Local and Global Stability of Host-Vector Disease Models

Transcription

1 Local and Global Stability of Host-Vector Disease Models Marc 4, 2008 Cruz Vargas-De-León 1, Jorge Armando Castro Hernández Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, México and Departamento de Matemáticas, Facultad de Ciencias, UNAM, México Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, IPN, México Abstract In tis work we deal wit global stability properties of two ost-vector disease models using te Poicaré-Bendixson Teorem and Second Metod of Lyapunov. We construct a Lyapunov function for eac Vector-Host model. We proved tat te local and global stability are completely determined by te tresold parameter, R 0. If R 0 1, te disease-free equilibrium point is globally asymptotically stable. If R 0 > 1, a unique endemic equilibrium point exists and is globally asymptotically stable in te interior of te feasible region. Key Words: Lyapunov Function, Dulac Function, Global Stability, LaSalle s invariant principle 1 Introduction It is ard to overstate te medical importance and burden of vector-transmitted infectious diseases malaria, Cagas disease, leismaniasis, dengue fever, lympatic filariasis, West Nile virus and te encepalitis viruses). Te key elements involved in uman vector-borne infectious diseases are te infectious microorganism virus, bacterium or parasite), te vector mosquitoes, tick or fly), and te reservoir from wic te vector obtains te infection. Control strategies for tese diseases sould be informed by an understanding of te complex dynamics of vector-ost interactions and te ways in wic te environments of bot te vector and ost intersect to produce uman disease. Te study of tese dynamics using matematical models as ad a staggering development in recent years, and as proven to be a valuable tool to understand 1 Corresponding autor. leoncruz82@yaoo.com.mx 1

2 epidemiological patterns and processes, provided tat models are as close as possible to real life situations and based on biological knowledge. In te case of dengue fever and West Nile virus, te matematical models we ave found in te literature propose compartmental dynamics in [3,4,5,6,7,8] and [2,10,11], respectively. We proved te global stability of ost-vector disease models by means te Poincaré-Bendixson teorem and second metod of Lyapunov. However it is difficult to construct a Dulac function and Lyapunov functions to establis te global stability of te equilibrium E. C. Vargas-De-León proved in [12], te global stability of endemic equilibrium of ost-vector disease models using te Lyapunov function: ), and LaSalle s invariant principle. Lx 1, x 2,...x n ) = n i=1 c ix i x i x i log xi x i 2 Host-Vector Disease Model In tis section, we study a Host-Vector Model wit constant population sizes in te ost and vector population and no immunity in ost population. We suppose tat we dispose of a ost population respectively of vector population) of size N resp. N v ) formed of Susceptibles S and Infectious I resp. S v and I v ). We consider a compartmental model tat is to say tat every population is divided into classes, and tat one individual of a population passes from one class to anoter wit a suitable rate. We use te SIS Susceptible-Infectious-Susceptible) model for te ost population and a SI Susceptible-Infectious) model for te vector population. Some vector-borne infectious diseases confer no immunity ten recovered wit te possibility of becoming susceptible. Tis type of disease can be modelled by tis model, i.e. malaria. Te dynamics of tis disease in te ost and vector populations are given by te following system of non-linear differential equations: ds di ds v di v = µn κs I v µs + φi = κs I v µ + φ)i = ηn v βs v I ηs v 1) = βs v I ηi v. Wit te conditions: N = S + I and N v = S v + I v. Te parameters are positive constants. Under te assumptions tat transmission term is mass action. Mass action assumes tat biting rates are limited by te densities of bot vectors and osts. Te vector infectious-to-ost susceptible transmission rate κs I v, were as te ost infectious-to-vector susceptible transmission rate βs v I. Te ost total population N is constant, µ is te per capita natural 2

