We consider SIR and SIRS models with differential mortality. Global stability
of equilibria is established by using Lyapunov's methodComment: page 2, one figur

Adda, Phillipe

Bichara, Derdei

English

arXiv

We consider SIR and SIRS models with differential mortality. Global stability of equilibria is established by using Lyapunov's method.On considère les modèles SIR et SIRS avec des mortalités différentes. La stabilité globale des équilibres est établie en utilisant les techniques de Lyapunov

Adda, Philippe

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HAL Id: hal-00675359https://hal.inria.fr/hal-00675359v2Submitted on 8 Mar 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Global stability for SIR and SIRS models withdifferential mortalityPhilippe Adda, Derdei BicharaTo cite this version:Philippe Adda, Derdei Bichara. Global stability for SIR and SIRS models with differential mortality.International Journal of Pure and Applied Mathematics, Academic Publishing Ltd, 2012, GLOBALSTABILITY FOR SIR AND SIRS MODELS WITH DIFFERENTIAL MORTALITY, 80 (3), pp.425-433. ￿hal-00675359v2￿ISSN0249-6399ISRNINRIA/RR--7894--FR+ENGRESEARCHREPORTN° 7894Mars 2012Project-Team MasaieGlobal stability for SIRand SIRS models withdifferential mortalityPhilippe Adda, Derdei BicharaRESEARCH CENTRENANCY – GRAND EST615 rue du Jardin BotaniqueCS2010154603 Villers-lès-Nancy CedexGlobal stability for SIR and SIRS models withdifferential mortalityPhilippe Adda, Derdei BicharaProject-Team MasaieResearch Report n° 7894 — Mars 2012 — 10 pagesAbstract: We consider SIR and SIRS models with differential mortality. Global stability ofequilibria is established by using Lyapunov’s method.Key-words: Nonlinear dynamical systems, global stability, Lyapunov methods.Stabilité globale pour les modèles SIR et SIRS avecmortalités différentesRésumé : On considère les modèles SIR et SIRS avec des mortalités differentes. La stabilitéglobale des équilibres est etablie en utilisant les techniques de Lyapunov.Mots-clés : Systèmes dynamiques non-linéaires, stabilité globale, méthodes de LyapunovGlobal stability for SIR and SIRS models with differential mortality 31 IntroductionThe SIR model is a classical model in mathematical epidemiology. Particularly Kermack andMcKendrick [8 ] use a SIR model to prove the existence of threshold. The model of Kermack andMcKendrick is without demography, i.e. without vital dynamics. The classic SIR models arevery important as conceptual models (similar to predator-prey and competing species models inecology). The SIR epidemic modeling yields the useful concept of the threshold.When a vital dynamic is introduced the asymptotic behavior changes. When the death ratesare equal in each compartments S,I and R, and equal to the birth rate, the global stability hasbeen solved in [ 6, 5 ]. Actually, the total population is constant, hence the system reduces toa two dimensional system. Then using phase plane methods (Poincaré-Birkhoff) and Lyapunovfunctions the global stability is obtained.Models with a variable total population size are often more difficult to analyze mathematicallybecause the population size is an additional variable which is governed by a differential equationThe global stability using Lyapunov functions of SIR model with a total constant populationis proved in [10 ]. However in this model the death rates of S and I are equal and the deathrate of the removed compartment is adjusted relatively to the death rate of S and the constantbirth rate. This adjustment is just done to have a constant total population. This is a little bitartificial. The model with constant population simplifies in two important ways :• The mass action lawSINreduces to a bilinear law β̃S I• The system is a two-dimensional system.In this paper we propose a more realistic model, with constant population. We