In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio R0 gives the threshold of the stability. If R0>1, the interior equilibrium is unique and globally stable, and if R0≤1, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.
Citation: Tsuyoshi Kajiwara, Toru Sasaki, Yoji Otani. Global stability of an age-structured infection model in vivo with two compartments and two routes[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11047-11070. doi: 10.3934/mbe.2022515
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In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio R0 gives the threshold of the stability. If R0>1, the interior equilibrium is unique and globally stable, and if R0≤1, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.
Until recent years, models for in-host infection had been considered with only one compartment. On the other hand, Qesmi et al. [1,2] proposed models for hepatitis B and C infections. Their models have two components of infection, for example, liver cells and blood. The model in [1] is an ordinary differential equation model, and that in [2] is an age-structured model. They used models which incorporate the effect of absorption of pathogens into uninfected cells, and showed that a backward bifurcation can occur under some conditions. In Kajiwara et al. [3], the global stability of the interior equilibrium for the same ordinary differential model in [1] is shown using a Lyapunov function under some condition on parameters.
Recently, a cell-to-cell infection route for within-host infection is also paid much attention to. For example, Hübner et al. [4] suggest that HIV infection is enhanced by cell-cell adhesions. For models with two routes of infection, virus-to-cell and cell-to-cell infections, the stability analysis is done (Pourbashash et al. [5] for an ordinary differential equation model, Lai et al. [6] for a model with time delay). Wang et al. [7] considered an age-structured model with two routes of infection, and constructed a Lyapunov functional for their model. Wu et al. [8] proposed a model that considers two infection routes and two virus strains. They transformed the system into integro-differential equations, and proved some local stability and persistence results.
Models that consider both a cell-to-cell infection route and two compartments were investigated by, for example, Cheng et al. [9] and Wu and Zhao [10]. Cheng et al. [9] extended the model of [7] to a 2-compartment model as in Qesmi et al. [1,2]. They formulated the model as an abstract Cauchy problem, and analyzed the model. They defined a quantity R0, which is similar to the basic reproduction number but is not equal to it. They showed that if the quantity R0 is greater than 1, there exists an interior equilibrium, and showed that a uniform persistence result holds if R0>1. They showed the global stability results only for some restricted cases. They treated the case where the infection route is unique for each compartment, and also treated the case where there exist two infection routes under some restriction on parameters by using the asymptotic stability theory. Cheng et al. [9] did not use a Lyapunov functional, and did not present a general result on the global stability for their model. Wu et al. [10] proposed models with two compartments, two infection routes, two virus strains and an age-structure. They showed some stability results, and proved a persistence result in the case R0>1. However they suggested the stability of the infection steady state by numerical simulations, and its mathematical proof was not given there.
In this paper, we formulate the model proposed in Cheng et al. [9] as an integral equation model. We show that the quantity Rm in Cheng et al. [9] is the type reproduction number (Roberts and Heesterbeek [11]) for the class of pathogens, and Rm determines the exact order relation between the basic reproduction number and 1. We prove qualitative properties, for example, asymptotic smoothness, and show a persistence result which is necessary to the definition and calculation of Lyapunov functionals. We follow the method in Smith and Thieme [12]. Moreover, we construct Lyapunov functionals for the cases R0>1 and R0≤1. We then show that the unique interior equilibrium is globally asymptotically stable (GAS) if R0>1, and the disease free equilibrium (DFE) is globally asymptotically stable if R0≤1, using an argument over the alpha-limit sets of total solutions in the compact attractor.
In this section, we present fundamental results of the model, the basic reproductive number and the type reproduction number, and compactness arguments.
We assume that the number of compartments is two, and assign numbers 1 and 2 to each compartment. We denote by Tj the amount of the uninfected cells in the jth compartment and denote by ij(t,a) the infection age density of the infected cells in the jth compartment. We denote by V the amount of the pathogens. Since blood circulates quickly, we assume that V is common for each compartment.
