Delay SIR Model with Nonlinear Incident Rate and Varying Total Population

Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior of SIR epidemic with nonlinear incident rate have been revisited We study the behavior of the model with time delay in which the total population size varies Lyapunov functions are constructed to establish the global asymptotic stability of all steady states of the model Numerical simulations are shown to confirm the main results Keywords Epidemic model, Nonlinear incidence rate, Delay, Stability, SIR R I INTRODUTION EENTLY, many research works on mathematical models in biology have focused on existence asymptotic stability of nonnegative equilibrium points through bifurcation analysis of the model []-[4] To make the model more realistic, many assumptions on the model have been imposed For example, in an SIR model, new population entering the system is not only due to the new born but communicable diseases may be introduced into a population by the arrival of infectives from outside the population, as seen in []-[5] Incidence rate also plays an important role in the dynamical modeling of infections diseases It has been suggested by several authors ([6]-[]) that the disease transmission process may have a nonlinear incidence rate In many epidemic models [6]-[8], the bilinear incidence rate SI, where S is the number of susceptible individuals, I is the number, of infective members the stard incidence rate SI N, where N is the total population size, are frequently used The bilinear incidence rate is based on the law of mass action This contact law is suitable for communicable diseases such as influenza so on, but not for sexually transmitted diseases It has been pointed out that R Ouncharoen is with the Department of Mathematics, Faculty of Science, hiang Mai University, 39 Huaykaew Road, Sutep, Muang, hiangmai, 5000 the entre of Excellence in Mathematics, HE, 38 Si Ayutthaya Road, Bangkok, 0400, THAILAND (corresponding author, phone: 6653-943-37; fax: 6653-89-80; e-mail: rujirao@cmuacth) S Daengkongkho was with the Department of Mathematics, Faculty of Science, hiang Mai University, 39 Huaykaew Road, Sutep, Muang, hiangmai, 5000 (e-mail: mild-melody@windowslivecom) T Dumrongpokaphan is with the Department of Mathematics, Faculty of Science, hiang Mai University, 39 Huaykaew Road, Sutep, Muang, hiangmai, 5000 the entre of Excellence in Mathematics, HE, 38 Si Ayutthaya Road, Bangkok, 0400, THAILAND (e-mail: thongchaid@cmuacth) Y Lenbury is with the Department of Mathematics, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 0400 the entre of Excellence in Mathematics, HE, 38 Si Ayutthaya Road, Bangkok, 4000 THAILAND (e-mail: yongwimonlen@mahidolacth) the stard incidence rate may give a good approximation if the number of available partners is large enough but not if everybody could not make more contacts than it's practically feasible After studying the cholera epidemic spread in 973, apasso Serio [9] introduced a saturated incidence rate g ( I ) S into epidemic models, where gi ( ) tends to a saturation level when I gets large, that is g( I) I ( I), where I measures the infection force of the disease ( I) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals This incidence rate seems more reasonable than the bilinear incidence rate SI, because it includes the behavioral change crowding effect of the infective individuals prevents the unboundedness of the contact rate by choosing suitable parameters We formulate the model based on the assumptions that the total population size does not change, no new population entered the system, none of the population dies the recovered population has permanent immunity cannot be infected again To make it more realistic we add a time delay in the infectious rate, which is an important key factor in the