Analysis of an SEIR Epidemic Model with a General Feedback Vaccination Law

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1 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. Analysis of an ER Epidemic Model wit a General Feedback Vaccination Law M. De la en. Alonso-Quesada A.beas and R. Nistal Abstract-Tis paper discusses and formulates a continuoustime ER -type epidemic model of pseudo-mass action type wit finitely distributed delays under a very general and in general time-varying vaccination control rule wic eventually generates feedback actions on te susceptible infectious and recovered subpopulations. A lot of particular vaccination laws can be got from te proposed general one. Te equilibrium points are caracterized and teir local stability properties discussed depending on te limits of te vaccination control gains provided tat tey converge asymptotically. Keywords- Epidemic models; distributed delays; ER model; feedback vaccination controls; equilibrium points.. NTRODUCTON H researc is concerned wit a ER epidemic T model subject to finitely distributed delays and eventual vaccination wic is of pseudo-mass action type in te sense tat te infective transmission rate does not depend directly on te total population [-3]. Te continuoustime model as te following caracteristics and properties: a) Te vaccination controls ave eventual feedback actions of te susceptible infected and recovered subpopulations and also an independent term wic ave in general time-varying gains wit a constant term plus an incremental one. Te independent term selection guarantees te non-negativity of te statetrajectory solution for all time so as to reflect real situations. Te structure of te vaccination control law is very general and it can be also implemented in te case wen te subpopulation numbers are not precisely known b) Te disease-free and te endemic equilibrium points are caracterized as well as teir local asymptotic stability properties in te case tat te vaccination controller gains converge asymptotically to limits. t is proved tat te infection is non-permanent and te state-trajectory solution converges asymptotically to te disease-free equilibrium point if te disease infective rate is under a certain maximum tresold. Manuscript received February 5; revised February 3 5. Tis work was supported in part by te panis Government for Grants DP-365and DP3-785-C3--R and to te Basque Government and UPV/EHU for Grant T378-. M.De la en. Alonso and R.Nistal are wit te DP of te UPV/EHU.Campus of Leioa Barrio arriena pain ( manuel.delasen@eu.es) A. beas is wir te UAB 893-Barcelona pain ( Asier.beas@uab.cat) Tere are new caracteristics in te sceme concerning te generality of te vaccination law related to te existing previous background..feedback VACCNATON EPDEMC MODEL Consider te following ER epidemic model wit a delayed-distributed transmission effect: t b V t t t f t d t t f t d b Et t Et b t t bv t t brt V t if V t t sat V t if V t () E () (3) R () V (5) V t k t Ŝ t k t Î t k t Ŝ t f Ît d k t 3 (6) were t E t t and R t are respectively te susceptible exposed infectious and recovered subpopulations at time t and V : is a non-negative real feedback vaccination control defined troug te real control gains ki : kmin kmax ; i 3 wit k max k min. Te functions Ŝt t t and Ît t t t wit m t M and m t M t are estimates of te susceptible and infectious subpopulations used for te vaccination law implementation wit m M m and m M m being known real constants wic are minimum and maximum relative per-unit errors of te estimates. Te following positive parameters parameterize te system ()-(6): b is te birt-rate of te population is te infectivity disease rate is te transition rate from te exposed subpopulation to te infectious one BN: N: (Print); N: (Online) WCE 5