We consider the following age-structured model with two compartments presented in Cheng et al. [9]:
dT1dt=f1(T1(t))−β11T1(t)V(t)−β12T1(t)∫∞0p1(a)i1(t,a)da,∂i1(t,a)∂t+∂i1(t,a)∂a=−(δ1(a)+m1)i1(t,a),dT2dt=f2(T2(t))−β21T2(t)V(t)−β22T2(t)∫∞0p2(a)i2(t,a)da,∂i2(t,a)∂t+∂i2(t,a)∂a=−(δ2(a)+m2)i2(t,a),dVdt=∫∞0q1(a)i1(t,a)da+∫∞0q2(a)i2(t,a)da−cV,i1(t,0)=β11T1(t)V(t)+β12T1(t)∫∞0p1(a)i1(t,a)da,i2(t,0)=β21T2(t)V(t)+β22T2(t)∫∞0p2(a)i2(t,a)da,T1(0)=T10>0,i1(0,a)=i10(a)∈L1([0,∞),R+),T2(0)=T20>0,i2(0,a)=i20(a)∈L1([0,∞),R+),V(0)=V0∈R+={x∈R|x≥0}. | (2.1) |
The constants βjk, mj and c are positive for each j=1, 2 and k=1, 2. For the growth function fj(x) of uninfected cells, we assume that fj(x) is a differentiable function with the properties
fj(0)>0,f′j(x)<0,fj(¯Tj)=0 |
for each j, where ¯Tj is a positive constant. We moreover assume that there exist constants Aj and Bj such that
fj(s)≦Aj−Bjs(s≧0). | (2.2) |
We note that the form fj(x)=hj−djx is often used. The functions pj(a)'s are the viral production rates of an infected cell with infection age a in the jth compartment, and qj(a)'s are the viral release rates of an infected cell with age a in the jth compartment. We assume that the non-negative functions pj(a), qj(a) and δj(a) are Lipschitz continuous, and that pj(a) and qj(a) are essentially bounded on (0,∞). For the definition of the Lyapunov functionals, we moreover assume that the functions apj:a↦apj(a) and aqj:a↦aqj(a) satisfy
apj,aqj∈L1([0,∞)). | (2.3) |
Define σj(a) by
σj(a)=e−∫a0(δj(b)+mj)db. |
Since δj(a) is continuous, σj(a) is differentiable. For j=1, 2, put
Jj[ij]=∫∞0pj(a)ij(a)da, |
where ij(⋅)∈L1([0,∞)) and put
Jj(t)=Jj[ij(t,a)]=∫∞0pj(a)ij(t,a)da, |
where ij(t,a) is an element of a solution of (2.1). The functions Jj(t) represents the forces of infection at each component. It is possible to integrate ij(t,a) along their characteristic curves:
ij(t,a)={σj(a)(βj1Tj(t−a)V(t−a)+βj2Tj(t−a)Jj(t−a))t≥a,σj(a)σj(a−t)ij0(a−t)t<a,(j=1,2). | (2.4) |
We note that the value of ij(t,a) can be recovered from Tj, V, Jj and the initial value ij0(⋅). We define a set ˜X by
˜X=R×L1([0,∞),R)×R×L1([0,∞),R)×R, |
with the ordinary product topology.
We take u=(T10,i10(⋅),T20,i20(⋅),V0)∈˜X, and use u as the initial condition. We translate the original differential equation model (2.1) into an integral equation model. First it holds
dTjdt=fj(Tj)−βj1Tj(t)V(t)−βj2Tj(t)Jj(t),(j=1,2),Jj(t)=∫∞0pj(a)ij(t,a)da=∫t0pj(a)σj(a)(βj1Tj(t−a)V(t−a)+βj2Tj(t−a)Jj(t−a))da+∫∞tpj(a)σj(a)σj(a−t)ij0(a−t)da,(j=1,2),dVdt=∫∞0q1(a)i1(t,a)da+∫∞0q2(a)i2(t,a)da−cV(t)=∫t0q1(a)(σ1(a)(β11T1(t−a)V(t−a)+β12T1(t−a)J1(t−a)))da+∫t0q2(a)(σ2(a)(β21T2(t−a)V(t−a)+β22T2(t−a)J2(t−a)))da+∫∞tq1(a)σ1(a)σ1(a−t)i10(a−t)da+∫∞tq2(a)σ2(a)σ2(a−t)i20(a−t)da−cV(t). | (2.5) |
Using the method of variation of the constant, we get the following integral equation model:
Tj(t)=Tj0+∫t0(fj(Tj(s))−βj1Tj(s)V(s)−βj2Tj(s)Jj(s))ds,(j=1,2),Jj(t)=∫t0pj(a)σj(a)(βj1Tj(t−a)V(t−a)+βj2Tj(t−a)Jj(t−a))da+∫∞tpj(a)σj(a)σj(a−t)ij0(a−t)da,(j=1,2),V(t)=e−ctV0+e−ct∫t0ecs(∫s0q1(a)(σ1(a)(β11T1(s−a)V(s−a)+β12T1(s−a)J1(s−a)))da+∫s0q2(a)(σ2(a)(β21T2(s−a)V(s−a)+β22T2(s−a)J2(s−a)))da+∫∞sq1(a)σ1(a)σ1(a−s)i10(a−s)da+∫∞sq2(a)σ2(a)σ2(a−s)i20(a−s)da)ds. | (2.6) |
Theorem 2.1. A local solution of (2.6) exists uniquely.
