On the global stability of infectious diseases models with relapse

 Musiktheorie

 3 views
of 12
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Description
"In this paper, we study the global stability conditions of two epidemiological model with relapse, and bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptible individuals into the community and that
Share
Tags
Transcript
  Abstraction & Application  9  (2013) 50 − 61 UADY On the global stability of infectious diseases models with relapse Cruz Vargas-De-Le´on Unidad Acad´emica de Matem´aticas,Universidad Aut´onoma de Guerrero,Guerrero, M´exico.leoncruz82@yahoo.com.mx Abstract In this paper, we study the global stability conditions of two epidemiological model with relapse,and bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptibleindividuals into the community and that the disease produces non-negligible death in the infectiousclass. The models are appropriate for tuberculosis, including tuberculosis in human and bovine, andfor human herpes virus. It is also included the dynamics of tobacco and alcohol use. We presentsthe construction of Lyapunov functions for the systems mentioned above using suitable combinationsof known functions, common quadratic and Volterra–type, and of a novel function we call compositeVolterra–type function. Keywords and phrases  : Epidemiological model, Relapse, Global stability, Lyapunov’s second method, LaSalle’s invarianceprinciple.2010  Mathematics Subject Classification  : 92D30, 34D23. 1 Introduction The relapse phenomenon in some infectious diseases is characterized by the acquisition of quiescent stateof the individuals that have previously been infected, and with subsequent relapse (or reactivation) to theinfectious state. This recurrence of infections including diseases such as bovine tuberculosis and humanherpes virus. These types of diseases can be modelled by  SIRI   epidemiological systems.Tudor [15] formulated one of the first epidemic models with relapse, this model incorporates, bilinearincidence rate and constant total population. This system was extended to include nonlinear incidencefunctions by Moreira and Wang [12]. Blower [3] developed a compartmental model for genital herpes,assuming standard incidence for the disease transmission and constant recruitment rate.Van den Driessche and Zou [5], developed a  SIRI   model in a constant population with standard incidenceand a general relapse distribution. Van den Driessche and coauthors, [6], formulated a  SEIRI   model for adisease with a general exposed distribution in a constant population.In [4, 14] modeling the dynamics of tobacco and alcohol use, respectively, as a contagious disease withrelapse. In the smoking and drinking models formulation proposed in [4, 14], assumed that the populationis constant, standard incidence and, linear and nonlinear relapse rates.In this paper, we present the construction of Lyapunov functions for two epidemiological models withrelapse, constant recruitment and disease-induced death.50  Cruz Vargas-De-Le´on  51 Λ ↓ S   βSI  −→ I   κI  −→  R  γR −→ I  ↓ µS  ↓ ( α + µ ) I  ↓ µR Figure 1: The transfer diagram for the  SIRI   modelIn recent years, Lyapunov  s  second method has been a popular technique to study global stability of epidemiological models. A Volterra-type Lyapunov function has been used in [2, 8] to prove global stability of the steady states of classic  SIS  ,  SIR  and  SIRS   epidemiological models with bilinear incidence rate. Vargas-De-Le´on in [16, 17] made some ingenious linear combination of functions, to obtain a suitable Lyapunovfunctions to prove global stability of steady states of   SIS  ,  SIR  and  SIRS   epidemiological models withbilinear and standard incidence rates, respectively.The most popular types of Lyapunov functions are the common quadratic and Volterra-type functions.The common quadratic functions are of the form W  ( x 