Global stability properties of a class of renewal epidemic models with variable susceptibility
aa r X i v : . [ m a t h . D S ] J u l Global stability properties of a class of renewal epidemic models withvariable susceptibility
Michael T. Meehan a , Daniel G. Cocks b , Emma S. McBryde a a Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, Australia b Research School of Science and Engineering, Australian National University, Canberra, Australia
Abstract
We investigate the global dynamics of a renewal-type epidemic model with variable susceptibility. Weshow that in this extended model there exists a unique endemic equilibrium and prove that it is globallyasymptotically stable when R >
1, i.e. when it exists. We also show that the infection-free equilibrium,which exists always, is globally asymptotically stable for R ≤ Keywords: global stability renewal variable susceptibility Lyapunov
1. Introduction
In a recent article, [21] investigated the asymptotic dynamics of a general class of renewal epidemic modelsfor which both the force of infection and infected removal rates are arbitrary functions of an individual’sinfection age [9, 4, 22, 2]. By identifying appropriate Lyapunov functionals of the form g ( x ) = x − − log x [12], the authors were able to establish that the infection-free and endemic system equilibria wereglobally asymptotically stable when the basic reproduction number R ≤ >
1, respectively (seealso [15]). Here we extend this investigation by considering a class of renewal epidemic models that accountfor variable susceptibility to infection among the susceptible cohort.The remainder of the paper is constructed as follow: in the next section we define the variable susceptibil-ity model and introduce the relevant model parameters. We also discuss the infinite-dimensional phase-spaceof our system and introduce several important definitions. Following this, in section 3, we derive expressionsfor the equilibrium solutions of the model system and determine the necessary and sufficient conditions fortheir existence. Then in section 4 we use the direct Lyapunov method to prove that the infection-free andendemic equilibria are globally asymptotically stable for R ≤ >
1, respectively.Both this work and the previous analysis [21] follow a long list of studies that have successfully invokedthe direct Lyapunov method to establish the global stability properties of dynamical system equilibria. Forrelevant references within the domain of epidemic modelling see, for instance, [13, 14, 12, 10, 11, 17, 18, 19,20, 15, 8, 1, 16, 3].
2. Model description
The general Kermack-McKendrick epidemic model [9], describing the spread of an infection through asusceptible population, can be written in terms of the following set of equations: dS ( t ) dt = λ − µS ( t ) − F ( t ) S ( t ) ,F ( t ) = Z ¯ τ A ( τ ) F ( t − τ ) S ( t − τ ) dτ (1)where S ( t ) is the fraction of the population that is susceptible to infection at time t , and F ( t ) is the force ofinfection. The quantity A ( τ ) appearing in the renewal definition of the force of infection (1) is the infectivity Preprint submitted to ArXiv Received: date / Accepted: date ernel and gives the expected contribution to F ( t ) for individuals who have been infected for τ units of time.The parameter ¯ τ is the maximum infection-age at which an individual remains infectious:¯ τ = sup { τ ≥ A ( τ ) > } . (2)We have also included the demographic parameters λ and µ , which give the constant birth/recruitment rateand per-capita death rate of susceptible individuals respectively, to generate a model that allows for endemicbehaviour. In this model all (susceptible) individuals are assumed to be equally susceptible to infection.In this article we would like to generalize the Kermack-McKendrick model (1) to account for varyinglevels of susceptibility among individuals in the S class. In particular we would like to decompose thelarger susceptible population S ( t ) into sub-populations S ( t, σ ) to which we assign an additional label σ toidentify their level of susceptibility. More precisely, we introduce the measure S ( t, σ ) to denote the densityof susceptible individuals with susceptibility index σ ∈ Σ, where Σ is a measurable space over a finite set :Total susceptible population (at time t ) = Z Σ S ( t, σ ) dσ < ∞ . For simplicity, we assume that the susceptibility index σ is a static label that is assigned at birth and thatit is not affected by aging or infection. We also