Global Dynamics of an SEIR Model with Infectious Force in Latent and Recovered Period and Standard Incidence Rate

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1 International Journal of pplie Physics an Mathematics Global Dynamics of an EIR Moel with Infectious Force in Latent an Recovere Perio an tanar Incience Rate Yanli Ma Department of Common Course nhui Xinhua University Hefei nhui 88 China. Corresponing author. Tel.: ; Manuscript submitte March 6; accepte eptember 6. oi:.776/ijapm bstract: In this paper we consier an epiemic moel with the infectious force in the latent an recovere perio an establish the EIR epiemic moel with stanar incience rate. Then we fin the basic reprouction number R which etermines whether the isease exists. By using Liapunov function metho we prove that the isease-free equilibrium E is globally asymptotically stable an the isease goes away when R. By Hurwitz criterion we also prove that E is unstable an the unique enemic equilibrium E is locally asymptotically stable when R. It is shown that when isease-inuce eath rate an elimination rate are zero E is globally asymptotically stable an the isease persists. Finally we give numerical simulation to illustrate the theoretical analysis. Key wors: Basic reprouctive number equilibrium stability EIR epiemic moel numerical simulation.. Introuction In our life there are a variety of infectious iseases. For a long time mathematical moels which escribe the population ynamics of infectious iseases have been playing an important role in a better unerstaning of isease control an epiemic pattern. In orer to preict the sprea of infectious isease among the areas the transmission ynamics of infectious isease is stuie by many epiemic moels in host populations. The ynamics of the classic IR or IR epiemic moels have been wiely stuie []-[]. However many infectious iseases such as pertussis R an so on incubate insie the population for a perio of time before they become infectious. o the systems that are more general than IR or IR types nee to stuy the role of incubation in the sprea of infectious isease. The present moel is of EIR or EIR class epening on whether the aaptive immunity is permanent or otherwise. Li [] analyze the global ynamics of a EIR moel with varying total population size. Fan [] iscusse the global stability of an EI epiemic moel with recruitment an a varying total population size. un [] stuie the global analysis of an EIR moel with varying population size an vaccination. Yi [] consiere an EIR epiemic system with nonlinear transmission rate. Zhang [5] stuie global asymptotic stability of a elaye EIR epiemic moel with saturation incience. Yan an Zhang [6] consiere an EIR epiemic system with nonlinear transmission rate. Liu [7] analyze the global stability of an EIR epiemic moel with age-epenent latency an relapse. However for malaria an some other infectious iseases the latent an immune perio may be infectious. The epiemic moel of EIR with infectious force in both infecte an recovere perio is rarely stuie in the paper. Motivate by literature []-[7] we consier an epiemic moel with the infectious force in the Volume 7 umber January 7

2 International Journal of pplie Physics an Mathematics latent an recovere perio an establish the EIR epiemic moel with stanar incience rate.. Moel Formulation The host population is ivie into four classes the susceptible latent infectious an recovere with sizes enote by t E t I t an R t respectively. The EIR moel having infectious force in the latent an recovere perio is epicte in the following system of ifferential equations E I R E I R E k E I E k I R I R. () where is constant recruitment rate of the population; the latent infecte an recovere perio respectively; are the rate of the efficient contact in are enote the rate of isease-cause eath of the expose an the infectious respectively; is the natural eath rate of the population; k k the elimination rate of the expose an the infectious respectively; is the transfer rates between the expose an the infectious; is the remove rate from the infective class to the recovere class. The total population size t which can be etermine by t t E t I t R t implies. From biological consierations we stuy () in the feasible region E I R R E I R where are R enotes the non-negative cone an its lower imensional faces. can be shown to be positively invariant with respect to (). et each of the ifferential equations on the right sie equal to zero in () we have the following system E I R. () n we have the following equation about F. Volume 7 umber January 7

3 International Journal of pplie Physics an Mathematics where F so the system () always has the isease-free equilibrium E in the interval Define the basic reprouction number of () as R F.. ince F R. F ( ) is monotone increasing an F so that F ( ) has only a positive root in () has an unique enemic equilibrium E E I R where E I R are etermine by ().. Global tability of the Disease-free Equilibrium Theorem. The isease-free equilibrium an E of () is globally asymptotically stable in if R E is unstable if R. The solutions to () starting sufficiently close to E in move away from E except that those starting on the invariant axis approach E along this axis. Proof. Constructing a Liapunov function V E I R Its erivative along the solutions of () with respect to t gives V t + + E I R E + E - I I R E I R R E I R R. Furthermore V if an only if E I R or R set in E I R R. Therefore the maximum compact invariant V is the singleton E when R. Laalle's invariance principle then implies that E is globally asymptotically stable in.. Local tability of the Enemic Equilibrium Theorem. The enemic equilibrium E of () is locally asymptotically stable if Proof. The Jacobian matrix of () at a point E R. is Volume 7 umber January 7

