GLOBAL STABILITY AND SPATIAL SPREAD OF THE NONLINEAR EPIDEMIC MODEL

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1 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN GLOBAL STABILITY AND SPATIAL SPREAD OF THE NONLINEAR EPIDEMIC MODEL Laid Chahrazed, PhD Rahmani Fouad Lazhar, Professor Department of Mathematics, Faculty of Exact Sciences, University Constantine, Algeria Abstract In this paper, we consider a non linear epidemic model SIQS by experiencing the disease; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time First, the local stabilities and global stability of the infection-free equilibrium and endemic equilibrium are analyzed, respectively Second, the endemic equilibrium is formulated in terms of the incidence rate, and after we stydy local and global asymptotic stability We study the stochastique systme by pertubing the contact rate, and we mainly use the theorie of Itô formula Finnaly we study the spatial spread model with traveling wave solution Keywords: Basic reproductive number, epidemic model, global stability, local asymptotically stable Introduction Classical epidemic models assume that the size of the total population is constant More recent models consider a population size variable to take into account a longer period with death and disease causing reduced reproduction Generally, a model contains a diseasefree equilibrium and one or multiple equilibria are endemic The stability of a disease-free status equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, which quantifies the number of secondary infections arise from infected in a population of sensitive In recent years the dynamics of the non linear epidemic models have received considerable attention in the refferences, in order to describe the effects of disease in differents models Motivated by the comment of the differents autors In this paper, we 67

2 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN discuss the equilibrium and global, local stability of the non linear SIQS epidemic model with constant paramters We have made the following contributions: The local and global stabilities of the infection-free equilibrium are analysed, respectively in section 3 Second the endemic equilibrium is formuled in terms of the insidence rate and the differents positive parametres and lacally, globallay asymptotic stabilities are found in section 4 and 5 3 We study the sochastic system by pertubieng the contact rate in section 6 4 Finally we considere the spatial spread of the ifectives and susceptibles in the section 7 then we analyse the system we travelling the wave solution in section 8 Model equations This paper considers the following SIQS nonlinear epidemic model: St = ( +γ β ) St +αit kstit +ν, I( t) =βs( t) ( +γ+α ) I( t) + ks( t) I( t ) +ρ, Q( t) =γi( t) γs( t) 3Q( t) () Where S (t) + I (t) + Q (t) =N (t) denotes the sizes of the population at time t; S (t), I (t) and Q (t) denote the sizes of the population susceptible to disease, of infective members, and members who have been in quarantine with the possibility of infection, respectively It is assumed that all newborns are susceptible The positive constants μ₁, μ₂, and μ₃ represent the death rates of susceptible, infectivity and quarantine, respectively Biologically, it is natural to assume that min{, 3} The positive constants μ and γ represent the birth rate (incidence rate) of the population and the recovery rate of infective, respectively The positive constants α, β are the average numbers of contacts infective for S and I The term γ S, indicate that an individual has survived from natural death in a recovery before becoming susceptible again The constant k is the rate of unknown members infected which is detected by the system The positive constants ν, ρ are the parameters of immigrations The initial condition of () is given as S( η) = ( η), I( η) = ( η), Q( η) = ( η), τ t, () 3 T Where = (,, 3), such that: S( η) = () = S, I( η) = () = I, Q( η) = () = Q (3) 3 68

3 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Let C denotes the Banach space C ([ τ, ], R ) of continuous functions mapping the interval [ τ, ] into R 3, with τ is latent time Consider the system without the parameters of immigration and study the stability of the system, and since Q (t) does not appear explicitly in the first two equations of system (), which is positive it means that the solution remains positive for any trajectory initialized to positive conditions We consider the system: S(t) = ( +γ β ) S(t) +αi(t) ks(t)i(t), I(t) =βs(t) ( +γ+α ) I(t) + ks(t)i(t), With the initial conditions (), (3) (4) Since N - N, and by integration we get: N Ne t, for all t (5) With initial conditions: S() S, I() I, Q() Q = = =, and N S I Q = + + (6) S Then we consider the system only in the region Ω={(S,I) R +,S+I N }, is positively invariant set of (4) The disease-free equilibrium and its stability Local stability An equilibrium point of system (4) satisfies ( ) +γ β S +αi ksi =, β S +γ+α I + ksi =, (3) It can be seen that, whenever all the seven associated parameters assume positive values, system (3) has a disease-free equilibrium of the form E = (,) T We start by analyzing the behavior of the system (4) near E The characteristic equation of the linearization of (4) near E is: ( + ) γ β -A α det = β ( + γ + α) -A (3) E, is locally asymptotically stable if and only if the trace of the Jacobin matrix near E is strictly negative and its determinant is strictly positive 69