assume that an individual’s infectivity is not related to theirsusceptibility such that we still have A ≡ A ( τ ). With these assumptions we generalize the model equationsgiven above to the system dS ( t, σ ) dt = λ ( σ ) − µS ( t, σ ) − η ( σ ) F ( t ) S ( t, σ ) ,F ( t ) = Z Σ Z ¯ τ A ( τ ) η ( σ ) F ( t − τ ) S ( t − τ, σ ) dτ dσ. (3)Here we have introduced the functions η ( σ ) and λ ( σ ) which respectively describe the relative susceptibilityand birth/recruitment of individuals with label σ . To ensure that the S compartment only contains indi-viduals that are capable of becoming infected, that all susceptibility states are continually replenished, andfinally that the evolution of the susceptibles S ( t, σ ) is sufficiently smooth, we impose the constraints λ ( σ ) , η ( σ ) ∈ L ∞ + (Σ) . Moreover, we assume that the maximum infection-age ¯ τ is finite:¯ τ < ∞ otherwise, the lack of compactness in the infinite case makes the problem much more difficult [7, 5].To simplify the analysis that follows, we introduce the mean susceptibility at the infection-free andendemic equilibria, defined respectively as η = Z Σ η ( σ ) S ( σ ) dσ (4)and ¯ η = Z Σ η ( σ ) ¯ S ( σ ) dσ (5)where S ( σ ) and ¯ S ( σ ) are the σ -distributions of susceptibles at the infection-free and endemic equilibriarespectively (see section 3). The extension to more general sets is relatively straightforward.
2o calculate the basic reproduction number for the model (3) we sum the expected contribution to theforce of infection A ( τ ) over all infection ages and multiply this quantity by the mean susceptibility of theinfection-free susceptible population: R = Z Σ η ( σ ) S ( σ ) dσ Z ¯ τ A ( τ ) dτ, = η Z ¯ τ A ( τ ) dτ. (6)Note that for the special (i.e. homogenous) case η ( σ ) = 1 we recover the familiar expression R = S R ¯ τ A ( τ ) dτ where S is the total number of susceptibles at the infection-free equilibrium [6].As discussed previously [21] we see that a full prescription of the model (3) requires a specification ofthe entire history of the susceptible population and the force of infection over time, from t = − ¯ τ up to thepresent ( t = 0). Additionally, for the variable susceptibility case we must also provide the σ -distributionof susceptibles over the measure space Σ. In this regard the present state of the system P = ( S , F ), isdescribed by a set of functions S and F , where F is defined over the interval [ − ¯ τ ,
0] and S is a (continuous)function from the time interval [ − ¯ τ ,
0] into L ∞ + (Σ). Hence, in order to ensure the necessary smoothness andcompactness properties of our system trajectories (see below), we choose the initial conditions S ∈ C ([ − ¯ τ , , L ∞ + (Σ)) and F ∈ L ( − ¯ τ , . (7)In general, our phase-space is the infinite-dimensional product topologyΩ = C ([ − ¯ τ , , L ∞ + (Σ)) × L ( − ¯ τ ,
0) (8)which is a Banach space which we assume takes the natural norm. With this choice of state-space, standardarguments show that the model (3) is well defined. Additionally, the model equations (3) induce a continuoussemiflow Φ t : Ω → Ω where the system trajectory is given by ( S t , F t ) ∈ Ω with S t ( s, σ ) = S ( t + s, σ ) , F t ( s ) = F ( t + s ) , s ∈ [ − ¯ τ , . Following the proof of Lemma 1 in [21], which invokes the smoothing properties of convolution integralsdescribed in [23], it is straightforward to show that if the infectivity kernel A is of bounded variation, i.e. A ∈ BV + ([0 , ¯ τ ]), we eventually have that S t ∈ C and F t ∈ AC + (that is, F t is absolutely continuous).Therefore, if we assume that A ∈ BV + , system trajectories generated by Φ t that originate in Ω are eventuallybounded and relatively compact. In this case the ω -limit set of (3) is non-empty and we may employthe infinite-dimensional form of LaSalle’s invariance principle [24, Theorem 5.17] to establish the globalasymptotic stability of our model equilibria.Finally, in the following, we decompose Ω into an “interior” and a “boundary” set b Ω and ∂ Ω, respectively.Here, we do not refer to topological concepts, but rather to the interpretation in view of our application. Forall initial values in the interior b Ω, the force of infection is non-zero. The elements in the boundary set ∂ Ω,in turn, do have a vanishing force of infection and therefore lead to trivial dynamics. To be more precise,we define b Ω = (cid:26) ( S , F ) ∈ Ω : ∃ a ∈ [0 , ¯ τ ] s.t. Z Σ Z ¯ τ A ( τ + a ) F ( − τ ) S ( − τ, σ ) dτ dσ > (cid:27) and ∂ Ω = Ω \ b Ω .