1 ,x 2 ,...x n ) =   ni =1 c i 2  ( x i − x ∗ i ) 2 , and the Volterra-type functions  W  ( x 1 ,x 2 ,...x n ) =   ni =1 c i  x i − x ∗ i  − x ∗ i  ln  x i x ∗ i  . We introduce a novel family of functions [17, 18]   W  ( x 1 ,x 2 ,...,x n ) =  c  ni =1 ( x i − x ∗ i ) −  ni =1 ( x ∗ i )ln  ni =1 ( x i )  ni =1 ( x ∗ i )  , we call this, composite-Volterra function.Generally speaking, within infectious disease models with bilinear incidence exist methods for construct-ing Lyapunov functions (see [7, 8, 9, 10, 11]), there are no systematic methods for constructing Lyapunovfunctions for epidemic models with standard incidence rate. In [17, 18] author reported the construction of Lyapunov functions in the classic epidemiological models with standard incidence.We will make ingenious linear combination of common quadratic, Volterra–type and composite–Volterra–type functions, to obtain a suitable Lyapunov functions to study the global stability conditions of two  SIRI  models with bilinear and standard incidence rates, respectively.In Sections 2 and 3, we shall construct Lyapunov functions for  SIRI   models. Section 4 includes thediscussion and concluding remarks. 2 The epidemiological model with relapse and bilinear incidence In most compartmental epidemiological models, the total population is often divided into several disjointclasses. The model divides the total population into the following sub-groups that are susceptible indi-viduals  S  , infectious individuals  I  , and individuals previously infectious but temporarily reverted to thenon-infectious state,  R . In this  SIRI   model, the flow is from the  S   class to the  I   class, and then eitherdirectly to the  R  class and then back to  I   class as shown in Fig. 1. The transfer diagram leads to thefollowing system of differential equations:  52  infectious diseases models  dS dt  = Λ − βSI  − µS,dI dt  =  βSI  − ( α + κ + µ ) I   + γR,  (2.1) dRdt  =  κI  − ( µ + γ  ) R. The parameters are positive constants. The constant Λ is the recruitment rate of susceptibles corre-sponding to births and immigration,  µ  is the natural death rate of population. The parameter  β   is thedisease transmission coefficient. In most classic disease transmission models, the disease-induced mortalityis neglected, we add the parameter  α , that is the disease related death rate. The parameter  κ  describesthe rate that the infectious individuals becomes no-infectious individuals and  γ   denotes the rate that theno-infectious individuals are reverted to the infectious state.The differential equation of the total population of (2.1) is: ddt ( S   + I   + R ) = Λ − µ ( S   + I   + R ) − αI.  (2.2)Thus the total population size may vary in time. In the absence of disease, the population size convergesto the steady state Λ /µ . We thus study (2.1) in the following feasible region:Ω =  ( S,I,R ) ∈ R 3+  :  S   ≥ 0 , I   ≥ 0 , R ≥ 0 , S   + I   + R ≤ Λ /µ  which can be shown to be positively invariant with respect to (2.1). Direct calculation shows that system (2.1)has two possible steady states in the non-negative octant  R 3+ : the disease–free steady state  E  0 = ( S  0 , 0 , 0)where  S  0 = Λ /µ , and a unique endemic steady state  E  ∗ = ( S  ∗ ,I  ∗ ,R ∗ ) with S  ∗ = Λ µR 0 , I  ∗ =  µβ   ( R 0 − 1) , R ∗ =  κµβ  ( γ   + µ ) ( R 0 − 1) .  (2.3)Where  R 0 ,  the basic reproduction number, is R 0  = ( µ + γ  ) βS  0 κµ + ( µ + γ  )( α + µ ) = ( µ + γ  ) β  Λ µ ( κµ + ( µ + γ  )( α + µ )) . 2.1 Global stability of disease–free steady state The global stability of the disease–free steady state  E  0 is proved by using common quadratic and linearLyapunov functions and LaSalle’s invariance principle. Theorem 1.  If   R 0  ≤ 1 , then the disease–free steady state   E  0 of (2.1) is globally asymptotically stable in   Ω .Proof.  V   :  ( S,I,R ) ∈ R 3+  :  S >  0  → R  by V  ( S,I,R ) = ( µ + γ  )2 S  0  ( S  − S  0 ) 2 + ( µ + γ  ) I   + γR.  (2.4)  Cruz Vargas-De-Le´on  53It is clear that at  E  0 the function  V  ( S,I,R ) reaches its global minimum in  R 3+ , and hence  V  ( S,I,R ) isa Lyapunov function. The derivative of (2.4) with respect to  t  along solution curves of (2.1) is given by V   ( S,I,R ) = ( µ + γ  ) S  0  ( S  − S  0 ) dS dt  + ( µ + γ  ) dI dt  + γ dRdt = ( µ + γ  ) S  0  ( S  − S  0 )(Λ − µS  − βSI  )+ ( µ + γ  )( βSI  − ( α + κ + µ ) I   + γR ) + γ  ( κI  − ( µ + γ  ) R ) , = ( µ + γ  ) S  0  ( S  − S  0 )( µS  0 − µS  − βSI  )+ ( µ + γ  ) βSI  − ( κµ + ( α + µ )( µ + γ  )) I. Using the expression βSI  ( S  − S  0 ) S  0  =  βI  ( S  − S  0 ) 2 S  0  + βI  ( S  − S  0 ) . We obtain, V   ( S,I,R ) = − ( γ   + µ )( µ + βI  )( S  − S  0 ) 2 S  0 + ( κµ + ( µ + γ  )( α + µ )) I    ( µ + γ  ) βS  0 κµ + ( µ + γ  )( α + µ )  − 1  , = − ( γ   + µ )( µ + βI  )( S  − S  0 ) 2 S  0  − ( κµ + ( µ + γ  )( α + µ )) I   (1 − R 0 ) . Therefore,  R 0  ≤  1 ensures that  V   ( S,I,R )  ≤  0 for all  S  , I  , R >  0, and that  V   ( S,I,R ) = 0 holds when S   =  S  0 and  I   = 0, or if   R 0  = 1 and  S   =  S  0 . It is easy to verify that the disease–free steady state  E  0 is thelargest invariant set in { ( S,I,R ) ∈ Ω :  V   ( S,I,R ) = 0 } , and hence by the LaSalle’s invariance principle [13],the steady state  E  0 is globally asymptotically stable. 2.2 Global stability of endemic steady state The globally asymptotic stability of the endemic steady state is proved by constructing a global Lyapunovfunction. We obtain the Lyapunov function of a suitable combination of common quadratic and Volterratype functions. Theorem 2.  If   R 0  >  1 , then the unique endemic steady state   E  ∗ of (2.1) is globally asymptotically stable in the interior of   Ω .Proof.  Define  L  : { ( S,I,R ) ∈ Ω :  S,I,R >  0 }→ R  by L ( S,I,R ) = ( S  − S  ∗ ) 2 2 S  ∗ +  I  − I  ∗ − I  ∗ ln  I I  ∗  +  γR ∗ κI  ∗  R − R ∗ − R ∗ ln  RR ∗  . This function is defined, continuous and positive definite for all  S  ,  I  ,  R >  0. It can be verified that thefunction  L ( S,I,R ) takes the value  L ( S,I,R ) = 0 at the steady state  E  ∗ , and thus, the global minimumof   L ( S,I,R ) occurs at the endemic steady state  E  ∗ . Since ( S  ∗ ,I  ∗ ,R ∗ ) is an endemic steady state point of (2.1), we have  54  infectious diseases models  Λ =  βS  ∗ I  ∗ + µS  ∗ , ( α + κ + µ ) =  βS  ∗ + γ R ∗ I  ∗ ,  (2.5)( µ + γ  ) =  κ I  ∗ R ∗ . Computing the derivative of   L ( S,I,R ) along the solutions of system (2.1), we obtain L  ( S,I,R ) = ( S  − S  ∗ ) S  ∗ dS dt  + ( I  − I  ∗ ) I dI dt  +  γR ∗ κI  ∗  1 −  R ∗ R   dRdt = ( S  − S  ∗ ) S  ∗ (Λ − βSI  − µS  )+ ( I  − I  ∗ )  βS  − ( α + κ + µ ) + γ RI   +  γR ∗ κI  ∗  1 −  R ∗ R  ( κI  − ( µ + γ  ) R ) . Using (2.5), we obtain L  ( S,I,R ) = − ( S  − S  ∗ ) S  ∗ ( µ ( S  − S  ∗ ) + β  ( SI  − S  ∗ I  ∗ ))+ ( I  − I  ∗ )  β  ( S  − S  ∗ ) + γ   RI   −  R ∗ I  ∗  +  γR ∗ κI  ∗  1 −  R ∗ R  κI  − κI  ∗  RR ∗  . Notice that SI  − S  ∗ I  ∗ =  S  ∗ ( I  − I  ∗ ) + I  ( S  − S  ∗ ) . Thus, L  ( S,I,R ) = − ( S  − S  ∗ ) S  ∗ ( µ ( S  − S  ∗ ) + β  ( S  ∗ ( I  − I  ∗ ) + I  ( S  − S  ∗ )))+ β  ( I  − I  ∗ )( S  − S  ∗ ) + γR ∗   RR ∗ −  I I  ∗ −  I  ∗ RIR ∗ + 1  + γR ∗   I I  ∗ −  RR ∗ −  IR ∗ I  ∗ R  + 1  , = − ( µ + βI  )( S  − S  ∗ ) 2 S  ∗ + γR ∗  2 −  I  ∗ RIR ∗ −  IR ∗ I  ∗ R  , = − ( µ + βI  )( S  − S  ∗ ) 2 S  ∗ − γR ∗   I  ∗ RIR ∗ −   IR ∗ I  ∗ R  2 . Therefore,  L  ( S,I,R )  ≤  0 for all  S,I >  0, where the equality  L  ( S,I,R ) = 0 holds only when  S   =  S  ∗ and  IR ∗ =  I  ∗ R . It is easy to see that the endemic steady state  E  ∗ is the only largest invariant set in { ( S,I,R ) ∈ Ω :  L  ( S,I,R ) = 0 } . Therefore, by LaSalle’s invariance principle [13], the endemic steady state E  ∗ is globally asymptotically stable in the interior of Ω.
Related Search
Similar documents
View more...
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks