Let be bounded non- The stability of the disease-free equilibrium. We investigate the stability of the The stability of the endemic equilibrium.

L EMMA 4.1. B RAUER , 2001. Let f t be a bounded non-negative function which satisfies an estimate of the form t f t f t f t a R a d ≤ + − ∫ a , where f t is a non-negative function with and is a non-negative function with Then . lim t f t →∞ = R a 1. R a da ∞ ∫ lim t f t →∞ = P ROOF . See [1]. It is also showed in [1] that the lemma is still true if the inequality in the lemma is replaced by sup t t a s t f t f t f s R a d − ≤ ≤ ≤ + ∫ a . 17 The following lemma is the extension of Brauer’s lemma. L EMMA

4.2. Let be bounded non-

negative functions satisfying , 1, 2 j f t j = 1 10 1 1 sup t t a s t f t f t f s R a d − ≤ ≤ ≤ + ∫ a , 2 20 1 2 sup t t a s t f t f t f s R a d − ≤ ≤ ≤ + ∫ a , where j f t is non-negative with and is non-negative with Then li . lim t j f t →∞ = j R a 1. j R a da ∞ ∫ m 0, 1, 2 t j f t j →∞ = = P ROOF . 1 10 20 1 2 1 2 sup{ , } sup sup{ , }sup{ , } t t a s t f t f t f t f s f s R a R a da − ≤ ≤ ≤ + ∫ 2 10 20 1 2 1 2 sup{ , } sup sup{ , }sup{ , } t t a s t f t f t f t f s f s R a R a da − ≤ ≤ ≤ + ∫ and hence, 1 2 10 20 1 2 1 2 sup{ , } sup{ , } sup sup{ , }sup{ , } t t a s t f t f t f t f t f s f s R a R a da − ≤ ≤ ≤ + ∫ From Lemma 4.1 we conclude that , and this is suffice to show that li 1 2 lim sup{ , } t f t f t →∞ = m 0, 1, 2 t j f t j →∞ = = .

4.1. The stability of the disease-free equilibrium. We investigate the stability of the

disease-free equilibrium for the case of 1 R . Consider the following inequalities. 1 sup t H H t a I s ds t H H t a H t a s t H e I s a β β β − − − − ≤ ≤ ∫ − ≤ ∫ ≤ ds I s 18 Hence we have, 1 sup t H H t a H H H H t I s ds a H H H t a H H H t a s t H I t I t B Q a e e da I t B Q a a I s e da β γ γ β − − − − − ≤ ≤ = ∫ + − ≤ + ∫ ∫ 19 And 1 1 sup 1 t H H t a H H H H t I s ds a H H H t a H H H t a s t H R t R t B Q a e e da R t B Q a a I s e da β γ γ β − − − − − ≤ ≤ = ⎡ ⎤ ∫ ⎡ ⎤ + − − ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ≤ ⎡ ⎤ + − ⎣ ⎦ ∫ ∫ 20 Moreover, since lim H I t t = → ∞ and 1 H a H H H B Q a a e da R γ β ∞ − = ∫ then using Lemma 4.1 we conclude that . lim t H I t →∞ = Next, let us see the expression which, if , can be written in the form 1 H a H H H B Q a a e da γ β ∞ − ⎡ ⎤ − ⎣ ⎦ ∫ S H H H R B Q a a β ∞ = ∫ da 0 S R R − . Hence, if then 1 S R + R 1 1 H a H H H B Q a a e da γ β ∞ − ⎡ ⎤ − ⎣ ⎦ ∫ Appendix 1. Furthermore, since then using Lemma 4.2 we conclude that lim H R t t = → ∞ lim t H R t →∞ = . Consequently, This shows that the disease-free equilibrium is globally stable. lim lim t H t H H H S t N t R t I t N →∞ →∞ = − − H = , 0, H H V H S I I N =