3. Equilibrium points
We now aim to evaluate the equilibrium σ -distributions of the susceptible populations both in the absence, S ( σ ), and presence, ¯ S ( σ ), of infection and determine conditions for their existence.3irst, we note that at equilibrium, the system (3) becomes0 = λ ( σ ) − µS ∗ ( σ ) − η ( σ ) F ∗ S ∗ ( σ ) , (9)and F ∗ = F ∗ Z Σ η ( σ ) S ∗ ( σ ) dσ Z ¯ τ A ( τ ) dτ. (10)From these equations it is straightforward to identify the infection-free susceptible distribution by settingthe force of infection F ∗ = 0 in (9): S ( σ ) = λ ( σ ) µ . (11)Hence, the infection-free equilibrium, P = ( S , F ) = ( λ ( σ ) /µ, λ ( σ ) ∈ L ∞ + ,we have that S > σ ∈ Σ.Next, we can find the endemic distribution ¯ S ( σ ) (for which ¯ F = 0) by re-arranging equation (9) to give¯ S ( σ ) = λ ( σ ) µ + η ( σ ) ¯ F . (12)Here we find that at the endemic equilibrium susceptible individuals are depleted from their infection-freedistribution according to their relative susceptibility, η ( σ ).It remains now to determine the conditions for the existence of ¯ S ( σ ), or equivalently, to determine thesign of the endemic force of infection ¯ F . We start by noting the identity1 = Z Σ η ( σ ) ¯ S ( σ ) dσ Z ¯ τ A ( τ ) dτ = ¯ η Z ¯ τ A ( τ ) dτ (13)which follows from (10) for the case ¯ F = 0.If we then substitute our solution for ¯ S ( σ ) (eq. (12)) into this expression we obtain1 = Z Σ η ( σ ) λ ( σ ) µ + η ( σ ) ¯ F dσ Z ¯ τ A ( τ ) dτ. (14)Here, given the definitions of η ( σ ) and A ( τ ), and the restrictions placed on them (e.g. both are non-negativefunctions), we observe that the right-hand side of (14) is a positive, strictly decreasing function of ¯ F . Further,from (6) and (11) we see that the right-hand side of (14) evaluated at ¯ F = 0 becomes Z Σ η ( σ ) λ ( σ ) µ dσ Z ¯ τ A ( τ ) dτ = R . Together, from these properties we can deduce that positive solutions to (14) exist if, and only if, R > P = ( ¯ S , ¯ F ) ∈ b Ω if, and only if, R >
1. For the boundary case R = 1, we find that ¯ F = 0 and the endemic and infection-free equilibria coincide, i.e. ¯ P = P ∈ ∂ Ω.Ultimately, our goal will be to establish that i) when R ≤ P ∈ ∂ Ω and ii) when R > P ∈ b Ω, except those that originate in ∂ Ωwhich approach P .
4. Global stability analysis
Theorem 1.
