29) k 0 k 1 1+ωN It follows from (29) that det(g) < 0. Since det(g) < 0, it follows that system (20), which has form of a system given in (24), can only have a unique solution Z = 0 (it corresponds to the DFE). Case(ii): α 0. By assumption, we have that Re(α) 0. Then, the remaining task is to show that the system has no non-trivial solution when Re(α) > 0. Clearly, we have that F i (α)+1 > 1, i =1, 2, 3. We then define F (α) = min{( F i (α)+ s 1 ),i =1, 2, 3}. Then, F (α) > 1 and hence <s. Since H has all nonnegative entries, by taking norms on both sides of (24), it follows from F (α) F (α) [1 + F i (α)] that Then, by using (26) and (19) in (30), we get F (α) Z H Z. (30) F (α) Z H Z shε 1 = sε 1, so that, Z s F (α) ε 1 <sε 1, which contradicts the fact that s is minimal. Hence, Re(α) < 0. Thus, the endemic equilibria, ε 1, is LAS. This completes the proof. 4.2 Global stability of EEP: Special case. As in the case of the local stability proof, the global stability of the endemic equilibrium ε 1, is explored for the special case with δ = 0 and λ as defined in (12). We claim the following. Theorem 4. The unique EEP, ε 1, of the reduced model (15) is GAS in D\D 0 whenever R 0 > 1, <π, β>σ 1 ω I and 2 E E R S + S R 0. E ER S SR

10 4000 Shulin Sun, Cuihua Guo and Chengmin Li Proof. Let 2 E Lyapunov function: E E R ER S S + S R SR 0. Consider the following V =(S S ln S)+(E E ln E)+a(I I ln I)+b(R R ln R), where the coefficients a and b are positive constants to be determined. Thus, V =(1 S S E I )Ṡ +(1 )Ė + a(1 E I ) I + b(1 R )Ṙ. (31) R Substituting the expressions of derivatives from system (2)at endemic steady state, gives V = S (2 S S S S )+[βs I 1+ωN (S ) 2 θr (1 R S R S +b(σ I I k θ R σ I I R R Setting the coefficients S βi + βs I ] 1+ωN 1+ωN + S R )+a(σ SR E E k 1 I σ E E I + k θr ) k 0 E E βsi E + k I 1 I ) + k 1+ωN 0E. (32) βs I θr a = + (1 + ωn )k 1 I k 1 I, b = θ, k θ (33) in (32),and simplifying, gives V = S (2 S S )+ βs I (3 S S S 1+ωN S I E IE E SI(1+ωN ) ) ES I (1+ωN) + S R). SR +θr (3 E R IR I E ) θr (2 E E R S ER I R IE E ER S The following inequalities hold: (34) 2 S S S S 3 S S 0, I E E SI(1+ωN ) 3 3 IE ES I 3 (1+ωN) 3 E R ER IR I R I E IE 0. 1+ωN 1+ωN 0, (35) Hence, using (35) and noting that 2 E E R S + S R 0,it E ER S SR follows from (34) that V 0. Therefore, by the Lyapunov function and the LaSalle s principle, every solution to the equation in the model (2), approaches the endemic equilibrium, ε 1,ast for R 0 > 1, <π, β>σ 1 ω I and 2 E E R S + S R 0. E ER S SR It is worth noting that the additional condition 2 E E R S S R SR E ER S + 0 is not necessary if the disease confers permanent immunity against re-infection (since, in this case, θ = 0, and the forth term of (34) vanishes). Numerical simulations of the reduced model (15), using the parameter values π =8,β =0.48,ω =1,=0.0027,σ E =0.5,σ I =0.2, β =0.48 and δ = 0 (so

11 Global analysis of an SEIRS model 4001 that R 0 > 1). The result obtained, depicted in Fig.2, shows convergence to the unique EEP ε 1. That is, all solutions satisfy that I(t) I, (t + ). Further extensive numerical simulations of the model (2) suggests the following conjecture. See Fig. 3, all solutions satisfy that I(t) I, (t + ). Conjecture 1. The endemic equilibrium of the model (2), given by ε 1,is GAS in D\D 0 whenever R 0 > 1. (a) 250 (b) I(t) I(t) ,000 1,500 t t Fig. 2: Simulations of model (15) with various initial conditions. π = 8,β = 0.48,ω =1, =0.0027,σ E =0.5,σ I =0.2. (a) δ =0,θ = 0 (so that R 0 > 1); (b) δ =0,θ =0.012 (so that R 0 > 1). (c) (d) I(t) I(t) ,000 1,500 t t Fig. 3: Simulations of model (2) with various initial conditions. π =8,β =0.48,ω = 1, =0.0027,σ E =0.5,σ I =0.2. (c) δ =0.0062,θ = 0 (so that R 0 > 1); (d) δ =0.0062,θ =0.012 (so that R 0 > 1). 5 Conclusions In this paper, an SEIRS deterministic model with saturating contact rate is designed. Some of the main findings of this study are:

12 4002 Shulin Sun, Cuihua Guo and Chengmin Li (i) The model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity; (ii) The model has a unique endemic equilibrium under certain conditions. Under these conditions, the endemic equilibrium is locally-asymptotically stable for a special case whenever the associated reproduction number exceeds unity. The endemic equilibrium is shown to be globally-asymptotically stable for the special case. (iii) Numerical simulations illustrate that whether the disease-induced death rate δ and The lose-immunity rate θ is zero or not, which does not affect the stability of the endemic equilibrium but affect the location of the endemic equilibrium. However, the analysis proof of this result is not easy which will be accomplished in future. From epidemiological point of view, this study shows that the disease being considered can be eliminated from the population whenever the associated reproduction number is brought to (and maintained at) a value less unity. The disease will persist in the community whenever the production number exceeds unity. References [1] M.Y.Li, J.S.Muldoweney, Global stability for SEIR model in epidemiology, Math Biosci, 125(1995), [2] S.Bowong, J.J.Tewa, Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate, Commun Nonlinear Sci Numer Simulat,15(2010), [3] C.Sun, Y.Lin and S.Tang, Global stability for a special SEIR epidemic model with nonlinear incidence rate, Chaos, Soliton. Fract., 33(2007), [4] D.Y.Melesse, A.B.Gumel, Global asymptotic properties of SEIRS model with multiple infectious stage, Journal of Mathematical Analysis and Applications, 366(2010), [5] J. Zhang, Z.E. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Math Biosci, 185(2003), [6] L.Esteva, A.B.Gumel and C.V.De Leon, Qualitative study of transmission dynamics of drug-resistant malaria, Math. Comput. Modelling, 50(2009), [7] L.Esteva, C.Vergas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math Biosci,167(2000),

13 Global analysis of an SEIRS model 4003 [8] H.W.Hethcote, H.R.Thieme, Stability of the endemic eauilibrium in epidemic models with subpopulations, Math Biosci,75(1985), [9] P.van den Driessche, J.Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models disease transmission, Math. Biosci,180(2002), Received: March, 2012