model that also involves immigration Then, our model has the following form: ds( t) S( t) I( t ) ( p) S( t), I( t ) di( t) S( t) I( t ) p ( ) I( t), I( t ) dr() t I( t) R( t), where St () is the number of individuals susceptible to the disease at time t, It () is the number of infective members at time t, Rt () is the number of members who have been removed from the possibility of infection through full immunity at time t, I ( t ) is the number of members who have been removed from the possibility of infection through full immunity at time t, Nt () is the total population size at time t, where N( t) S( t) I( t) R( t), p is the fraction of infectives, 0 p, is a constant flow of new members into the whole population per unit time which can be susceptible or infective, is the average number of contacts () Issue 4, Volume 7, 03 369

per infective member per day, is the time delay during which the infectious agents, or germs, use as incubation time before a human who has been infected can spread the infection to a susceptible human, ( It ( )) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals, is the natural death rate of the population, is the recovery rate of infective individuals, is the death rate due to the disease, all parameters are positive The paper is arranged as follows In Section II, the basic results are given In Section III, the stability is analyzed by making use of parts of Kuang s work [] In Section IV, numerical results are illustrated by computer simulations The conclusion is provided in the final section II BASI RESULTS We rewrite the system () into the form ds( t ) SI ( p ) S, I di( t ) SI p I, I dr( t ) I R, where S S( t), I I( t), I I( t ), R R( t) The initial conditions of () are given as S( ) ( ), I( ) ( ), R( ) ( ), 3 ( ) 0, [,0], (0) 0, i,,3, i where ( ), ( ) ( ) 3 are continuous on [, 0] First, we consider the steady states of the system () These steady states ( S, I, R ) are determined analytically by setting S I R 0 Then, we obtain ( p) ( ( N )) S, ( )( N ) I ( N ), R ( N ) i () (3) where N S I R Letting N ( N )( ) FN ( ), ( ( N )) we then have 0 p F( N )( N ) (4) If the system () has a solution where N (0, ], then the system () has a physically feasible equilibrium solution By substitution, the following can be easily shown Proposition If p 0, then the system () has a unique disease-free equilibrium point, namely E0 (, 0, 0) Proposition If R0, (i) p 0 R0, or (ii) 0 p N (0, ), or (iii) p Then the system () has an endemic equilibrium point Proof: (i) From system (), if p 0, the steady state E ( S, I, R ) is determined analytically by setting S I R 0 R0 Then we obtain S ( ) ( ), I ( ) [ ( )] 0, R [ ( )] [ ( )] 0 (ii) onsider the system () when 0 p N (0, ) Multiplying both sides of (4) by ( N ), Then, we obtain F N p ( N ) ( ) 0 p H( N ) F( N ), ( N ) dh( N ) df( N ) p dn dn ( N ) Since all parameters are positive, Let N (0, ) df( N ) dh ( N ) 0, then 0 This means dn dn increasing function of N Now, consider the limit of as N N 0 We obtain N 0 N 0 HN ( ) is an lim H ( N ) lim F ( N ) p HN ( ) Since lim F( N ), we have lim H( N ) 0 As N 0, we obtain p lim H( N ) F( ) lim N N ( N ) Since p lim N ( N ) N 0 we have lim H( N ) 0 N Thus, HN ( ) has a unique positive root N (0, ) which means that the endemic equilibrium point will exist is unique Let E ( S, I, R ) be the endemic equilibrium point of the system () where 0 p N S I R (0, ), so that ( p) ( ( N )) S, ( )( N ) I ( N), R ( N)] (iii) From the system (), if p we have the steady state E3 ( S3, I3, R3 ) where S3 0, I3 R3 Issue 4, Volume 7, 03 370