suppose,which is more or less observable, that the natality compensates for the mortality. Our model candeal with different death rates, and particularly with a over-mortality from the disease.We denote R0 the basic reproduction number. It is defined as the expected number of newcases of infection caused by a typical infected in a population susceptible [3 , 16]. We prove inthis paper the global stability of disease free equilibrium (DFE) if R0 ≤ 1 and that there existsa unique endemic equilibrium (EE) if R0 > 1, which is globally asymptotically stable on thedomain minus the stable manifold of the DFE.The stability analysis of classical SIR model is well know since 1976 [5 , 7 ]. The reason wasthat study of stability for these models reduce to the study of 2-dimensionnal systems, hencephase methods can be used: Poincaré-Bendixon theorem. Periodic orbits are ruled out usingDulac criteria or a condition of Busenberg and Van Den Driessche [ 2].In the recent litterature, the Lyapunov method is successfuly used to prove the global stabilityof endemic equilibrium. This method consists to find one function, called Lyapunov function andusually denoted by V , positive definite and its derivative along trajectories is negative definite.If the derivative V̇ is only negative , the LaSalle’s invariance principle extend the Lyapunovmethod in particular cases. This Lyapunov function is very difficult to exhibit. However, theclass of Lyapunov functionV =n∑i=1ai (xi − x̄i log xi)is used. This function has a long history of application to Lotka-Voltera models and was dis-covered by Voltera himself, although he did not use the vocabulary and the theory of Lyapunovfunctions. In 2002, Korobeinikov and Wake use this type of function to prove the global stabilityfor SIR, SIRS and SIS models [10 ] and in 2004, for SEIR and SEIS model [9 ] and give asimply proof of the result of Li and Muldowney [15 ].RR n° 78944 Adda & BicharaWe give a brief outline of the paper. In section 2, we formulate the model and study theglobal stability of the DFE if R0 ≤ 1. In section 3, we study the global stability of endemicequilibrium if R0 > 1. In section 4, we deduce results of the next section to SIRS model. Infinally we conclude in section 5.InriaGlobal stability for SIR and SIRS models with differential mortality 52 Model formulationWe consider a population N divided into classes of susceptible, infectious and removed indi-viduals, with numbers at time t denoted by S(t), I(t) and R(t) respectively, that is N =S(t)+I(t)+R(t). We assume that there is no vertical transmission, then all offsprings are suscep-tibles. We assume that the natality Λ compensates for the deaths. Then Λ = µ1S + µ2I + µ3R.The parameter γ is the rate of recovery. Note that, in our model the disease confers a permanentimmunity. The parameter β is the effective per capita contact rate of infective individuals. Wemodelize the contact by the classical law of mass action mass. We have the following flow graph:S I RβSI/Nμ1 μ2 μ3ΛγΙThe dynamic of this model is given by the following system:Ṡ = Λ− β S IN − µ1 Sİ = β S IN − µ2I − γ IṘ = γ I − µ3R(1)Which reduces to Ṡ = −β S IN + µ2 I + µ3Rİ = β S IN − µ2 I − γ IṘ = γ I − µ3R(2)The population size is constant, so that S+I+R = N , then we can omit the equation of removedpopulation. We obtain the two-dimensional system:{Ṡ = −β SIN + µ2I + µ3 (N − S − I)İ = β SIN − (µ2 + γ) IFor simplicity we can consider the prevalence, i.e. the proportions.If we denote SN ,IN , the susceptible and infectious fractions, again by S and I. Then thesystem (2) is reduced to {Ṡ = µ3 + (µ2 − µ3)I − µ3S − βSIİ = βSI − (µ2 + γ) I(3)RR n° 78946 Adda & BicharaWe have 0 ≤ S, 0 ≤ I and S + I ≤ 1. The biological domain of this two-dimensional system isthe standard simplex.The set Ω = {(S, I) : S ≥ 0; I ≥ 0;S + I ≤ 1} is a positively