4.2. The stability of the endemic equilibrium.

The endemic equilibrium appears only if . Let us see the perturbations of , , H H H S I R 1 R H I and H R , respectively, by and . Define u t v t H H I t I u t = + and substitute this quantity into equation 4 to obtain the following calculations. [ ] 1 t H H t a H H H t I u s ds a H H I u t I t B Q a e e da β γ − − + − + = ⎛ ⎞ ∫ + − ⎜ ⎟ ⎝ ⎠ ∫ 1 1 1 t t H H H t a t a H H H H t H t a H H H H H t I ds u s ds a H H I a a H H H t u s ds I a a H H u t I I t B Q a e e e da B Q a e e da I t B Q a e e e da β β γ β γ β β γ − − − − − − ∞ − − − − − = − + ⎛ ⎞ ∫ ∫ + − ⎜ ⎟ ⎝ ⎠ = − − + ⎛ ⎞ ∫ + − ⎜ ⎟ ⎝ ⎠ ∫ ∫ ∫ 1 1 1 H H H H H t H t a H H H I a a H H H t t I a H H t u s ds I a a H H u t B Q a e e da I t B Q a e da B Q a e e e da β γ β β β γ − ∞ − − − − − − = − − + − − ⎛ ⎞ ∫ + − ⎜ ⎟ ⎝ ⎠ ∫ ∫ ∫ 1 1 1 sup H H H t H t a H H H H H H H H H I a a H H H t t u s ds I a a H H I a a H H H t I a a H H H t a s t u t B Q a e e da I t B Q a e e e da B Q a e e da I t B Q a e ae u s β γ β β γ β γ β γ β − ∞ − − − − − ∞ − − − − − ≤ ≤ = − − + ⎛ ⎞ ∫ + − ⎜ ⎟ ⎝ ⎠ ≤ − − + + ∫ ∫ ∫ t da ∫ Hence, we have 1 sup H H H H H H I a a H H H t t I a a t a s t H H H u t B Q a e e da I t u s B Q a e ae da β γ β γ β ∞ − − − − − ≤ ≤ ≤ − − + + ∫ ∫ By defining 1 f t u t = , 1 H H I a a H H H R a B Q a e ae H β γ β − = − , and 10 1 H H H I a a H H H t f t B Q a e e da I β γ ∞ − − = − − + ∫ t , then we have 1 10 1 1 sup t t a s t f t f t f s R a d − ≤ ≤ ≤ + ∫ a . We see that and it can be shown that is non-negative with 10 lim t f t →∞ = 1 R a 1 1 R a da ∞ ∫ see Appendix 2. Then by Lemma 4.1 we have , means that 1 lim t f t →∞ = lim t H H I t I →∞ = . Next, define and substitute these quantities into equation 5 to obtain the following calculations: H H R t R v t = + [ ] 1 1 t H H t a H H H t I u s ds a H H R v t R t B Q a e e da β γ − − + − + = ∫ ⎡ ⎤ + − − ⎣ ⎦ ∫ 1 1 1 1 1 1 t t H H H t a t a H H H H t H t a H H H H H t I ds u s ds a H H I a a H H H t u s ds I a a H H v t R R t B Q a e e e da B Q a e e da R t B Q a e e e da β β γ β γ β β γ − − − − − − ∞ − − − − − = − + ⎛ ⎞ ∫ ∫ ⎡ ⎤ + − − ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎡ ⎤ = − − − + ⎣ ⎦ ⎛ ⎞ ∫ ⎡ ⎤ + − − ⎜ ⎟⎣ ⎦ ⎝ ⎠ ∫ ∫ ∫ 1 1 1 1 1 1 H H H H H H t H t a H H H I a a H H H t t I a a H H t u s ds I a a H H v t B Q a e e da R t B Q a e e da B Q a e e e da β γ β γ β β γ − ∞ − − − − − − − ⎡ ⎤ = − − − + ⎣ ⎦ ⎡ ⎤ − − − ⎣ ⎦ ⎛ ⎞ ∫ ⎡ ⎤ + − − ⎜ ⎟⎣ ⎦ ⎝ ⎠ ∫ ∫ ∫ 1 1 1 1 1 1 1 H H H t H t a H H H H H H H H I a a H H H t t u s ds I a a H H I a a H H H t I a H H H v t B Q a e e da R t B Q a e e e da B Q a e e da R t B Q a e a β γ β β γ β γ β β − ∞ − − − − − ∞ − − − ⎡ ⎤ = − − − + ⎣ ⎦ ⎛ ⎞ ∫ ⎡ ⎤ + − − ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎡ ⎤ ≤ − − − + ⎣ ⎦ + ∫ ∫ ∫ sup H t a t a s t e u γ − − ≤ ≤ ⎡ ⎤ − ⎣ ⎦ ∫ s da Hence, we have 1 1 sup 1 H H H H H H I a a H H H t t I a a t a s t H H H v t B Q a e e da I t u s B Q a e a e da β γ β γ β ∞ − − − − − ≤ ≤ ⎡ ⎤ ≤ − − − + ⎣ ⎦ ⎡ ⎤ + − ⎣ ⎦ ∫ ∫ By defining 1 f t u t = , 2 f t v t = , and 2 1 H H H I a a H H H R a B Q a e a e β β − ⎡ = ⎣ γ − ⎤ − ⎦ 20 1 1 H H H I a a H H H t f t B Q a e e da I β γ ∞ − − ⎡ ⎤ = − − − + ⎣ ⎦ ∫ t , then we have 2 20 1 2 sup t t a s t f t f t f s R a d − ≤ ≤ ≤ + ∫ a . We can show that in Appendix 3 that 20 lim t f t →∞ = and is non-negative with 2 R a 2 1 R a da ∞ ∫ . Then by Lemma 4.2 we have 2 lim t f t →∞ = , means that . lim t H R t R →∞ = H Finally, since lim t H N t →∞ is a constant, , and lim t H R t R →∞ = H H lim t H I t I →∞ = then is globally stable. , , H H H S I R

V. Concluding Remarks