A Local Stability III STABILITY ANALYSIS In what follows, we analyze the model system in terms of its stability relying in parts on the work of Kuang s [] Theorem 3 Let R0 If p 0 (i) R0, then the disease-free equilibrium point E 0 is locally asymptotically stable for any time delay 0 (ii) R0, then the disease-free equilibrium point E 0 is unstable for any time delay 0 (iii) R0, the disease-free equilibrium point E 0 is stable Here, R 0 is the average number of secondary infections generated by one primary infection in a susceptible population Proof: To discuss the local asymptotic stability of the disease-free equilibrium point E0 (, 0, 0), let us consider the following coordinate transformation x( t) S( t) S, y( t) I( t) I, z( t) R( t) R, where ( S, I, R ) denotes any equilibrium of the system () Hence, we have the corresponding linearized system of () of the form I S x ( t) ( ) x( t) y( t ), I ( I ) I S y ( t) x( t) y( t ) y( t), I ( I ) z ( t) y( t) z( t), The associated transcendental characteristic equation of (5) at the disease-free equilibrium point E0 ( S, I, R ) (,0,0) becomes ( )[ ( ) ( e ) ] 0 (6) It is clear that (6) has a characteristic root, which is always negative since 0 Now, we consider the transcendental polynomial equation ( ) ( e ) 0 (7) (i) We will show that if R0 then E 0 is locally asymptotically stable onsider the case 0 in (7) We have ( ) 0 Since R0, from the Routh-Hurwitz criterion, all roots of (7) have negative real parts for 0 If (7) has pure imaginary roots i for some 0 0, then by substituting i into (7), we then have sin cos (8) ( ) cos sin (9) (5) Squaring both sides of (8) (9) adding, we get 4 ( ) 0 Since R0, then we have 0 0 From the Routh-Hurwitz criterion, we obtain 0 This contradiction means that any solution of (7) must have negative real part Hence, the disease-free equilibrium point E 0 is locally asymptotically stable for any time delay 0 (ii) Next, we will show that if R0, then E 0 is unstable Let G( ) ( ) ( ) e Note that, since R0, we have that G(0) 0 G( ) as It follows from continuity of the function G( ) on (, ) that the equation G( ) 0 has at least one positive solution Hence, the characteristic (7) has at least one positive real solution Hence, E 0 is unstable This proves the conclusion of (ii) (iii) Next, we will show that if R0, then E 0 is stable If R0, the transcendental polynomial (7) becomes G( ) ( ) ( ) e 0 (0) It is clear that 0 is a simple root of G( ) We further show that other solutions of (0) must have negative real parts In fact, if (0) has an imaginary solution, u i for some u 0, 0 0, then by substituting u i into (0), we have u u ( ) u e [( u )cos sin ] () u u ( ) e [( u )sin cos ] () Squaring both sides of () () then adding we get [ u ( ) u ] [ u ( ) ] [( u ) ] (3) Simplifying the above inequality, we then have ( u )[( u ) u ] u( u ) 0 Since, u,,, are all positive, the above inequality is not true Hence, the solutions of (0) have negative real parts except for the solution 0 Hence, E 0 is stable for any time delay 0 This proves the conclusion of (iii) Theorem 4 If p 0 R0, then the endemic equilibrium point E is locally asymptotically stable for any time delay 0 Issue 4, Volume 7, 03 37

Proof: The associated transcendental characteristic equation of system (5) at ( S, I, R ) E ( S, I, R ) becomes ( )[ ( A ) A B ( ) e ] 0, (4) where A I ( I), B S ( I) It is clear that is always negative Now, we consider ( A ) A B ( ) e 0 (5) For 0, we have that ( A B ) A B 0 Substituting S ( ) ( ) I ( ) [ ( )] we then have A B I ( ) ( I) 0 A B I ( ) ( I) 0 It follows that any solution of (5) has negative real part for 0 If (5) has pure imaginary solution i for some 0 0, then we have A B cos B sin, (6) A B cos B sin (7) Squaring both sides of equations (6) (7) adding, we get 4 ( A B ) A B 0 Letting z, we then have z ( A B ) z A B 0 (8) onsider I ( I) S A B 4 I ( I) ( ( I) S)] ( ) [( ) η + ωi βs ηβω β ση + βω β ση = σω + β η σω + β It is readily seen that if R0, then ( I ) S 0 So, A B 0 I S A B [ ( ( ) ) I ( I) I ( )] 0 I It follows that any solution of (8) has negative