invariant compact set for (3). Thesystem is well posed.The basic reproduction ratio is given byR0 =βµ2 + γ.2.1 Stability of DFESystem (3) has a disease free equilibrium state, which is given by (S∗, 0) = (1, 0) .Theorem 2.1 If R0 ≤ 1 then the DFE is globally asymptotically stable on Ω.Proof:We consider the Lyapunov-LaSalle function V (S, I) = I. We have:V̇ = İ= βSI − (µ2 + γ) I= I (R0S − 1) (µ2 + γ)≤ 0Furthermore V̇ = 0 if I = 0 or S = S∗ and R0 = 1. Hence the largest invariant set containedin the set L ={(S, I) ∈ Ω / V̇ (S, I) = 0}is reduced to the DFE. Since we are in a compactpositively invariant set, by the LaSalle’s Invariance Principle [12 , 1] , the DFE is globalyasymtotically stable in Ω.Remark 2.1 Unlike Lyapunov’s theorems, LaSalle’s principle does not require the function V (x)to be positive definite . If the largest invariant set M , contained in the set E of points where V̇vanishes, is reduced to the equilibrium point, i.e. if M = {x0}, the LaSalle’s principle allows toconlude that the equilibrium is attractive. But a drawback of Lasalle’s principle, when significant,is that it proves only the attractivity of the equilibrium point. It is well known that in thenonlinear case attractivity does not imply stability. But when the function V is not positivedefinite, Lyapunov stability must be proven. This is why LaSalle’s principle is often misquoted.Some additional condition enables, with LaSalle’s principle, to ascertain asymptotic stability.To obtain stability from LaSalle’s principle some additional work is needed. The most completeresults, in the direction of Lasalle’s principle to prove asymptotic stability, have been obtained byLaSalle himself (LaSalle:[13 ], in 1968, completed in 1976 [14 ] .)InriaGlobal stability for SIR and SIRS models with differential mortality 73 Global Stability of endemic equilibriumAn equilibrium for system (3), different from the DFE, is given by(S̄, Ī), whereS̄ =µ2 + γβ=1R0and Ī =µ3µ3 + γ(1− 1R0)This equilibrium is in the simplex, i.e., 0 ≤ S̄, 0 ≤ Ī and S̄ + Ī ≤ 1 iff R0 > 1.Clearly 0 ≤ Ī is equivalent to R0 ≥ 1. Now we can writeS̄ + Ī =γR0 + µ3γ + µ3When R0 = 1 this equilibrium coincides with the DFE. Then there is an unique equilibrium inthe interior of the simplex iff R0 > 1.Theorem 3.1 If R0 > 1, the DFE is unstable and there exists a unique endemic equilibrium(S̄, Ī) and this endemic equilibrium is globally asymptotically stable on the domain Ω\ [0, 1]×{0}.In other words on the simplex minus the stable manifold of the DFEProof:When R0 > 1 the instability of the DFE comes from [3 ].Let Ω1 the set defined by Ω1 ={(S, I)/ S ≥ µ2 − µ3β, I ≥ 0, S + I ≤ 1}. The set Ω1 is acompact positively invariant. We Consider on◦Ω1 the Lyapunov function defined byV (S, I) =(S − S̄)− µ3 + γβlog−µ2 + µ3 + βS−µ2 + µ3 + βS̄+(I − Ī)− Ī log IĪIt is easy to verifiy that V is definite positive, that is V (S, I) ≥ 0 and V (S̄, Ī) = 0 if and only ifRR n° 78948 Adda & Bichara(S, I) = (S̄, Ī). His derivative along trajectories of (3) is given by:V̇ (S, I) = Ṡ − (µ3 + γ)µ3 + (µ2 − µ3)I − µ3S − βSI−µ2 + µ3 + βS+βSI − (µ2 + γ) I − Ī (βS − (µ2 + γ))= Ṡ − (µ3 + γ)(µ3 − µ3S)−µ2 + µ3 + βS+ (µ3 + γ) I +βSI − (µ2 + γ) I − Ī (βS − (µ2 + γ))= µ3 (1− S)− (µ3 + γ)(µ3 − µ3S)−µ2 + µ3 + βS− Ī (βS − (µ2 + γ))= µ3 (1− S)[1− µ3 + γ−µ2 + µ3 + βS]− Ī (βS − (µ2 + γ))= µ3 (1− S)(−βS̄ + βS−µ2 + µ3 + βS)− µ3µ3 + γ(1− S̄) (βS − βS̄)= −µ3β(S̄ − S) [ 1− S−µ2 + µ3 + βS− 1− S̄µ3 + γ]= −µ3β(S̄ − S) [ 1− S−µ2 + µ3 + βS− 1− S̄−µ2 + µ3 + βS̄]= − βµ3µ3 + γ[−µ2 + β + µ3−µ2 + β + βS] (S − S̄)2≤ 0Then we conclude V̇ is semi-definite positive. Then the endemic equilibrium is stable by Lya-punov theorems. We prove the attractivity of endemic equilibrium using Lasalle’s principle.The set on which V̇ = 0 is given by E ={(S, I) ∈◦Ω1 / S = S̄}. Then on this set, we haveṠ = µ3 + (µ2 − µ3)I − µ3S − βS̄I = 