4 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN E, is locally asymptotically stable -( + )-( α+ β)< ( + β γ )( + γ + α) αβ > (33) As long as condition (33) holds the disease-free equilibrium of system (4) is unique and stays locally asymptotically stable R Let us define the basic reproduction number of the infection as αβ + γ = + + β + α + β + β (34) Lemma If R <, then the disease-free point E is locally asymptotically stable; Stable if R =, and unstable if R > Global stability Theorem If R <, then the disease-free point E is globally asymptotically stable in Ω Proof Let (, ) S I ΩSince I(t) =βs(t) +γ+α I(t) + ks(t)i(t), Then I(t) ( +γ+α ) I(t) so by integration, we get t I(t) I e +γ+α, for every t (35) If R < then ( +γ+α ) > Hence, I (t) converges to zero The second equation from (4) gives ( ) t S ( ) S I ksi ( ) S Ie +γ+α = +γ β +α +γ β +α (36) Integrating the above inequality, we obtain ( +γ+α) t +β γ αi + ( +β γ) ( +γ+α) ( +β γ)t ( )t S(t) S e e e, (37) + αi mt S(t) S e, ( +β γ) ( +γ+α) With m min{, } (38) = +β γ +γ+α (39) So if R <, S (t) converges to zero 7

5 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Lemma [7] Let D be a bounded interval in R and h: (t₀, ) D R, be bounded and uniformly continuous function, let x: (t₀, ) D R be a solution: x = h t,x (3) Which is defined on the whole interval (t₀, ), then (i)liminf h t,x limsuph t,x, (i)liminf h t,x limsuph t,x, (3) Where, x = liminf x t,x = limsup x t (3) Theoreme Let R₀=, then the disease-free point E is globally asymptotically stable in Ω Proof Let ( S, I, Q) Ω di dt From the second equation to the system () we obtain I(t) I e +γ+α t Since R₀=, then αβ ( γ )( α β ) ( β )( α β) = + + +, and So I (t) is a positive and non-increasing function, hence: I t t l, [ ) By the application of lemma to the third equation to the system (), we get: γi 3Q I 3Q γ (33) Therefore γl γl Q Q 3 3 (34) So γl limq(t) Q = Q = = Q( ) (35) 3 7

6 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Applying the same method using lemma we deduce that lims(t) = S( ) (36) Since R₀=, the point E is stable Hence for every >, there exist η () such that if + + η( ) (37) S I Q Then for every t, S(t), I(t), et Q(t) We get l, for every > therefore l= Finally for all (, ) S I Ω lims(t) = S,lim I(t) = I,et limq(t) = Q Endemic equilibrium and its locally asymptotical stability From the previous section it is follows that when the trivial equilibrium E₀ of system (3) is locally asymptotically stable, then the endemic equilibrium does not exist When R >, system (3) has a unique non-trivial equilibrium * * * E = ( S, I ) T other than the diseasefree equilibrium, where E + α + γ β + γ + γ α + γ β, k k + γ k k + γ k * = By introducing System (3) is centered at T (4) = *, = *, who are the small perturbations x t S t S y t I t I * E and its linear part reads α + γ β + γ + γ x t = x t + y t + γ + γ ( + γ )( + γ α) β ( + γ) y ( t) = x ( t) + y ( t) + γ + γ The characteristic equation of (3) at * E is ( + ) ( + )( + ) α γ β γ γ -A det + γ + γ = ( + γ )( + γ α) β ( + γ) A γ γ + + * E, is locally asymptotically stable if and only if the trace of the matrix is strictly negative and its determinant is strictly positive (4) (43) 7