real part, which contradicts the fact that z This shows that all solutions of the characteristic (5) have negative real parts for any time delay 0 By using the similar technique, the following theorem can be shown Theorem 5 If 0 p N (0, ), I S ( I ) 0 I S ( I ) 0 then the endemic equilibrium point E is locally asymptotically stable for any time delay 0 Proof: The characteristic equation of (5) at ( S, I, R ) E ( S, I, R ) becomes ( )[ ( A ) A B ( ) e ] 0, (9) where A I ( I ) B S ( I ) Now, we consider ( A ) A B ( ) e 0 (0) For 0, we have ( A B ) A B 0 Observe that ( I ) I A B 0 I S S ( I ) I A B 0 I It follows that any solutions of (0) have negative real parts for 0 If (0) has a purely imaginary solution i for some 0 0, then we have A B cos B sin () ( A ) B cos B sin () Squaring both sides of () (), then adding, we have 4 ( A B ) A B 0 Letting z, we then have z ( A B ) z A B 0 (3) We observe that S A B [( ( I ) ) I I S ( ( I ) )] 0 I I I S A B [ ( ) ] I ( I) I S [ ( I ) ] 0 I I Therefore, all solutions of (3) have negative real parts, which contradicts the fact that z This shows that all solutions of (0) have negative real parts for 0 Theorem 6 If p, then the endemic equilibrium point E 3 is locally asymptotically stable Issue 4, Volume 7, 03 37

Proof: If p, the characteristic equation at E (0, I, R ) becomes 3 3 3 λ + σ λ + η + σ + βi 3 λ + η βi 3 + σ + ωi 3 + ωi 3 = 0 4 It is easy to see that any solutions of (4) have negative real parts B Global Stability We now consider the following equation with time delay au u cu, u( ) ( ) 0, [,0), (0) 0, (5) u where u u( t), u u( t ) The parameters ac, are positive constants 0 Note that (5) always has a trivial equilibrium u 0 Moreover, if a c, then (5) has also a unique positive equilibrium u a c c For (5), we have the following result Lemma 7 If a c, then the positive equilibrium u a c c of (5) is globally asymptotically stable; if a c, then the trivial equilibrium (0,0) of (5) is globally asymptotically stable Proof: See [9] Proposition 8 Any solution ( S( t), I( t), R( t )) of system () satisfies lim sup( S( t) I( t) R( t)) t Proof: Since N ( t) S ( t) I ( t) R ( t), we have lim sup Nt ( ) t Proposition 9 Any solution ( S( t), I( t), R( t )) of () with initial conditions in (3) is defined on [0, ) remains positive for all t 0 Proof: Define c K { (, 3 ) 0, 0, 3 0} From Proposition 8, we see that K attracts all solutions of () For any (, 3 ) K, let ( S( t), I( t), R( t )) be the solution of () with the initial function We claim that for any t 0, S() t In fact, if there is a t 0 such that S( t ) St ( ) 0, then we have that S( t) I( t ) S ( t) ( p) S( t) It ( ) S( t) I( t ) 0 It ( ) Here, we have used S( t ) This is a contradiction to St ( ) 0 The claim is proved Hence, K is positively invariant with respect to the system () Theorem 0 If p 0 R0 then the disease-free equilibrium point E 0 is globally asymptotically stable Proof: Let ( S( t), I( t), R( t )) be any positive solution of () When p 0, ( S( t), I( t), R( t )) is the solution of system () with the initial conditions (3) By Proposition 8, we have limsup S( t) If R0 then We may t choose 0 sufficiently small satisfying ( ) (6) Hence, for 0 sufficiently small satisfying (6), there is a T 0 such that if t T, then S() t When p 0, we derive from the second equation of (), for t T, that ( ) I I ( t) I( t) I onsider the following auxiliary equation u ( t) [ ( ) u ( u )] u( t) From inequality (6), ( ) by Lemma 7, it follows that lim ut ( ) 0 t By comparison we find that limsup It ( ) 0 Hence there is a, t T, T T, such that if t T then It () We find, from the third equation of (), that, for t T, R ( t) I( t) R( t) R( t) By comparison it follows that lim sup Rt ( ) 0 When p 0 t, we find, from the first equation of (), for t T, that St () S ( t) S( t) By comparison, it follows that ( ) lim inf St ( ) t ( ) Letting 0, we obtain lim inf S( t) Therefore, t lim S( t) t If p 0 R0, then E 0 is locally asymptotically stable We conclude that E 0 is globally asymptotically stable Issue 4, Volume 7, 03 373

