GLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT

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1 Electronic Journal of Differential Equations, Vol (2015), No. 304, pp SSN: URL: or ftp ejde.math.txstate.edu GLOBAL STABLTY OF SR MODELS WTH NONLNEAR NCDENCE AND DSCONTNUOUS TREATMENT ME YANG, FUQN SUN Abstract. n this article, we study an SR model with nonlinear incidence rate. By defining the Filippov solution for the model and constructing suitable Lyapunov functions, we show that the global dynamics are fully determined by the basic reproduction number R 0, under certain conditions on the incidence rate and treatment functions. When R 0 1 the disease-free equilibrium is globally asymptotically stable, and when R 0 > 1 the unique endemic equilibrium is globally asymptotically stable. 1. ntroduction The asymptotic behavior of SR models with the general nonlinear incidence rate have been studied by many researchers; see for example 5, 6, 7, 9, 11, 12 and the references therein. One of the basic SR epidemic models is described as follows: ds = µ f(s(t), (t)) µs(t), d = f(s(t), (t)) (µ + σ)(t), dr = σ(t) µr(t), where the host population size is constant and is divided into three classes: susceptible, infectious and recovered. We denote these classes by S(t), (t) and R(t), respectively (that is, S + + R = 1). The positive constant µ represents the birth/death rate and the positive constant σ represents the recovery rate. The function f(s, ) denotes the nonlinear incidence rate. t is assumed that all newborns are susceptible, and the immunity received upon recovery is permanent. A number of works display the threshold behavior of the model. That is, for a basic reproduction number R 0 of some form, the disease dies out for R 0 1, whereas it is permanent for R 0 > 1. Note that treatment is an essential measure for the precautions taken of some disease (e.g. measles, phthisis, influenza). n 10, the SR model with a limited resource for treatment is studied. n 11, the SR model with a constant removal rate 2010 Mathematics Subject Classification. 34A36, 34A60, 34D23. Key words and phrases. Filippov solution; discontinuous treatment; reproduction number; asymptotic stability. c 2015 Texas State University. Submitted November 3, Published December 16,

2 2 M. YANG, F. SUN EJDE-2015/304 of infective individuals is analyzed. Many classic epidemic models with treatment are modelled by continuous systems. However, it is known that the therapeutic measures vary from each period of the transmission. For example, a small number of infectives can not get treatment timely and completely during the early spread for lacking of high attention paid by the society. After a period of time, when people are aware of the seriousness, the therapeutic measures would be enhanced and improved sharply. The treatment function related to the number of the infective should be piecewise continuous. Thus, it is meaningful to introduce discontinuous treatment into classical infectious disease model to reinforce the efficacy on prevention and treatment of the epidemic. n this article, we consider the following 2-dimensional SR model with a class of nonlinear incidence rate of S(t)g((t)) and discontinuous treatment: d ds = µ S(t)g((t)) µs(t), = S(t)g((t)) (µ + σ)(t) h((t)), (1.1) where the function h() denotes the removal rate of infective individuals because of the treatment of infectives. The initial condition for system (1.1) is S(0) = S 0 0, (0) = 0 0, (1.2) The purpose of our study is to show the threshold behavior of system (1.1) using stability theory based on the Filippov solution 1, 2, 3, 4. The article is organized as follows. n Section 2, some elementary assumptions on the functions g and h will be given, and the basic reproduction number R 0 is provided. After defining the Filippov solution for a given initial condition, the equilibrium points are discussed. The local and global stability of equilibrium points are analyzed in Sections 3 and 4, respectively. 2. Basic reproduction number and Equilibrium To define the basic reproduction number R 0 and indicate the existence of equilibriums, we give some hypotheses. (H1) h((t)) = ϕ(), where ϕ : R + R + is piecewise continuous and monotone nondecreasing; that is ϕ is continuous apart from a countable number of isolated points {ξ k }, ϕ(ξ + k ) > ϕ(ξ k ) holds, where ϕ(ξ k ) and ϕ(ξ+ k ) represent the left and right limits of ϕ at {ξ k }, respectively, and have only finite number of discontinuous points in any compact subset of R +. We assume that ϕ is continuous for = 0. (H2) g(0) = 0, g () > 0 and g () 0 for 0. Furthermore, we assume that the function φ() = g()/ is bounded. Remark 2.1. (1) We modify the definition of treatment rate to be h((t)) = ϕ(), which means that the treatment rate is proportional to the number of the infectives. (2) Because of (H2), we have that φ() is a monotone decreasing function on > 0. (3) By the assumptions, it is easy to find that system (1.1) always has a diseasefree equilibrium point E 0 = (1, 0).