The infection-free equilibrium point P of the system (3) is globally asymptotically stable in Ω for R ≤ . However, if R > , solutions of (3) starting sufficiently close to P in Ω move away from P , except those starting within the boundary region ∂ Ω which approach P . roof of Theorem 1. To verify theorem 1 we define the forward invariant set D = Φ ¯ τ (Ω). Importantly, anytrajectory that originates in Ω enters D either at, or before t = ¯ τ and, from (3), we have that S (0) > S , F ) ∈ D .Now, consider the Lyapunov functional U : D → R + defined by U ( S , F ) = Z Σ S ( σ ) g (cid:18) S (0 , σ ) S ( σ ) (cid:19) dσ + Z Σ Z ¯ τ ξ ( τ ) η ( σ ) F ( − τ ) S ( − τ, σ ) dτ dσ (15)where g ( x ) = x − − log x and ξ ( τ ) = η Z ¯ ττ A ( ρ ) dρ. (16)We note that the kernel ξ has the following properties: ξ (0) = R , ξ (¯ τ ) = 0 and dξ ( τ ) dτ = − η A ( τ ) . (17)Importantly, the functional U ( S , F ) ≥ P .Evaluating the Lyapunov functional U ( S , F ) along system trajectories ( S t , F t ) we then have U ( S t , F t ) = Z Σ S ( σ ) g (cid:18) S t (0 , σ ) S ( σ ) (cid:19) dσ + Z Σ Z ¯ τ ξ ( τ ) η ( σ ) F t ( − τ ) S t ( − τ, σ ) dτ dσ, = Z Σ S ( σ ) g (cid:18) S ( t, σ ) S ( σ ) (cid:19) dσ + Z Σ Z ¯ τ ξ ( τ ) η ( σ ) F ( t − τ ) S ( t − τ, σ ) dτ dσ where in the second line we have re-introduced the notation S t ( s, σ ) = S ( t + s, σ ) and F t ( s ) = F ( t + s ).Next, in order to compute derivatives of U we rewrite the integral in the second term such that U ( S t , F t ) = Z Σ S ( σ ) g (cid:18) S ( t, σ ) S ( σ ) (cid:19) dσ + Z Σ Z tt − ¯ τ ξ ( t − s ) η ( σ ) F ( s ) S ( s, σ ) dτ dσ. (18)Differentiating the first term in our Lyapunov functional U with respect to time gives: ddt Z Σ S ( σ ) g (cid:18) S ( t, σ ) S ( σ ) (cid:19) dσ = Z Σ (cid:18) − S ( σ ) S ( t, σ ) (cid:19) dS ( t, σ ) dt dσ, = Z Σ (cid:18) − S ( σ ) S ( t, σ ) (cid:19) ( λ ( σ ) − µS ( t, σ ) − η ( σ ) F ( t ) S ( t, σ )) dσ, = Z Σ (cid:18) − S ( σ ) S ( t, σ ) (cid:19) ( λ ( σ ) − µS ( t, σ )) dσ − F ( t ) Z Σ η ( σ ) (cid:0) S ( t, σ ) − S ( σ ) (cid:1) dσ, = − µ Z Σ S ( t, σ ) (cid:18) − S ( σ ) S ( t, σ ) (cid:19) dσ − F ( t ) Z Σ η ( σ ) S ( t, σ ) dσ + η F ( t ) . (19)Note that in the final line we have substituted in the identities λ ( σ ) = µS ( σ ) and η = R Σ η ( σ ) S ( σ ) dσ .5ext, we differentiate the second term in U and use the properties of ξ (see equation (17)) to get ddt Z Σ Z tt − ¯ τ ξ ( t − s ) η ( σ ) F ( s ) S ( s, σ ) dτ dσ = Z Σ (cid:20) ξ (0) η ( σ ) F ( t ) S ( t, σ ) − ξ (¯ τ ) η ( σ ) F ( t − ¯ τ ) S ( t − ¯ τ , σ )+ Z tt − ¯ τ dξ ( t − s ) dt η ( σ ) F ( s ) S ( s, σ ) dτ (cid:21) dσ, = R F ( t ) Z Σ η ( σ ) S ( t, σ ) dσ − η Z Σ Z tt − ¯ τ A ( t − s ) η ( σ ) F ( s ) S ( s, σ ) dτ dσ, = R F ( t ) Z Σ η ( σ ) S ( t, σ ) dσ − η F ( t ) . (20)Finally, combining (19) and (20) yields dU ( S t , F t ) dt = − µ Z Σ S ( t, σ ) (cid:18) − S ( σ ) S ( t, σ ) (cid:19) dσ − (1 − R ) F ( t ) Z Σ η ( σ ) S ( t, σ ) dσ, ≤ . (21)We emphasize that we know for a trajectory ( S t , F t ) ∈ D ⊂ Ω, that for t > ¯ τ we have