Permanence (or persistence) is an important property of dynamical systems other systems in epidemiology, biology, ecology, so on A more basic important question to ask is whether or not those involved populations will be alive well in the long run Proposition Suppose that p 0 R0, then for any solution ( S( t), I( t), R( t )) of system () with initial conditions (3), we have that lim inf S t σ + ω t + ωσ + β + σ d, β ση lim inf I t t + η ωσ + β e ητ d, lim inf R t γd t + σ d 3 Proof: Let ( S( t), I( t), R( t )) be any solution of () When p 0, ( S( t), I( t), R( t )) is the solution of () with the initial conditions (3) By Proposition 8, it follows that lim t + sup I t Hence, for ε > 0 sufficiently small, σ there is a T > 0 such that if t > T, I t < /σ + ε When p 0, we therefore find, from the first equation of (), for t > T + τ, that Thus, S t σ + β σ + ε + ω S t σ + ε σ + ω lim inf S t t + ωσ + β + σ d We now show that lim inf I t d t + For t 0, define a differentiable function t I u V t = I t + βs du 7 t τ + ωi u alculating the derivative of V t along the solutions of () when p 0, we obtain V t = βi τ S t S + ωi τ βs + η I t 8 + ωi t For any 0 < q <, we have that β ση qi < I = S < η ωσ + β qi β + qi ω + σ Let qi β/ + qi ω + σ k There is a constant ρ sufficiently large such that S < /k e k ρτ S We now claim that it is impossible that I t qi for all t ρτ Otherwise, if I t qi for all t ρτ, when p 0, we have, from the first equation of (), that, for t ρτ + τ, S t σ + βqi + ωqi S t 9 It then follows that, for t > ρτ + τ, ds t + σ + βqi / + ωqi S t We have S t > e k t ρτ τ /k Hence, for t > ρτ + τ, S t > k e k ρτ = S > S 30 Noting that I t qi < I, it follows from (8) (30) that, for t > ρτ + τ, V t βi τ S t S + ωi + βs η I t τ + ωi onsider η + ω βs β σω + β η = + ωi + ω β ση η σω + β η η ηβ + ωβ = η = 0, ηβ + ωβ We then have V t > βi τ + ωi τ S S 3 Setting j = min I θ + ρτ + τ, θ τ,0 we claim that I t j for all t > ρτ + τ Otherwise, if there is a T 0 such that I t j for ρτ + τ t ρτ + τ + T, I ρτ + τ + T = j I ρτ + τ + T 0, it follows from inequality (30) the second equation of () when p 0 that, for t = ρτ + τ + T, I t = βs t I t τ + ωi t τ ηi t or βs I t > η j > 0 + ωi This is a contradiction Hence, I t j for all t ρτ + τ Accordingly, for t ρτ + τ, it follows from inequality (3) that V t > βj + ωj S S, which yields V t + as t + It follows from (7) that there is a T > 0 such that if t > T, V t σ + βs τ σ + ω A contradiction occurs Hence, the claim is proved, that is, it is impossible that I t qi for all t ρτ By the claim, we are left to consider two possibilities First, I t qi for all t sufficiently large Second, I t oscillates about qi for all t sufficiently large We now show that I t qd for all t sufficiently large The conclusion is obvious for the first case For the second case, let t < t be sufficiently large such that I t = I t = qi, I t < qi, t < t < t If t t τ, when p 0, it follows from the second equation of () that I t > ηi t, which yields, for t < t < t, Issue 4, Volume 7, 03 374