3 mortality rate of ost. Te recovered individuals are removed from te susceptible class S ) at te rate, φ. Te vector total population N v is constant, η is te per capita natural mortality rate of vector. We tus study system 1) in te following feasible region: Γ = { S, I, S v, I v ) R 4 + : S, I, S v, I v 0, S + I = N, S v + I v = N v }, were Γ is positively invariant wit respect to 1). Direct calculation sows tat system 1) as two possible equilibrium points in R 4 +: te disease-free equilibrium point E 0 S 0, I0, S0 v, I 0 v ), were S 0 = N, I 0 = 0, S0 v = N v, I 0 v = 0, and a unique endemic equilibrium point E S, I, S v, I v ) wit S = N Hµ+φ)R 0+κN v) κn v+µ+φ)r 0, I = ηµ+φ)r0 1) βκn, v+µ+φ) S v = NvκNv+µ+φ) µ+φ)r 0+κN v) and I v = Nvµ+φ)R0 1) µ+φ)r 0+κN v). Let R 0 = κβn N v ηµ+φ) is tresold parameter. Te quantity R 0 = R 0 is called te basic reproductive number of te disease, since it represents te average number of secondary cases tat one case can produce if introduced into a susceptible population. Ten, we ave proved te following proposition: Proposition 1. System 1) always as te disease free equilibrium E 0 N, 0, N v, 0). If R 0 > 1 tere is also a unique endemic equilibrium E S, I, S v, Iv ) in te interior of Γ. Te uman and vector populations remain constant in Γ, terefore, witout loss of generality, since S = N I y S v = N v I v, system 1) in te invariant space Γ can be written as te equivalent two-dimensional non-linear system of ODEs: di di v = κn I )I v µ + φ)i 2) = βn v I v )I ηi v We tus study system 2) in te following feasible region: Ω = { I, I v ) R 2 + : 0 < I N, 0 < I v N v } and te equilibrium points are defined: te trivial equilibrium point P 0 0, 0), and an interior equilibrium point P I, I v ) wit I = ηµ+φ)r0 1) βκn v+µ+φ), and I v = Nvµ+φ)R0 1) µ+φ)r 0+κN v). 3

4 2.1 Asymptotic stability analysis In tis subsection, we study te local stability properties of 2). Teorem 1. i) If R 0 < 1, ten te disease-free equilibrium point P 0 is locally asymptotically stable. ii) If R 0 > 1, P 0 is unstable and P is locally asymptotically stable. Proof We study te local stability of te trivial equilibrium point P 0. Te Jacobian matrix J = I, I v ) of system 2) is given by J = κi v + µ + φ) κn I ), βn v I v ) βi + η) Ten, te local stability of P 0 is governed by te eigenvalues of te matrix µ + φ) J P 0 = κn, βn v η and te caracteristic polynomial of J P 0 is: τ 2 + µ + φ + η)τ + ηµ + φ)1 R 0 ) = 0. 3) Te two eigenvalues of 3) ave negative real parts if and only if te coefficients of 3) are positive, and tis occurs if and only if R 0 < 1. Terefore P 0 is locally asymptotically stable for R 0 < 1. If for R 0 > 1 te equilibrium P 0 becomes an unstable yperbolic point. Tis proves Teorem 1. i). Te local stability of P is given by te Jacobian evaluated in tis point: κiv + µ + φ) κn I ) J P =, βn v Iv ) βi + η) tat can be rewritten as κiv + µ + φ) J P = µ + φ) I I v η I v I βi + η) wen we take into account te identities: κn I ) = µ + φ) I I and βn v I v v ) = η I v I wic are obtained from equations of system 2) wit te rigt-and side equal to zero. Hence, te caracteristic polynomial of te linealized system is given by, Φτ) = τ 2 + κi v + βi + µ + φ + η)τ + βi κi v + µ + φ) + ηκi v. 4) 4

5 Te coefficients of 4) are positive, ten te two eigenvalues of J P ave negative real part. Terefore P is locally asymptotically stable for R 0 > 1. Tis proves Teorem 1. ii). 2.2 Global stability of interior equilibrium of 2) In tis subsection, we study te global stability of interior equilibrium of system 2) using te Poicaré-Bendixson Teorem. Teorem 2. If R 0 > 1, ten P is globally asymptotically stable in te interior of Ω. Proof Denote te rigt-and side of 2) by f, g) and coose a Dulac function as Ten we ave Df) I DI, I v ) = 1 I I v. 5) + Dg) I v = κn I 2 βn v Iv 2, wen I, I v > 0. Tus, system 2) does not ave a limit cycle in te interior of Ω. Ten, it is easy to see tat P is globally stable in te interior of Ω. 2.3 Global stability of systems 1) In tis subsection, we study te global stability properties of system 1) using te Second Metod Lyapunov. Teorem 3. i) If R 0 1, ten te disease-free equilibrium point E 0 is globally asymptotically stable in Γ. ii)if R 0 > 1, ten E is globally asymptotically stable in te interior of Γ. Proof of Teorem 3 i): Define te Lyapunov function U : {S, I, S v, I v ) Γ : S, S v > 0} R by US, I, S v, I v ) = + S N N log S µ + φ) βn v N ) + I S v N v N v log S ) v + I v. 6) N v Ten U is C 1 on te interior of Γ, E 0 is te global minimum of U on Γ, and US 0, I0, S0 v, I 0 v ) = 0. Te time derivative of U computed along solutions of 1) is 5

6 du = du = + 1 N ) ds + di µ + φ) + 1 N ) v dsv S βn v S v + di v, 1 N ) [µn κs I v µs + φi ] + [κs I v µ + φ)i ] S µ + φ) 1 N ) v [ηn v βs v I ηs v ] + [βs v I ηi v ]. βn v S v From I = N S, S 0 = N and S 0 v = N v, we ave du = µ + φ) S N ) 2 S µ+φ) S v N v) 2 βn v S v µ+φ) βn v 1 R 0 )I v 0. Terefore, if R 0 1, du 0 for all S,I,S v, I v > 0. Note tat, du = 0 if and only if S = S 0, S v = Sv 0 and I v = 0, or if R 0 = 1, S = S 0 and S v = Sv. 0 Terefore te largest compact invariant set in { S, I, S v, I v ) Γ : du = 0} is te singleton {E 0 }, were E 0 is te disease-free equilibrium point. LaSalle s invariant principle [9] ten implies tat E 0 is globally asymptotically stable in Γ. Tis proves Teorem 3 i). Proof of Teorem 3 ii): Define te Lyapunov function Define L : {S, I, S v, I v ) Γ : S, I, S v, I v > 0} R by LS, I, S v, I v ) = c 1 S S S log S ) + c 2 I I I log I ) S + c 3 S v S v S v log S v S v I ) + c 4 I v I v I v log I v I v ), 7) were c 1 = c 2 = βs vi and c 3 = c 4 = κs I v. Ten L is C 1 on te interior of Γ, P is te global minimum of L on Γ, and LS, I, S v, I v ) = 0. Te time derivative of L computed along solutions of system 1) is ) ) dl = βs vi 1 S ds + βs S vi 1 I di I ) ) + κsi v 1 S v dsv S v + κs Iv 1 I v div I v ) dl = βs vi 1 S [µn κs I v µs + φi ] S ) + βsvi 1 I [κs I v µ + φ)i ] I ) + κsi v 1 S v [ηn v βs v I ηs v ] S v ) + κsi v 1 I v [βs v I ηi v ] I v 6