3 EJDE-2015/304 GLOBAL STABLTY OF SR MODELS 3 Then we define the basic reproduction number R 0 for model (1.1) as R 0 = g (0) µ + σ + ϕ(0). We shall assume that (H1) and (H2) hold in the rest of this article. Let us recall the definition of the Filippov solution to system (1.1) with initial condition (1.2). Definition 2.2 (1, 2, 3, 4). A vector function (S(t), (t)) defined on 0, T ), T (0, +, is called the Filippov solution of (1.1)-(1.2), if it is absolutely continuous on any subinterval t 1, t 2 of 0, T ), S(0) = S 0, (0) = 0, and satisfies the differential inclusion ds = µ Sg() µs, (2.1) d Sg() (µ + σ) coh(), for almost all of t 0, T ), where coh() is the interval h( ), h( + ), and h( ) and h( + ) represent the left and right limits of function h( ) at, respectively. By noting that the equilibrium (S, ) of (2.1) satisfies 0 = µ S g( ) µs, 0 S g( ) (µ + σ) coh( ), from the measurable selection theorem (see 2), there exists a unique constant such that ξ = S g( ) (µ + σ) coh( ) µ S g( ) µs = 0, S g( ) (µ + σ) ξ = 0. The following proposition shows that for positive initial values, the solution of (1.1) is positive and is bounded. T = implies that the solution exists globally. Proposition 2.3. Let (S(t) and (t)) be the unique solution of (1.1)-(1.2). Then S(t) and (t) are positive for all t > 0. Moreover, this solution is bounded and thus exists globally. Proof. By the fact ds/() S=0 > 0 and S 0 0, we obtain the positivity of S(t). Since coh(0) = {0}, and ϕ is continuous at = 0, there exists δ > 0 such that ϕ() is continuous for < δ. Also, the second differential inclusion in (2.1) can be rewritten as d Sg() = (µ + σ) ϕ(), (2.2) for < δ. f 0 = 0, we derive that (t) = 0, for all t 0, T ). f 0 > 0, then we claim (t) > 0, for all t 0, T ). Otherwise, let t 1 = inf{t (t) = 0} 0, T ). Because of the continuity of (t) on 0, T ), there exists θ > 0 such that t 1 θ > 0 and 0 < (t) < δ for t t 1 θ, t 1 ). ntegrating both sides of (2.2) from t 1 θ to t 1 gives R t t 0 = (t 1 ) = (t 1 θ)e 1 θ Sg(ξ) ξ (µ+σ) ϕ(ξ)dξ > 0, which is a contradiction. Thus, we have (t) > 0 for t 0, T ). Since semi-continuous set-valued mapping (S, ) (µ Sg() µs, Sg() (µ+ σ) coh()) has compact and convex image, we have that (1.1) has a solution

4 4 M. YANG, F. SUN EJDE-2015/304 (S(t), (t)) defined on 0, t 0 ), which satisfies the initial condition (1.2). From (2.1), we know that d(s + ) µ µs (µ + σ) coh(), for any ν coh(). f S + > 1, then we have µ µ(s + ) σ ν 0. Thus, 0 S + max{s 0 + 0, 1}, which yields the boundedness of (S(t), (t)) on 0, t 0 ). Moreover, (S(t), (t)) is defined and bounded on 0, + ). The following result concerns the existence and uniqueness of an endemic equilibrium. Proposition 2.4. f R 0 > 1, then there exists an unique endemic equilibrium E. Proof. We look for solutions (S, ) of the differential inclusion ds/(d) = 0 and 0 d/(): S µ = µ + g( ), 0 µg( ) µ + g( ) (µ + σ) coϕ( ). Let L() = µg() (µ + σ) coϕ(). µ + g() f R 0 > 1, then g (0) > µ + σ + ϕ(0) so that L(0) = g (0) (µ + σ) > ϕ(0) 0. A direct calculation gives L µ () = µ + g() 2 2 µg () µg() g 2 () µ = µ + g() 2 2 g ()µ µg (ξ) g 2 (), where, ξ (0, ). Observe that g(0) = 0 and g () 0 for > 0, we have L () < 0. That is, L() is monotonously decreasing on > 0. Since ϕ() is nondecreasing for > 0, and L() 0 for (µ + σ)µ + g()/µg(), the set { L() ϕ( + ), > 0} should be bounded. Let Ĩ = sup{ L() ϕ(+ ), > 0}, then L(Ĩ) ϕ(ĩ ). We claim that L(Ĩ) ϕ(ĩ ), ϕ(ĩ+ ). Suppose on the contrary, there exists a constant δ > 0 such that L(Ĩ + δ) > ϕ(ĩ + δ) = ϕ((ĩ + δ)+ ), which contradicts to the definition of Ĩ. This leads to L(Ĩ) ϕ(ĩ ), ϕ(ĩ+ ) and Ĩ is a positive solution to L() coϕ(). Next, we prove the uniqueness of E. Let 1 = Ĩ and 2 1 is another solution to L() coϕ(). There exist η i coϕ i () (i = 1, 2) satisfying µm 1 (µ + σ) η 1 = 0, (µ + m 1 ) 1 µm 2 (µ + σ) η 2 = 0, (µ + m 2 ) 2 where we denote m i = g( i ) (i = 1, 2) for convenience such that µ = (µ + σ + η 1 ) ( 1 + µ m 1 ) 1, µ = (µ + σ + η 2 ) ( 1 + µ m 2 ) 2.