I t > I t e η t t > qi e ητ β ση = q η σω + β e ητ = qd If t t > τ, we obtain I t qd for t t, t + τ We now claim that I t qd for all t t + τ, t, Otherwise, there is a T > 0 such that I t qd for t t t + τ + T, I t + τ + T = qd I t + τ + T 0 On the other h, when p = 0, it follows from the second equation of (), for t 0 = t + τ + T, that I t 0 = βs t + τ + T I t + T ηi t + ωi t + T + τ + T βs qd η > 0, + ωi a contradiction Hence, I t qd for t t, t Since the interval t, t is chosen arbitrarily, we conclude that I t qd for all t sufficiently large for the second case Since 0 < q < is chosen arbitrarily, we have that lim inf I t d t + We now show that lim t + inf R t d 3 Since I t d, we have I t d ε for large t for any sufficiently small ε > 0 Thus the third equation of () gives dr t γ d ε σr t for large t, which implies that lim t + inf R t γd /σ By using the similarly techniques, the following propositions can be shown Proposition If 0 < p < N 0, /σ, then for any solution ( S( t), I( t), R( t )) of system () with initial conditions (3), we have that p σ + ω lim inf S t t + σω + β + σ e, lim inf I t t + α σn e ητ e, lim inf R t γe t + σ e 3 Proposition 3 Suppose that p, then for any solution ( S( t), I( t), R( t )) of system () with initial conditions (3), we have that lim inf S t > 0, t + lim inf I t t + η e ητ l, lim inf R t γ t + σ l l Theorem 4 If p 0 R 0 >, then the endemic equilibrium point E is globally asymptotically stable Proof: We define the function x f x = + ωx When p 0, from the first the second equations of () at E, we have = σs + βs f I 3 Let ηi = βs f I 33 g x = x ln x V S t = g V I t = g τ 0 V E t = g S t S I t I I t s I We will study the behavior of the Lyapunov functional V E+ t = βf I V I S + βs f I V I + V E 34 We note that g: R + R +0 has the strict global minimum g = 0 Since V S = V I = V E = 0 so that V E+ t = 0 if only if S t /S = I t /I = I t s /I = for all s 0, τ Hence V E+ t 0 By Proposition 8 Proposition, solutions are bounded above bounded away from zero for large time Without loss of generality, we may assume that, when p 0 the solution of system () satisfies these bounds for all t 0 Thus, V E+ t is defined for all t 0 Let v = S, v S = I I Additionally, let Then ds v 3 = I τ I F v 3 = f I v 3 f I = f I τ f I dv E+ = βf I dv I S + βs f I dv I + dv E = σ S S + + F v βf I SS v 3 v F v 3 v v 3 + ln v 3 ln v By adding subtracting the quantity ln v F v 3, we obtain dv E+ = σ S S g g v F v 3 βf I SS v v + F v 3 v 3 + ln v 3 ln F v 3 Since f I > 0 g: R + R +0 So that g /v 0 g v F v 3 /v 0 We see that dv E+ 0 if F v 3 v 3 + ln v 3 ln F v 3 Q v 3 We consider F v 3 v 3 + ln v 3 ln F v 3 : = Q v 3 find the critical point We have dq v 3 = 0 dv 3 F v 3 F v = 0 3 v 3 where Issue 4, Volume 7, 03 375

v 3 F v 3 = + v 3ωI, + ωi v 3 + ωi v 3 F v 3 = + ωi + ωi v 3, F v 3 = + ωi v 3 v 3 + ωi Substituting F v 3 F v 3, we have v 3 + ωi v 3 = 0 We see v 3 = is the solution of Q v 3 or ωi v 3 + ωi v 3 = 0 We see v 3 = 0 or v 3 = /ωi is the solution of Q v 3 but it is impossible because we are considering an endemic equilibrium point When 0 < v 3 <, Q v 3 is positive when v 3 > Q v 3 is negative, so that v 3 = is the solution of Q v 3 So, F v 3 v 3 + ln v 3 ln F v 3 0 with equality only if v 3 = together with the fact that g 0 with equality only if the argument is By the invariance principle [0], solutions limit to M, the largest invariant subset of dv E + = 0 We note that dv E+ is only zero if v = v = v 3 = In particular, this requires that, for any solution in M, we have S t = S, I t = I R t = R for all t so M consists of the single point E Thus, we see that all solutions approach the endemic equilibrium E By Theorem 4, E is locally asymptotically stable by Proposition 8 Proposition, when p = 0, the system () is permanent, allowing us to conclude that E is, in fact, globally asymptotically stable By using the similarly techniques, the following theorem can be shown Theorem 5 If 0 p N 0, /σ, then the endemic equilibrium point E is globally asymptotically stable Theorem 6 If p =, then the endemic equilibrium point E 3 is globally asymptotically stable Proof: onsider V E3 t = S t 35 By Proposition 8 Proposition 3, solutions are bounded above bounded away from zero for large time Without loss of generality, we may assume that, when p =, the solution of () satisfies these bounds for all t 0 Thus, V E3 t is defined for all t 0 Then