7 Using I = N S, I = N S, µn = κs I v + µ + φ)s, µ + φ) = κ S I v I, ηn v = βsvi + ηs v and η = β S v I I, to rewrite tis, we get v dl = µ + φ)βs vi βκs I S vi v S S )2 S ηκsi v S v Sv) 2 S v ] 0. [ S + S v + S I I v S S v S I Iv + S viv I SvI v I 4 Tus, using te aritmetic geometric means inequality, one can see tat dl is less or equal to zero wit equality only if S = S, I = I, S v = Sv and I v = Iv. Terefore te largest compact invariant set in { S, I, S v, I v ) Γ : dl = 0} is te singleton {E }, were E is te endemic equilibrium point. LaSalle s invariant principle [9] ten implies tat E is globally asymptotically stable in te interior of Γ. Tis proves Teorem 3 ii). 3 Host-Vector Disease Model wit Transmission Horizontal in Host Population In tis section, we formulate a model wit constant population sizes wit transmission orizontal in te ost population. Te Cagas diseases involve transmission by vectors and blood transfusion. Te following system is a simple model for Cagas disease: ds di ds v di v = µn κs I v λs I µs = κs I v + λs I µi = ηn v βs v I ηs v 8) = βs v I ηi v Te variables and parameters are te same as in te previous model. We tus study system 8) in Γ. Te ost infectious-to-ost susceptible transmission rate λs I. For system 8) te region Γ is positively invariant. Our next result concerns te existence of equilibrium points. We observe tat E 0 N, 0, N v, 0) is te disease-free equilibrium point. From te equations of system 8) wit te rigt-and side equal to zero, can be seen tat te endemic equilibrium point E S, I, S v, Iv ) wit 0 < I < N and 0 < Iv < N v must satisfy te following relations S = µβi +η) κβn v+λη+λβi, S v = ηnv βi +η and I v = βηnvi βi +η)η. and I is solution of te following quadratic polynomial 7

8 P I ) = λβi 2 + µβ + λη + βκn v λβn )I µηr 0 1), 9) were R 0 = N κβn v+λη) ηµ. We are looking for solutions of equation 9) between zero and N. For tis, notice tat P 0) < 0 if and only if R 0 > 1, and terefore a necessary and sufficient condition to ave a unique zero 0 < I < N is tat P N ) > 0. Ten, we ave proved te following proposition: Proposition 2. System 8) always as te disease free equilibrium E 0 N, 0, N v, 0). If R 0 > 1 tere is also a unique endemic equilibrium E S, I, S v, Iv ) in te interior of Γ. Teorems 1, 2 and 3, old for system 8) as well. Teorems 1 is proved wit an analysis made similar to te system 2) using te same Dulac function 5) to establis te global stability of a two-dimensional model as te equivalent fourdimensional system 8). Teorems 2 and 3 are proved using te second metod of Lyapunov based in te same Lyapunov functions 6) and 7) to establis te global stability of te two equilibrium points of four-dimensional model 8). 4 Discussions and Conclusions Tis paper presents a matematical study on te global stability of standard ost-vector models using a Dulac functions and Lyapunov functions. It is establised in Teorem 3 tat R 0 is a sarp tresold parameter and completely determines te global stability of 1) and 8) in te feasible region. If R 0 1, te disease-free equilibrium point is globally asymptotically stable in te feasible region and te disease will die out. If R 0 > 1, a unique endemic equilibrium point is globally asymptotically stable in te interior of te feasible region and te disease persists. Te results of tis work indicate tat te Lyapunov functions of te form Lx 1, x 2,...x n ) = n i=1 c ix i x i x xi i log x ) can be especially useful for compartmental ost-vector models wit any number of i compartments. References [1] N. T. J. Bailey, Te Matematical Teory of Infectious Diseases, Griffin, London, [2] C. Bowman, A. B. Gumel, P. van den Driessce, J. Wu, H. Zu, A matematical model for assessing control strategies againstwest Nile virus. Bulletin of Matematical Biology, 67, , [3] K. Dietz, Transmission and control of arbovirus diseases in: D. Ludwig et al. Eds.), Epidemiology, Proceedings of te Society for Industrial and Applied Matematics, Piladelpia, PA,

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