4 International Journal of pplie Physics an Mathematics a a J E 5 a a a a a a. m m m m a where a a a a5 m m E I R. Its characteristic equation is et E I R I I J E where. o the I is the unit matrix E m characteristic equation become to b b b a where b b m m m b m m. It is easy to calculate bb b a. ccoring to Hurwitz criterion the enemic equilibrium E of () is local asymptotical stability. 5. Global tability of the Enemic Equilibrium Let =t the system () becomes E I R E E I R E I E I R I R. where () k k k k k k. The equation for the total population is k E k I. Volume 7 umber January 7

5 International Journal of pplie Physics an Mathematics We use as a variable in place of the variable to give the following system E E I R E I R E I EI R I R k E k I. () The system () is equivalent to (). From biological consierations we iscuss () in the close set = T E I R R E I R. We enote by T an T the bounary an the interior of T in R respectively. For () the global stability of the enemic equilibrium E is consiere when k k. ince as we can obtain the following limit system E E I R E I R E I EI R I R. We make the change of variable x E I R y E z I then the following (5) is equivalent to the above system x x y z x x y y z x x y z yz. (5) Theorem. Consier the following system n x f x f C R n xt R. (6) where T is an open set if the system (6) satisfies the following conitions: () The system (6) exists a compact absorbing set K T an has a unique equilibrium P in T ; () P is local asymptotically 5 Volume 7 umber January 7

6 International Journal of pplie Physics an Mathematics stable; () The system (6) satisfies a Poincare-Benixson criterion; () perioic orbit of (6) is asymptotically orbitally stable. Then the only equilibrium P is the globally asymptotically stable int. Theorem. sufficient conition for a perioic orbit orbitally stable with asymptotic phase is that the linear system asymptotically stable where f t f. The system (6) calle the secon compoun system of the orbit P P t :t of (6) to be asymptotically f zt P t z t t is the secon aitive compoun matrix of the Jacobian matrix Pt. Lemma. ny perioic solution to (5) if it exists is asymptotically orbitally stable. Proof. uppose that the solution x y z T. The perioic orbit is system y J P y perioic linear system of the ifferential system x t y t z t is perioic of least perio f t is of such that P P t :t. We have the secon compoun x J P x in the perioic solution is the following X ax xy Z Y X ay xz Z ay x Z. (7) where a x y z x a x y z x a x y z a x x J P a x x 6 Volume 7 umber January 7

7 International Journal of pplie Physics an Mathematics a x x J P a x. a x uppose that X t Y t Z t is a solution to (7). Let sup V X Y Z x y z X y Y Z z. From the conition () of Theorem we can know there exists constant X Y Z X Y Z R x y z V X Y Z x y z ifferential inequalities D X t a X t x Y t Z t such that P. Direct calculations lea to the following (8) Using (9) an () having D Y t X t a Y t x Z t (9) D Z t a Y t x Z t. () y z X t Y t Z t D Y t Z t From (6) an (9) leaing to where D V y z y y z z y z. () t supg g V t () x y g y z y y z z Rewriting the last two equations of (5) obtaining z x xz y () g. () xz y y y x y x x y (5) y z z z. (6) ubstituting (5) into () an (6) into () having 7 Volume 7 umber January 7

8 International Journal of pplie Physics an Mathematics g y y x y z x x y y z (7) sup g t g t y y Thus g. (8) y y supg t g tt ln y t t an in turn that V X Y Z x y z X Y Z limv t. From () we obtain as t. s a result the secon compoun system (5) is asymptotically stable an the perioic solution asymptotical orbital stability by Theorem. Lemma. The system (5) is uniformly persistent when R. Proof. et G P when R the stable set bounary of. It also implies that the stable set s G x t y t z t is is just containe in the -axis an thus in the s G is isolate in. Then when R the system (5) satisfies the conitions of Theorem of [8] namely (a) the maximal compact invariant set G bounary of is isolate an (b)the stable set s G the system (5) is uniformly persistent in when R. Lemma. The system () is competitive when Proof. et x x E x I x R in the of G is containe in the bounary of. Therefore R an k k. the system () is replace by x x x x x x x x x x x. x x x x x x x x x x x Furthermore the system has the following form x x t I C t. x x x x x R I enotes the unit matrix where Ct is a vector function an t. The off-iagonal entries in this matrix are non-negative the system as a whole is quasimonotone. Thus it 8 Volume 7 umber January 7

9 International Journal of pplie Physics an Mathematics can be verifie that the system () is competitive with respect to the partial orering efine by the orthant K EI R R E I R. R the enemic equilibrium E of () is globally asymptotically stable when Theorem 5. If k k. 6. The umerical imulation.5.5 E I R Fig.. Variational curves of E I an R with t when R E R I Fig.. Variational curves of E I an R with t when R.7. 9 Volume 7 umber January 7

10 International Journal of pplie Physics an Mathematics et the parameters.6.5 k.8 n set five initial values k When set.8.5 an. the basic reprouction number R.55. The isease-free equilibrium E is global asymptotical stability in as shown in the follow Fig.. When set.7.5 an.5 the basic reprouction number R.7. The enemic equilibrium E is global asymptotical stability in as shown in the Fig.. 7. Conclusions This paper has iscusse an EIR epiemic moel with infectious force in latent an recovere perio. The basic reprouctive number R is ientifie. If R the isease-free equilibrium E is globally asymptotically stable in the feasible region so that the isease ies out. If R the enemic equilibrium E is globally asymptotically stable in the interior of the feasible region an once the isease appears it eventually persists at the unique enemic equilibrium level. It is obtaine that function of an by the formula of R is a monotony increase R. The smaller the infectious force in latent an recovere perio the more etrimental to the control an elimination of the infectious isease. Therefore the infecte patients the latent patients an recovere patients are all controlle for the isease with the infectious force in latent an recovere perio. In this paper we stuy the asymptotic stability of the EIR moel with infectious force in latent an recovere perio an enriches the research work of the ynamics of infectious iseases. In aition there is a lot of work waiting for us to stuy in this fiel. It is ifficult for the pulse moe of the epiemic moel to be thought of. We will stuy these problems in the future. cknowlegment This work is supporte by the atural cience Founation of nhui (KJ5) an the atural cience Founation of nhui (KJ58). References [] Wang J. Zhang Z. & Jin Z. (). nalysis of an sir moel with bilinear incience rate. onlinear nalysis: Real Worl pplications () 9. [] Kanhway K. & Kuri J. (). How to run a campaign: Optimal control of I an IR information epiemics. pplie Mathematics an Computation () [] Muroya Y. Huaixing L. & Kuniya T. (). Complete global analysis of an IR epiemic moel with grae cure an incomplete recovery rates. Journal of Mathematical nalysis an pplications () [] Yanan Z. & Daqing J. (). The threshol of a stochastic IR epiemic moel with saturate incience. pplie Mathematics Letters () 9 9. [5] Qun L. Qingmei C. & Daqing J. (5). The threshol of a stochastic elaye IR epiemic moel with Volume 7 umber January 7

11 International Journal of pplie Physics an Mathematics temporary immunity. Physica : tatistical Mechanics an its pplications 5(5) 5 5. [6] Zengyun H. Zhiong T. & Long Z. (). tability an bifurcation analysis in a iscrete IR epiemic moel. Mathematics an Computers in imulation 97() 8 9. [7] Rui X. & Yanke D. (). elaye IR epiemic moel with saturation incience an a constant infectious perio. Journal of pplie Mathematics an Computing 5( ) 9 5. [8] Zhixing H. Wanbiao M. & higui R. (). nalysis of IR epiemic moels with nonlinear incience rate an treatment. Mathematical Biosciences 8(). [9] Qianqian C. & Qiang Z. (5). Global stability of a iscrete IR epiemic moel with vaccination an treatment. Journal of Difference Equations an pplications () 7. [] Zhichao J. & Wanbiao M. (5). Permanence of a elaye IR epiemic moel with general nonlinear incience rate. Mathematical Methos in the pplie ciences 8() [] Zhichao J. Wanbiao M. et al. (999). Global ynamics of a EIR moel with varying total population size. Mathematical Biosciences 6() 9. [] Meng F. Michae l Y. & Wang K. (). Global stability of an EI epiemic moel with recruitment an a varying total population size. Mathematical Biosciences 7() [] Chengjun. & Ying-Hen H. (). Global analysis of an EIR moel with varying population size an vaccination. pplie Mathematical Moelling () [] a Y. Qingling Z. Kun M. Dongmei Y. & Qin L. (9). nalysis an control of an EIR epiemic system with nonlinear transmission rate. Mathematical an Computer Moelling 5(9) [5] Tailei Z. & Zhiong T. (8). Global asymptotic stability of a elaye EIR epiemic moel with saturation incience. Chaos olitons & Fractals 7(5) [6] Qingling Z. Chao L. & Xue Z. (). nalysis an control of an EIR epiemic system with nonlinear transmission rate. Complexity nalysis an Control of ingular Biological ystems () 5. [7] Lili L. Jinliang W. & Xianning L. (5). Global stability of an EIR epiemic moel with age-epenent latency an relapse. onlinear nalysis: Real Worl pplications () 8 5. [8] Michael Y. Graef R. Liancheng W. et al. (999). Global ynamics of a EIR moel with varying total population size. Mathematical Biosciences 6() 9. Yanli Ma was born in nhui of China on October he finishe her bachelor's egree in mathematics an computing science from Huaibei ormal University the Department of Mathematics Huaibei nhui China in 6. he finishe her master's egree in applie mathematics from Xi'an Jiao Tong University College of cience Xian hanxi China in 9. Her research interest lies in applie mathematics biology mathematics ynamics of infectious iseases an computer simulation. o far in 9 she is working as a teacher at epartment of common course nhui Xinhua University Hefei nhui China. he has publishe several papers in national Journals. Her current research interest lies in the stuy of ifferential ynamic system. Volume 7 umber January 7