we have βi τ VE 3 t = S + σ 0 + ωi τ We obtain that VE 3 t = 0 if only if S = 0 or βi τ / + ωi τ + σ = 0, but it is impossible in the case that βi τ / + ωi τ + σ = 0 So that VE 3 t = 0 if only if S = 0 By the invariance principle [0], solutions limit to M, the largest invariant subset of dv E3 = 0 Thus S t = 0 for all t When p =, from the second equation of () S = 0, we further have that I t = ηi t Thus, lim I t = t + η From the third equation of () lim t + I t = /η, we have R t = γ ση Hence, the invariance of M implies that I t = /η R t = γ/ση for all t So M consists of the single point E 3 Thus, we see that all solutions approach the endemic equilibrium point E 3 By Theorem 6, E 3 is locally asymptotically stable by Proposition 8 Proposition 3, when p the system () is permanent, allowing us to conclude that E 3 is globally asymptotically stable IV NUMERIAL SIMULATIONS The qualitative behavior of the three population variables, namely the susceptibles St (), the infectives It () the recovered Rt (), are illustrated by numerical simulations, carried out in this section by using a Matlab program In Figure, we show a computer simulation of the system () subject to the initial conditions S(0) 4, I(0) 3, R(0) 3, with parameters p 0,, 000, 0, 0, 005, 007 5 In this case, we get R0 009 p 0 which ensures that the disease-free equilibrium point E 0 is globally asymptotically stable In Figure, we show a computer simulation of the system () subject to the initial conditions S(0) 4, I(0) 3, R(0) 3, with parameters p 0,, 00, 0, 0, 005, 007 5 We have R 0 8 satisfying the conditions in Theorem 4, R0 p 0, which ensures that E 0 will lose its stability while E become globally asymptotically stable In Figure 3, we show a computer simulation of the system () with different values of R 0 subject to the initial conditions S(0) 4, I(0) 3, R(0) 3 In Figure 3(a), we used parameters p 03,, 08, 5, 003, 0, 0 5 In this case, we get R0 666 E (754,58,3874) Since 0 p N 47304, I S ( I) 557 0 I S ( I ) 9088 0 Therefore, E is globally asymptotically stable, as seen in Figure 3(a) In Figure 3(b) we used parameters p 07,, 00, 5, 0, 04, 003 5 In this case, we get R 0 037 E (853,69,4538) Since 0 p N 66, I S ( I ) 6 0 9 I S ( I ) 9989 0 Therefore, E is globally asymptotically stable, as seen in Figure 3(b) We observe that different values of R 0 ( R0 or R0 ) does not effect a stability of E as long as those conditions in Theorem 5 are satisfied Issue 4, Volume 7, 03 376

In Figure 4, we show a computer simulation of the system () with different values of R 0 subject to the initial conditions S(0) 4, I(0) 3, R(0) 3, In Figure 4(a), we used parameters p,, 008, 0, 0, 04, 03 5 In this case we get R 0 E3 (0, 5,0) Since p, E 3 is globally asymptotically stable, as seen in Figure 4(a) In Figure 4(b) we used parameters p,, 00, 0, 04, 005, 007 5 In this case, we get R 0 09 E3 (0, 3846, 048) Therefore, E 3 is globally asymptotically stable, as seen in Figure 4(b) We observe that different values of R 0 ( R0 or R0 ) does not effect the stability of E 3 as long as those conditions in Theorem 6 are satisfied Fig Numerical simulation of the system () subject to the initial conditions S ( 0) 4, I(0) 3, R(0) 3, with parameters p 0,, 000, 0, 0, 005, 0 07 5 Hence, R 0 009 The solution time series tend to E 0 (5,0,0), where N 5, as predicted in Theorem 0 in which E 0 is globally asymptotically stable Fig 3 Numerical simulation of the system () subject to the initial conditions S ( 0) 4, I(0) 3, R(0) 3, (a) p 03,, 08, 5, 003, 0, 0 5 such that R 0 666 N 47 304 The solution curves tend to E (754,58,3874) as predicted in Theorem 5 in which E is globally asymptotically stable (b) p 07,, 00, 5, 0, 04, 0 03 5 such that R 0 037 N 9 66 The solution curves tend to E (853,69,4538) as predicted in Theorem 5 in which E is globally asymptotically stable Fig Numerical simulation of the system () subject to the initial conditions S ( 0) 4, I(0) 3, R(0) 3, with parameters p 0,, 00, 0, 0, 005, 0 07 5 Hence, R 0 8 The solution time series tend to E (55,045,03), where N 8 568 as predicted in Theorem 4 in which E is globally asymptotically stable Issue 4, Volume 7, 03 377