0, then I =µ3 − µ3S̄βS − µ2 + µ3= Ī. Furthermore the largestinvariant set contained in the set{(S, I) ∈◦Ω1 / V̇ (S, I) = 0}is reduced to the endemic equilib-rium. Hence (S̄, Ī) is attractive. Then EE is GAS on◦Ω1.If S ≤ µ2 − µ3β, we have:Ṡ = µ3 + (µ2 − µ3)I − µ3S − βSI= µ3 (1− S) + (µ2 − µ3 − βS) I> 0Then Ṡ > 0. Furthermore all trajectories in◦Ω \◦Ω1 enter in◦Ω1. Then the set◦Ω1 is absorbant.Hence the EE is GAS on◦Ω .In the boundary S = 0 et S + I = 1, the vector field is strictly pointing inside Ω. Only theS-axis is invariant. The endemic equilibrium is GAS on Ω\ {(S, I) : I = 0; 0 ≤ S ≤ 1}. This endthe proof.InriaGlobal stability for SIR and SIRS models with differential mortality 94 SIRS ModelIn this section, we consider a SIRS model with different mortality. With the same notation thatabove, we have the following system:Ṡ = Λ− β S IN − µ1 S + νRİ = β S IN − µ2I − γ IṘ = γ I − (µ3 + ν)R(4)Which reduces to Ṡ = −β S IN + µ2 I + (µ3 + ν)Rİ = β S IN − (µ2 + γ) IṘ = γ I − (µ3 + ν)R(5)The system (5) is exactly as system (2) where µ3 is replaced by µ3 + ν.5 ConclusionIn this contribution, we have proved the global stability of SIR and SIRS models with differentialmortality by Lyapunov methods. Our results encompass and improve the results of [10 ].References[ 1] N. P. Bhatia and G. P. Szegö. Stability Theory of Dynamical Systems. Springer-Verlag,1970.[ 2] S. Busenberg and P. van den Driessche. A method for prouving the nonexistence of limitcycles. J. Math. analysis applic., 172:463–479, 1993.[3 ] O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz. On the definition and the compu-tation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneouspopulations. J. Math. Biol., 28:365–382, 1990.[ 4] B. S. Goh. Global stability in many-species systems. Amer.Naturalist, (11):135–143, 1977.[5 ] Herbert W. Hethcote. Qualitative analyses of communicable disease models. Math. Biosci.,28:335–356, 1976.[ 6] Herbert W. Hethcote. Three basic epidemiological models. In Applied mathematical ecology(Trieste, 1986), volume 18 of Biomathematics, pages 119–144. Springer, Berlin, 1989.[7 ] Herbert W. Hethcote. The mathematics of infectious diseases. SIAM Rev., 42(4):599–653(electronic), 2000.[8 ] W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory ofepidemics. Proc. Royal Soc. London, 115:700–721, 1927.[9 ] A. Korobeinikov and P. Maini. Lyapunov functions and global properties for SEIR and SEISmodels. Math. Med. Biol., 21:75–83, 2004.[10 ] A. Korobeinikov and G.C. Wake. Lyapunov functions and global stability for SIR, SIRS,and SIS epidemiological models. Appl. Math. Lett., 15(8):955–960, 2002.RR n° 789410 Adda & Bichara[ 11] J. P. LaSalle. The stability of dynamical systems. Society for Industrial and Applied Math-ematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability ofnonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Seriesin Applied Mathematics.[12 ] J. P. LaSalle and S. Lefschetz. Stability by Liapunov’s direct method. Academic Press, 1961.[13 ] J.P. LaSalle. Stability theory for ordinary differential equations. J. Differ. Equations,41:57–65, 1968.[14 ] J.P. LaSalle. Stability of nonautonomous systems. Nonlinear Anal., Theory, Methods Appl.,1(1):83–91, 1976.[15 ] M. Y. Li and J. S. Muldowney. Global stability for the SEIR model in epidemiology. Math.Biosci., 125:155–164, 1995.[ 16] P. van den Driessche and J. Watmough. Reproduction numbers and sub-threshold endemicequilibria for compartmental models of disease transmission. Math. Biosci., pages 121–142,2002.InriaRESEARCH CENTRENANCY – GRAND EST615 rue du Jardin BotaniqueCS2010154603 Villers-lès-Nancy CedexPublisherInriaDomaine de Voluceau - RocquencourtBP 105 - 78153 Le Chesnay Cedexinria.frISSN 0249-6399

Global stability for SIR and SIRS models with differential mortality

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