F t ∈ C ([ − ¯ τ , U is a proper Lyapunov function on the domain D .From (21) we see that the derivative ˙ U ( t ) = 0 if and only if S t (0 , σ ) = S ( σ ) and either (a) R = 1 or(b) F t (0) = 0. Therefore, the largest invariant subset in Ω for which ˙ U = 0 is the singleton (cid:8) P (cid:9) . Giventhat the system orbit is eventually precompact, by the infinite-dimensional form of LaSalle’s extension ofLyapunov’s global asymptotic stability theorem [24, Theorem 5.17], the infection-free equilibrium point P is globally asymptotically stable in Ω for R ≤ R > F t (0) >
0, the derivative ˙
U > S ( t, σ ) is sufficiently close to S ( σ ). In thiscase, solutions starting sufficiently close to the infection-free equilibrium point P leave a neighbourhood of P , except those starting in ∂ Ω. Since ˙ U ≤ P as t → ∞ . Theorem 2. If R > the endemic equilibrium point ¯ P is globally asymptotically stable in b Ω (i.e. awayfrom the boundary region ∂ Ω ).Proof of Theorem 2. Recall from the proof of theorem 1 that when R >
1, the force of infection F ( t ) isbounded away from zero for t >
0. Therefore, when R >
1, the interior region b Ω is forward invariant,i.e. Φ t : b Ω → b Ω. Hence, in analogy with theorem 1, for R > b D = Φ ¯ τ ( b Ω) where S , F > S , F ) ∈ b D .In this case, we introduce the Lyapunov functional W : b D → R + defined as W ( S , F ) = Z Σ ¯ S ( σ ) g (cid:18) S (0 , σ )¯ S ( σ ) (cid:19) dσ + Z Σ Z ¯ τ κ ( τ )¯ v ( σ ) g (cid:18) F ( − τ ) S ( − τ, σ )¯ F ¯ S ( σ ) (cid:19) dτ dσ (22)where g ( x ) has been defined previously in (16),¯ v ( σ ) = η ( σ ) ¯ F ¯ S ( σ ) (23)6nd κ ( τ ) = ¯ η Z ¯ ττ A ( ρ ) dρ. (24)Similar to before, the kernel κ has the following properties: κ (0) = 1 , κ (¯ τ ) = 0 and dκ ( τ ) dτ = − ¯ ηA ( τ ) . (25)Following the same steps as in theorem 1, we evaluate the Lyapunov functional W along system trajec-tories: W ( S t , F t ) = Z Σ ¯ S ( σ ) g (cid:18) S ( t, σ )¯ S ( σ ) (cid:19) dσ + Z Σ Z tt − ¯ τ κ ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ. Next, we differentiate each term in W ( S t , F t ) with respect to time to get ddt Z Σ ¯ S ( σ ) g (cid:18) S ( t, σ )¯ S ( σ ) (cid:19) dσ = Z Σ (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dS ( t, σ ) dt dσ, = Z Σ (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) ( λ ( σ ) − µS ( t, σ ) − η ( σ ) F ( t ) S ( t, σ )) dσ, = − µ Z Σ S ( t, σ ) (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dσ + Z Σ ¯ v ( σ ) (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dσ − F ( t ) Z Σ η ( σ ) (cid:0) S ( t, σ ) − ¯ S ( σ ) (cid:1) dσ, (26)where in the final line we have used the identity λ ( σ ) = µ ¯ S ( σ ) + η ( σ ) ¯ F ¯ S ( σ ) = µ ¯ S ( σ ) + ¯ v ( σ ) . Similarly, differentiating the second term and substituting in the properties of κ ( τ ) (eq. (25)) gives ddt Z Σ Z tt − ¯ τ κ ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ = Z Σ ddt Z tt − ¯ τ κ ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ = Z Σ (cid:20) κ (0)¯ v ( σ ) g (cid:18) F ( t ) S ( t, σ )¯ F ¯ S ( σ ) (cid:19) − κ (¯ τ )¯ v ( σ ) g (cid:18) F ( t − ¯ τ ) S ( t − ¯ τ , σ )¯ F ¯ S ( σ ) (cid:19) + Z tt − ¯ τ dκ ( t − s ) dt ¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds (cid:21) dσ, = Z Σ ¯ v ( σ ) g (cid:18) F ( t ) S ( t, σ )¯ F ¯ S ( σ ) (cid:19) dσ − ¯ η Z Σ Z tt − ¯ τ A ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ. Next, we substitute in the definition of g ( x ) and collect like terms to get ddt Z Σ Z tt − ¯ τ κ ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ = F ( t ) Z Σ η ( σ ) (cid:0) S ( t, σ ) − ¯ S ( σ ) (cid:1) dσ − Z Σ ¯ v ( σ ) (cid:20) log (cid:18) F ( t ) S ( t, σ )¯ F ¯ S ( σ ) (cid:19) − ¯ η Z tt − ¯ τ A ( t − s ) log (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds (cid:21) dσ. η Z tt − ¯ τ A ( t − s ) log (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds ≤ log (cid:20) ¯ η ¯ F ¯ S ( σ ) Z tt − ¯ τ A ( t − s ) F ( s ) S ( s, σ ) ds (cid:21) ≤ log (cid:18) ¯ ηG ( t, σ )¯ F ¯ S ( σ ) (cid:19) where G ( t, σ ) = Z tt − ¯ τ A ( t − s ) F ( s ) S ( s, σ ) ds. (27)Substituting this result back in we then have ddt Z Σ Z tt − ¯ τ κ ( t − s )¯ v ( σ ) g (cid:18) F ( s ) S ( s, σ )¯ F ¯ S ( σ ) (cid:19) ds dσ ≤ F ( t ) Z Σ η ( σ ) (cid:0) S ( t, σ ) − ¯ S ( σ ) (cid:1) dσ + Z Σ ¯ v ( σ ) log (cid:18) ¯ ηG ( t, σ ) F ( t ) S ( t, σ ) (cid:19) dσ. (28)Combining (26) and (28) then yields dW ( S t , F t ) dt ≤ − µ Z Σ S ( t, σ ) (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dσ + Z Σ ¯ v ( σ ) (cid:20) − ¯ S ( σ ) S ( t, σ ) + log (cid:18) ¯ ηG ( t, σ ) F ( t ) S ( t, σ ) (cid:19)(cid:21) dσ. In order to demonstrate that the expression on the right-hand side is indeed non-positive, we first add andsubtract the expression Z Σ ¯ v ( σ ) log (cid:18) ¯ S ( σ ) S ( t, σ ) (cid:19) dσ to get dW ( S t , F t ) dt ≤ − µ Z Σ S ( t, σ ) (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dσ − Z Σ ¯ v ( σ ) (cid:20) g (cid:18) ¯ S ( σ ) S ( t, σ ) (cid:19) + log (cid:18) ¯ ηG ( t, σ ) F ( t ) ¯ S ( σ ) (cid:19)(cid:21) dσ. (29)Secondly, we add a zero term: Z Σ ¯ v ( σ ) (cid:20) − ¯ ηG ( t, σ ) F ( t ) ¯ S ( σ ) (cid:21) dσ = ¯ F (cid:20)Z Σ η ( σ ) ¯ S ( σ ) dσ − ¯ ηF ( t ) Z Σ η ( σ ) G ( t, σ ) dσ (cid:21) , = ¯ F [¯ η − ¯ η ] , = 0to the right-hand side to finally obtain dW ( S t , F t ) dt ≤ − µ Z Σ S ( t, σ ) (cid:18) − ¯ S ( σ ) S ( t, σ ) (cid:19) dσ − Z Σ ¯ v ( σ ) (cid:20) g (cid:18) ¯ S ( σ ) S ( t, σ ) (cid:19) + g (cid:18) ¯ ηG ( t, σ ) F ( t ) ¯ S ( t, σ ) (cid:19)(cid:21) dσ. (30)8ince g ( x ) ≥ dW/dt ≤
0. Moreover, from (30) we see that the largest invariant subset in b Ωfor which ˙ W = 0 consists only of the endemic equilibrium point ¯ P . Therefore, since the orbit is eventuallyprecompact, by LaSalle’s extension to Lyapunov’s asymptotic stability theorem [24, Theorem 5.17], theendemic equilibrium point ¯ P is globally asymptotically stable.
5. Acknowledgements
The authors would like to gratefully acknowledge Prof. Johannes M¨uller for providing several key sug-gestions during the preparation of this manuscript.
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