AKNOWLEDGMENT This research was supported by hiang Mai University Fig 4 Numerical simulation of the system () subject to the initial conditions S ( 0) 4, I(0) 3, R(0) 3, (a) p,, 008, 0, 0, 04, 0 3 5 such that R 0 N 5 The solution curves tend to E 3 (0,5,0) as predicted in Theorem 6 in which E 3 is globally asymptotically stable (b) p,, 00, 0, 04, 005, 007 5 such that R 0 09 N 437 The solution curves tend to E 3 (0,3846,048) as predicted in Theorem 6 in which E 3 is globally asymptotically stable REFERENES [] W M, M Song, Y Takeuchi, Global stability of an SIR epidemic model with time delay, Applied Mathematics Letters, vol 7, 004, pp 4-45 [] F Brauer Pven den Driessche, Models for transmission of disease with immigration of infective, Mathematical Biosciences, vol 7, 00, pp 43-54 [3] G Li, W Wang, Z Jin, Global stability of an SEIR epidemic model with constant immigration, haos Solutions Fractals, vol 30, 006, pp 0-09 [4] T Dumrongpokaphan, R Ouncharoen, T Kaewkheaw, Stability analysis of epidemic model with varying total population size constant imigration rate, hiang Mai Journal of Science, In press, 03 [5] M E Kahil, Population dynamics: a geometrical approach of some epidemic models, WSEAS Transactions on Mathematics, vol 0, 0, pp 454-46 [6] onnell Mcluskey, Global stability for an SIR epidemic model with delay nonlinear incidence, Real world applications, vol, 00, pp 306-309 [7] R Xu, Z Ma, Global stability of a SIR epidemic model with nonlinear incidence rate time delay, Nonlinear Analysis: Real world applications, vol 0, 009, pp 375-389 [8] P Pongsumpun, I M Tang, Limit cycle chaotic behaviors for the transmission model of plasmodium vivax malaria, International Journal of Mathematical Models Methods in Applied Sciences, vol, 008, pp 563-570 [9] V apasso G Serio, A generalization of the Kermack- Mckendrick deterministic epidemic model, Mathematical Biosciences, vol 4, 978, pp 4-6 [0] Y Pei H Wang, Rich dynamical behaviors of a predator-prey system with state feedback control a general functional responses, WSEAS Transactions on Mathematics, vol 0, 0, pp 387-397 [] S Boonrangsiman K Bunwong Hopf bifurcation dynamical behavior of a stage structured predator sharing a prey, International Journal of Mathematical models Methods in Applied Sciences, vol 6, 0, pp 893-900 [] Y Kuang, Delay differential equations with applications in population dynamics, Academic Press, New York, 993 [3] Li, Dynamics of stage-structured population models with harvesting pulses, WSEAS Transactions on Mathematics, vol, 0, pp 74-8 [4] Y Wang D Huang, Dynamic behavior in a hiv infection model for the delayed immune response, WSEAS Transactions on Mathematics, vol 0, 0, pp 398-407 V ONLUSION We have studied investigated the behavior of the disease-free equilibrium point all the endemic equilibrium points their local global stability We also obtained sufficient conditions to ensure that each equilibrium point is locally or globally asymptotically stable The existence of E, E or E 3 depends on the values of p, N or R 0 as described In epidemiology, the basic reproduction number of an infection, R 0, is the average number of secondary infections generated by one primary infection in a susceptible population In other words, it determines the number of people infected by direct or indirect contact with a single infected person R 0 is useful because it helps us to determine whether or not an infectious disease can spread through a population Issue 4, Volume 7, 03 378