2 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. is te transition rate from te infectious subpopulation to te recovered one. Note tat is te average time tat an infectious individual stays at tis stage before recovering completely. t is assumed tat f : and tat f d f d (7) One gets te following dynamics for te total N t t E t t R t t : population t t N b N t (8) wic implies tat at any existing equilibrium point N lim N t. A result wic guarantees te linear t structure of te feedback vaccination law is te following: Teorem. Te vaccination law is given by t k t k t f Ît d Ŝ t k t Î t k t V 3 t (9) so tat it does not enter te saturation zone at any time if k max max ki t i Ŝ t f s Î t s ds t t f s s t s ds t () A sufficient conditions for te above condition to old is: kmax t f st sds M M wic olds under te stronger constraint: t () k max t t f s t s ds. EQULBRUM PONT () Te disease-free and endemic equilibrium points are caracterized and discussed in te following: Teorem. Te disease-free equilibrium point of ()-(6) is x T k E R k k k k (3) and te corresponding equilibrium vaccination value and total population are respectively k k V R k lim k i t lim t t k i t and N R if ; 3 lim t i t T () and tat for te disease-free equilibrium point (3) one as k k k max ki i f k for k ten 5 if k. t (5) k and Teorem 3. Te following properties old: k satisfy (i) A necessary condition for an equilibrium point x E R T to exist being an endemic equilibrium (tat is E ) of ()-(6) is tat te infectivity disease rate is large enoug b b satisfying. (ii) A necessary condition for suc an endemic equilibrium point to exist under a saturation-free equilibrium vaccination b b V. V is tat BN: N: (Print); N: (Online) WCE 5

3 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. (iii) Te exposed and infectious subpopulations of te endemic equilibrium point satisfy te constraints: b E b b b b b b (6) (iv) f b k b k ten tere is a unique feasiblein te sense tat all its components are positive endemic equilibrium point. Te local stability of te disease-free equilibrium point is now caracterized. Teorem. Te following properties old: (i) Te delay-free disease-free equilibrium point of te deterministic ER model ()-(6) under a linear limiting control satisfying te conditions of Teorem is locally uniformly asymptotically stable. n te presence of distributed delay te system is still locally uniformly asymptotically stable if te transfer matrix s A Ĥ s is in RH wit H norm s A Ĥ s s T were Ĥ s e e e fˆ s 3 were s denotes te Laplace transform argument s L f t T e T e b k3 A bk3 3 and b fˆ b b k b k 3 k k bk 3 k b b b (7) (ii) A sufficient condition for Property (i) to old is tat te infective disease rate be small enoug to satisfy / fˆ s s A. Remark. Note tat since all te eigenvalues of s A negative ten RH te reproduction number R p is defined as: A are. n te delayed case s A Ĥ s sup i A Ĥi T Rp e3 R wit R R e 3 being te tird unity vector in te canonical basis of R and i. f R p / fˆ s s A equivalently if ten te disease-free equilibrium point is locally asymptotically stable. f R p te infection propagates. Note tat R p is equivalent to T s e s A Ĥ s Bˆ 3 being bounded real (i.e. cur wit real coefficients) and to te transfer function Bˆ s Ŝ s to be strictly positive real [5-6]. Bˆ s V. NUMERCAL MULATON WTH FURTHER ANALY Tis section contains some numerical examples illustrating te teoretical results introduced in te BN: N: (Print); N: (Online) WCE 5

4 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. previous sections. Te subsequent extended stocastic version of te deterministic model ()-(6) is stated by modifying ()-() as follows: E t b V t t t f t d t w t t t f t d b Et E t t E t b t t (7) E w t w t R t R w t (8) 3 3 (9) t bv t t br t t R () were w i are mutually independent standard Wiener processes i.e. mutually independent definite integrals from zero to time t of a zero mean unit variance wite Gaussian stocastic processes tat is w i t Edwi t w t E i ; i 3 for t wit E denoting expectation te functions t w t i ; i 3 are almost everywere surely continuous and i ; i 3 are real parameters. Te parameters of te model are given by b 5. 5 days. days.66 days 3. 5 days and. Te function t t f is defined by t f f for. 5 / N t for t t in similarity wit te standard incidence rate for delay-free models. is te system in te absence of vaccination for te deterministic case. Te system gets te endemic equilibrium values. 3 E and R. 58 By comparing Figures and we can see tat bot te deterministic and stocastic systems possess te same equilibrium points. Te vaccination control law given by (5)-(6) is now applied to te system in order to eradicate te illness from te population. Te control parameters k( t ) k. 6 an stability abscissa for k ( t ) k3 and k( t ) k. 7 provide A of.6. Moreover 3 is selected to be k3 ( t ) k3.. Populations R E 5 5 Figure. Deterministic endemic equilibrium point in te absence of vaccination.8 normalization constant guaranteeing f ( ) d. Te initial conditions are. 5. E. and R. so tat te initial total population N. Te constant values ( t ). 5 is Populations R ( t ). and R ( t ) R. are used for simulation purposes. Te Wiener processes parameters are and.. Figure sows te final values acieved by te trajectory of te.. E 5 5 Figure. tocastic endemic equilibrium point in te absence of vaccination BN: N: (Print); N: (Online) WCE 5

5 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. Te control parameter function k ( t ) does not ave to satisfy any special requirement tat cannot be accomplised by using te oters. Terefore it will be fixed to zero k ( t ) for te sake of simplicity. Te last control parameter function k 5 ( t ) is potentially timevarying since its purpose is to guarantee tat te control law always lies witin ( t ) V (te linear feedback condition). t can be seen in Figure 3 tat te disease is removed asymptotically from te population since te exposed and infectious subpopulations converge to zero. Figure displays te time evolution of te vaccination. t is confined to te interval by te action of te control function t depicted in Figure 5. t can be seen k 5 in Figure 3 tat R. 75 wic is exactly te equilibrium value of te vaccination as Figure reveals. Tese results also old in te stocastic case. Tus Figure 6 displays te system s trajectory wen a Wiener process is added to te system dynamics wile Figure 7 sows te corresponding vaccination function. Terefore we can see in Figures 6 and 7 tat te disease is asymptotically removed te percentages of susceptible and immune correspond to tose selected beforeand and te vaccination function converges to te value of immune at equilibrium. Te solution of te ER model under te standard independent Wiener processes (7)- () and te vaccination feedback law (5)-(6) of te given class is given by t t t x t A x A x v x w t were te evolution operator is given by: d w t A / t t t t t t t e wit s s d w w t / t t e d t Populations E R 5 5 Figure 3. tate trajectory wen te feedback control law is applied. Deterministic case Vaccination Figure. Vaccination law. Deterministic case w t t e / t t e w so tat A t and d t e E E E t d E s s were is te identity matrix Te evolution operator t follows for a Wiener- type forced differential process of te form BN: N: (Print); N: (Online) WCE 5

6 Proceedings of te World Congress on Engineering 5 Vol WCE 5 July London U.K. dx t A x t dt x t d wt Ft t [] wit omogeneous part dx t A xt dt xt d wt. We can get after some calculations te subsequent result: Teorem 5: lim E x t x if Teorem related to te t deterministic version of te ER model. Figure 7. Vaccination law. tocastic case free equilibrium point is lost ACKNOWLEDGMENT Te autors are very grateful to te panis Government for Grants DP-365and DP3-785-C3--R and to te Basque Government and UPV/EHU for Grant T378- AOTEK -PE3UN39 and UF /7. Figure 5. Evolution of k 5 t Figure 6. tate trajectory wen te feedback control law is applied. tocastic case REFERENCE [] M. Keeling and P. Roani Modeling nfectious Diseases in Humansand Animals Princeton University Press New Jersey 8. [] Epidemic Models: Teir tructure and Relation Data Denis Mollison Editor Publications of te Newton nstitute Cambridge University Press Cambridge 995. [3] M. De la en A. beas and. Alonso-Quesada Feedback linearization-based vaccination control strategies for true-mass action type ER epidemic models Nonlinear Analysis: Modelling and Control Vol. 6 No. 3 p [] L. C. Evans An ntroduction to tocastic Differential Equations American Matematical ociety Providence Rode sland 3. [5] P. Dorato L. Fortuna and G. Muscato Robust control for unstructured perturbations: An introduction Lecture Notes in Control and nformation ciences pringer-verlag Vol. 68 Heidelberg Germany 99. [6]. Boyd V.Balakrisnan and P. Kabamba On computing te H norm of a transfer matrix Proceedings of te American Control Conference pp Atlanta 988. BN: N: (Print); N: (Online) WCE 5