7 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN * E, is locally asymptotically stable α < β ( + β γ )( + γ α) αβ > The global stability for endemic equilibrium Here, we restraint to the case when = =, we have in this case we have the system: = ( +γ β ) +α = S S I ksi P (S,I), I =βs ( +γ+α ) I + ksi = P (S,I), Theorem3 Dulac Critere We have D₁={S(t)>, I(t)>, S+I }is the region connexe of plan to phase If the function exist: (44) (5) ( DP) ( DP), S = P( S, I), I = P ( S, I) (5) S + I The orbites in D₁ are not closes Theorem4 If R₀>, then the endemique-disease point Proof Take a Dulac function [], ( DP ) ( DP ) α β + = S I S I DSI (, ) * E is globally asymptotically stable = for S, I> We have SI < (53) Hence, according to Dulac criterion, the system (4) has not periodic orbits Since (53) admit only two equilibriums E₀ and E * When R₀> and E₀ is unstable, hence by Poincar- Binedixon theorem [], * E is globally asymptotically stable The stochastic system by perturbing the contact rate We limit ourselves here to perturbing only the contact rate so we replace k by k+ σ W (t), where W(t) is white noise (Brownian motion) The system (4) is transformed to the following Itô stochastic differential equations: ( ) ds = +γ β S +αi ksi σ SIdW, di =βs +γ+α I t + ksi +σsidw, dq =γi γs 3Q (6) 73

8 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN In this section, we will prove, under some conditions, that E is globally exponentially mean square and almost surely stable, and for this purpose, we need the following Theorem: Theorem5 The set Ω is almost surely invariant by the stochastic system (6) Thus if (,, ) P( SIQ,, ) Ω = S I Q Ω, then [ ] Proof The system (6) implies thatdn - Ndt, hence Since ( S, I, Q) Ω N Ne t, for all t We have I(t) t Ie +γ+α, for every t, then ( ) +γ+α > Hence, I (t) converges to zero Because, S >, I >, andq >, then there exist > such that S >, I >, andq > Considering: { t S t I t Q t } for τ = inf, (), (), () >, (6) (63) t { t St It Qt } τ = limτ inf,,, Let V ( t) = log log log St () + It () + Qt () (64),t, Then, using Itô formula we have, for all t and T [ τ ] I T dv () t = ( + β γ) α + ki T + I T dt + σi T dw T S T S T + ( + γ + α) β ks T + S T dt σs T dw T I T S T I T γ Q T γ Q T dt (65) dv () t ( α + β ) + ki ( T ) + I ( T ) + S ( T ) dt ( I ( T ) S ( T )) dw ( T ) + σ (66) 74

9 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN For all T [ τ ],t, with L = α + β, therefore ( ) dv ( t ) LdT + σ I T S T dw ( T ) (67) Hence T ( ) V ( t ) LT + σ I u S u dw ( u ) (68) E ( V () t ) With proposition 76 in [7] σ T I u S u dw ( u ) is a mean zero process then: LT (69) For all t, EV ( t τ ) L( t τ ) Lt (6) Moreover V ( t τ ), S ( t τ ), I ( t τ ), andq ( t τ ) thus, EV ( t τ ) EV () t χ + E V () t χ (6) [ t] [ > t] τ τ EV ( t τ) EV ( t ) χ[ ] P ( τ t ) log τ t (6) Where for all t, P χd is the indicator function of a subset D Combining (69) with (6) gives ( t) τ where P ( τ ) = Lt log Tending ɛ to zero, we obtain for all t, P( t) τ =, from Spatial spread model We consider the spatial spread of the infectives and susceptibles By modeling the model (4), we adding the term S represent the interaction between the same kind of species in the susceptible population We obtain the following model: S = ( +γ β) S ksi +αi S + d S, t I =βs ( +γ+α ) I + ksi + d I, t Where S=S(x,t) and I=I(x,t) S, susceptible densities respectively (7) I represent the diffusion of the infective and Traveling wave solution We seek a constant shape travelling wave solution of (7) by setting S x, t = S( z), I x, t = I( z), z = x ct, (7) Where c is the wave speed, substitute (7) into (7) we get 75

10 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Sz Sz c = ( +β γ ) Sz + ksziz α Iz + S( z) d, z z I( z) I( z) c = β Sz + ( + γ + α) Iz ksziz d, z z (73) This assumption is a legitimate one because the infective population is very active in infecting other individuals in the total population and it is capable of moving more but the susceptibles is not so Therefore we assume that d is negligible compared to d, and then we analyze (73) equations Hence, with d =, and ' Sz z = S we can rewrite (73) as three ordinary differential ' ( +β γ ) S+ ksi α I+ S, S = c ' I = T, ' ( +γ+α) I ksi βs ct T =, d (74) In the (S, I,T) phase space there are three steady states, namely +β γ (,,) (,,),(S,I,T), +γ+α β (S,I,T) = (m+ m )e, *, k k S With γ ( ) (75), m,m are the constants We expect the traveling wave front solution to be the form (,,)to(s,i,t) and from +β γ (,,)to(s,i,t) conditions: Therefore we have to seek solution S, I of (74) with the following boundary S =,I =,S = S,I = I, +β γ S( ) =,I( ) =,S( ) = S,I( ) = I, (76) (77) 76