5 EJDE-2015/304 GLOBAL STABLTY OF SR MODELS 5 Thus, we have where 0 = (µ + σ)( 1 2 ) + η 1 ( 1 2 ) + 2 (η 1 η 2 ) ( 1 + µ(µ + σ) ) ( 2 η1 1 + µ η ) 2 2 m 1 m 2 m 1 m 2 = (µ + σ) + η H 1 + µ(µ + σ)h 2 + µh 3 ( 1 2 ), H 1 = (η 1 η 2 )/( 1 2 ), H 2 = ( 1 /m 1 2 /m 2 )/( 1 2 ), H 3 = (η 1 1 /m 1 η 2 2 /m 2 )/( 1 2 ). (2.3) Note that H 1 0 for the monotonicity of ϕ, H 2 0 for the monotonicity of x/g(x), and H 3 0 for the monotonicity of x/g(x) and ϕ(). This yields a contradiction for (2.3). Hence, the uniqueness of E is proved. 3. Local stability of equilibria n this section, we prove the following results, which guarantee the local asymptotical stability of the disease-free equilibrium, and the endemic equilibrium of (1.1). Theorem 3.1. f R 0 < 1, then the disease-free equilibrium E 0 of (1.1) is locally asymptotically stable. Proof. The Jacobian matrix of (1.1) at E 0 = (1, 0) is µ g J 0 = (0) 0 g. (0) (µ + σ) ϕ(0) Let λ 1 and λ 2 be the characteristic roots of J 0. We obtain λ 1 + λ 2 = µ + (R 0 1)µ + σ + ϕ(0) < 0, λ 1 λ 2 = µ(r 0 1)µ + σ + ϕ(0) > 0, for R 0 < 1. This proves that E 0 is locally asymptotically stable. Theorem 3.2. f R 0 > 1 and ϕ is differential at, then E 0 is unstable and E is locally asymptotically stable. Proof. The Jacobian matrix of system (1.1) at E is µ g( J = ) S g ( ) g( ) S g ( ) (µ + σ) ϕ( ) ϕ ( ). Observe that system (1.1) at E can be rewritten into the form 0 = µ S g( ) µs, 0 = S g( ) (µ + σ) ϕ( ). Let α 1 and α 2 be the characteristic roots of J. According to the second equality of the above system, since g () < 0, we can obtain that α 1 + α 2 = S g ( ) S g( ) µ g( ) ϕ ( ) = S g ( ) g( ) µ g( ) ϕ ( )

6 6 M. YANG, F. SUN EJDE-2015/304 where ξ (0, ). = S g ( ) g (ξ) µ g( ) ϕ ( ) < 0, α 1 α 2 = µ S g( ) µs g ( ) + g( )(µ + σ) + ϕ ( ) + ϕ( ) + µϕ ( ) = µs g (ζ) g ( ) + g( )(µ + σ) + ϕ ( ) + ϕ( ) + µϕ ( ) > 0, where ζ (0, ). Hence, E is locally asymptotically stable. 4. Global stability of equilibria n this section, we discuss the global stability of disease-free equilibrium E 0 and endemic equilibrium E. The following theorem indicates that the disease can be eradicated in the host population if R 0 1, while R 0 > 1, the disease is permanent. Theorem 4.1. f R 0 1, then E 0 is globally asymptotically stable. Proof. We move E 0 to the origin firstly by setting x = S 1. Then, system (2.1) becomes dx = µx xg() g(), d xg() + g() (µ + σ) coϕ(). We now consider the Lyapunov function V 1 (x, ) = x 2 /2 +. We can calculate that V 1 = (x, 1). Then, the constant l, which means that the set L l = {(x, ) R 1 R 1 V 1 (x, ) l} can be arbitrarily large. Let G(x, ) = µx xg() g() xg() + g() (µ + σ) coϕ() t is clear that semi-continuous and set-valued mapping G has compact and convex image. For each ν = (ν 1, ν 2 ) G(x, ), there exists a measurable function η(t) coϕ() corresponding to (x(t), (t)); that is, ν = µx xg() g() xg() + g() (µ + σ) η(t) We now calculate the derivative of V 1 (x, ) by V 1, ν = µx 2 x 2 g() + g() (µ + σ) η(t) g() = µx 2 x 2 g() + (µ + σ) η(t). t follows from Remark 2.1 that Using η(t) > ϕ(0), we have g()/ lim 0 + g()/ = g (0)... V 1, ν µx 2 x 2 g() + µ + σ + ϕ(0)(r 0 1). Since R 0 1, we claim V 1, ν 0. Observe that if R 0 < 1, then Z V1 = {(x, ) R 1 R 1 V 1, ν = 0} = {(0, 0)}. That is, if R 0 = 1, then Z V1 = {(0, 0)} {(0, ) η(t) = ϕ(0), 0}. n addition, if x = x(t) 0, then = (t) 0. Thus, {(0, 0)} is the largest weak invariant subset of Z V1 L l for all l > 0, which