11 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Let us consider only (74) with (77) and the analysis of (74) with (76) is analogue We linearize (74) about the point are the roots of +β γ (,,) then determine the eigenvalues A, which μ +β-μ-γ k μ-μ-β+γ α -A - c cμ c det A = β μ(μ +α+γ)-k ( μ-μ-β+γ ) c A d μd d (78) Which are the roots of the characteristic polynomial, P A = A + fa ga + h 3, with f g h = c = + d = μ+γ-μ-β, c k-μ μ-μ-β+γ -μ(μ +α+γ), μd ( k ) + β μ +β-μ-γ μ +β-μ-γ μ +α+γ β α μ cμd (79) Using the theory of cubic polynomials we can easily see that it s possible to have three real roots To get stability to small linear perturbations we use the Routh-Hurwitz conditions for the roots P(A) to have negative real parts This holds if c + d μ+γ-μ-β >, c ( k ) μ +β-μ-γ + μ +β-μ-γ μ +α+γ β α μ > cμd β ( k ) ( k-μ)( μ-μ -β+γ) -μ(μ +α+γ) c ( μ+γ-μ -β) ( ) ( ) ( ) μ +β-μ-γ + μ +β-μ-γ μ +α+γ β αβμ + < μd d c cμd (7) It is obvious that if the first and the second inequalities of (7) hold, the third equation of (7) does not hold unless d < Therefore in order to have a traveling wave 77

12 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN front solution which approach the steady state (S, I,T) require that d < and (7) must hold, otherwise we have instability in an oscillatory manner as z, we Conclusion In this paper, we considered stability of the epidemic model SIQS We showed that if R <, the disease-free equilibrium is locally asymptotically stable, if R =, instable whereas if R > the endemic equilibrium is locally asymptotically stable Then we resolve the system SIQS epidemic model References: R M Anderson, G F Medley, R M May and A M Johnson, A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J Math Appl Med Biol 3, p 9-63, 986 Asif Yokus and D Kaya, A numerical comparison for coupled Bossiness equations by using the ADM, Dynamical Systems and Applications, Proceedings, Antalya, Turkey, P73-736, 4 Bailley N T J Some stochastic models for small epidemics in large populations, Appl Statist 3, p 9-9, 964 Bailley N T J The mathematical theory of infection diseases and its application, Appl Statist 6, p 85-87, 977 Bartlett M S An Introduction to Stochastic Processes, 3rd ed Cambridge University Press, New York, 978 Batiha, M S M Noorani and I Hashim, Numerical solutions of the nonlinear integrodifferential equations, Int J Open Probl Compt Math, p 34-4, 8 Isham V Mathematical Modeling of the Transmission Dynamics of HIV Infection and AIDS J R Statist Soc A 5, Part, p5-3, 988 Isham V Stochastic Models For Epidemics with special References to AIDS The Annals of Applied Probability Vol3 N, p -7, 993 Jin Z, Zhien M and Maoan, H Globale stability of an SIRS epidemic model with delay, Acta Matimatica Scientia 6 B 9-36, 6 Lounes R, Arazoza, H A Two-Sex Model for the AIDS-Epidemic Application to the Cuban National Programme on HIV-AIDS Second Conference on Operation Research, Habana, 3-5 October995, Cuba,

13 European Scientific Journal February 3 edition vol9, No6 ISSN: (Print) e - ISSN Lounes R, Arazoza, H Modeling HIV Epidemic Under Contact Tracing The Cuban Case Journal of theoritical Medecine Vol, p 67-74, Lounes R, Arazoza, H A Non-Linear Model for a Sexually Transmitted Disease with contact tracing IMA J MJath Appl Med Biol 9, p -34, Lounes R, Arazoza, H What percentage of the Cuban HIV-AIDS Epidemic is known? Rev Cubana Med Trop; 55() p3-37, 3 Luo Q and Mao M, Stochastic population dynamics under regim switching, JMath Anal Appl334 p 69-84, 7 Perto L Differential Equations and Dynamical Systems nd edition, Springer, New York, 996 Wang K, Wang W, Liu X Viral infection model with periodic lytic immune response Chaos, Solitons &Fractals; 8: p9-99, 6 Xiao D and Ruan, S Global analysis of an epidemic model with nonmonotone incidence rate, Math Bio, V8, No P 49-49, 7 79