7 EJDE-2015/304 GLOBAL STABLTY OF SR MODELS 7 leads to the global asymptotical stability of {(0, 0)} due to the invariance theorem (see 2), and thus E 0 is globally asymptotical stable. Theorem 4.2. f R 0 > 1, then the unique endemic equilibrium E is globally asymptotical stable. Proof. Denote η = S g( )/ (µ + σ) coϕ( ). We consider S V 2 = S S /τdτ + ln, ε where ε is a small unspecified parameter. Note that V 2 is well-defined and continuous for all S > ε and > 0, and V 2 = (1 S /S, 1 /). Let µ Sg() µs F (S, ) =. Sg() (µ + σ) coϕ() t is easy to check that F is a semi-continuous and set-valued mapping, which has compact and convex image. Thus, for any ω = (ω 1, ω 2 ) F (S, ), there exists a measurable function η(t) coϕ() such that Thus, it follows that ω = µ Sg() µs Sg() (µ + σ) η(t) V 2, ω = µ Sg() µs µ S S + S g() + µs + Sg() (µ + σ + η) Sg() + (µ + σ + η) S = µs + S g( ) µs µs + S g( ) S + S g() + µs S g( ) η + η Sg() S + g( ) η + η = µs ( 1 S ) S S S S g( ) S g( ) S S + S g() S g( ) + (η η) Sg() (η η) = µs ( 1 S )( ) S 1 S + 2S g( ) S g( ) S S S Sg() S g( ) g( ) g() + S g() S g( ) + S g( ) g( ) g() + (η η)( ) = µs (1 S S )(1 S S ) + S g( ) 3 S + S g( ) 1 + g() g( ) + g( ) g() = µs ( 1 S )( ) S 1 S + S g( ) 3 S S + S g( ) g() g( ) S. Sg() S g( ) g( ) g() + (η η)( ) Sg() S g( ) S g( ) g() 1 + (η η)( ). g( ) g()

8 8 M. YANG, F. SUN EJDE-2015/304 One can see that ( S )( S ) S S Since the arithmetic mean is greater than or equal to the geometric mean, we have S S + Sg() S g( ) + g( ) g() 3. The monotonicity of g() and φ() (see Remark 2.1) gives Then we have for all > 0. (2.1) 0. dv 2 g() g( ) 1 for 0 < <, 1 g() g( ) for. g() g( ) g( ) g() 1 0, Since (η η)( ) 0 for the monotony of ϕ, we obtain Clearly, we see that dv2 = 0 holds only when S = S, = and that E is the only equilibrium of that system. This implies the globally asymptotical stability of E. References 1 Aubin, J. P.; Cellina, A.; Differential nclusions, Springer-Verlag, Aubin, J. P.; Frankowska, H,; Set -valued analysis, Birkauser, Filippov, A. F.; Differential equations with discontinuous right-hand sides (n Russian), Matemat.Sbornik., Filippov, A. F.; Differential equations with discontinuous right-hand side, Mahematics and its applications (Soviet series), Korobeinikov, A.; Lyapunov functions and global stability for SR and SRS epidemiological models with non-linear trasmission, Bull. Math. Biol, 68 (2006), Korobeinikov, A.; Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), Korobeinikov, A.; Maini, P. K.; Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), Li, M. Y.; Muldowney, J. S.; van den Driessche, P.; Global stability of SERS models in epidemiology, Can. Appl. Math. Quort., 7 (1999), van den Driessche, P.; Watmough, J.; Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), Wang, W.; Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), Wang, W.; Ruan, S.; Bifurcation in an epidemic model with constant removal rate of the infectives, Math. Anal. Appl, 291 (2004), Yoichi, E.; Yukihiko, N.; Yochiaki, M.; Global stability of SR epidemic models with a wide class of nonlinear incidence rates and disrtibuted delays, Discrete and continuous dynamical systems series B, 15 (2011), Mei Yang College of Science, Tianjin University of Technology and Education, Tianjin , China address: yangmei @yeah.net Fuqin Sun College of Science, Tianjin University of Technology and Education, Tianjin , China address: sfqwell@163.com