Proof. The proof is similar to that of Proposition 1 in [13]. We use the Banach fixed point theorem in the space
Hτ={(T1,T2,J1,J2,V)∈C[0,τ]5;|T1−T10|≦M,|T2−T20|≦M,|J1−J10|≦M,|T2−T20|≦M,|V−V0|≦M} |
where M and τ are positive numbers, τ being chosen sufficiently small later. We can define the integral operator from H to itself by (2.6), if we take τ small enough. Moreover the operator defines the contractive map for small enough τ. Thus we can use the Banach fixed point theorem to complete the proof.
Let u=(T10,i10(⋅),T20,i20(⋅),V0)∈˜X. We denote by (0,τ(u)) the maximum existence interval of the solution u of (2.6) such that u(0)=u0.
The following theorem is proved in Cheng, Dong and Takeuchi [9] (Lemma 1). We note that the equations for ij and V are common.
Theorem 2.2. Let u0=(T10,i10(⋅),T20,i20(⋅),V0) and assume Tj0>0, ij0(⋅)∈L1([0,∞),R+)) and V0≥0. Then for 0<t<τ(u), the all components of solution u with u(0)=u0 take nonnegative values.
We give a complete metric on (0,∞) which gives the usual topology on it as in [13] and put
X=(0,∞)×L1([0,∞),R+)×(0,∞)×L1([0,∞),R+)×R+ |
with the product topology.
By a standard method using differential inequalities, we can show that each solution is bounded as long as it exists.
Lemma 2.3. The positive orbit of each bounded subset of X is bounded.
Proof. This lemma is shown by using differential inequalities as in [13]. We consider
Wj(t)=Tj(t)+∫∞0ij(t,a)da |
for i=1,2. Then, by (2.4), we have
Wj(t)=Tj(t)+∫t0βj1Tj(t−a)V(t−a)σj(a)da+∫t0βj2T(t−a)Jj(t−a)σj(a)da+∫∞0σj(b+t)σj(b)ij0(b)db. |
From this, we can obtain
dWjdt=fj(Tj(t))−∫∞0(δj(a)+mj)ij(t,a)da≦fj(Tj(t))−mj∫∞0ij(t,a)da. |
Thus, using (2.2), we can show that Wj is bounded, and hence Tj and ij are also bounded.
By the equation for V in (2.1) and by the fact that ij is bounded in L1, it is easy to show that V is bounded, because qi's are assumed to be essentially bounded.
By Lemma 2.3, each solution to (2.1) is bounded as long as it exists. Then it holds τ(u)=∞ for each u∈X. As in Kajiwara et al. [13], we can define a semiflow on the phase space X corresponding to the solutions of the equation.
Definition 2.4. We define a semiflow {St}t≥0 on X satisfying St(u)=u(t) for each u∈X, where u(t) is the solution of (2.1) with u(0)=u.
As in [13], {St}t≥0 is a continuous semiflow on the phase space X.
Following Cheng et al. [9], put
Mj=∫∞0pj(a)σj(a)da,Nj=∫∞0qj(a)σj(a)da,(j=1,2), |
and put
R1=R11+R12,whereR11=β11N1c¯T1,R12=β12M1¯T1,R2=R21+R22,whereR21=β21N2c¯T2,R22=β22M2¯T2,Rm=R111−R12+R211−R22forR12<1,R22<1. |
We consider them using the notion of type reproduction numbers. We assume the population is divided into n host-types, and firstly focus on one of the host-types. The infection spreads from one infected individual of the first host-type around other host-types, and finally produces infected individuals of the first host-type. The average number of the secondary infected individual of the first host-type is called the type reproduction number [11,14].
To consider the next generation matrix and the type reproduction number, we call the class of pathogens, the class of infected cells in compartment 1 and the class of infected cells in compartment 2, 0th-class, first-class and second-class respectively. We denote by tpq (p,q=0, 1, 2) the average number of pathogens or infected cells directly created in the pth class from qth class at DFE. Put
K=[t00t01t02t10t11t12t20t21t22]. |
Then K is the next generation matrix (NGM), and the spectral radius of K is the basic reproductive number R0 of the model.
Lemma 2.5. For the elements of K, we have
t00=0,t12=0,t21=0,t10t01=R11,t20t02=R21,t11=R12,t22=R22. |
Proof. It is trivial that t00=0, t12=0 and t21=0.
The quantity t10t01 is the average number of pathogens newly created in compartment 1 from a pathogen at DFE. At time t, the population size of pathogens which exist at t=0 is written as V(t)=V0e−ct. We denote the number of pathogens by ˜V(t) which are newly created in compartment 1 until time t. Since ˜V(0)=0, it holds
˜V(∞)=∫∞0d˜Vdtdt=β11∫∞0∫t0q1(a)σ1(a)¯T1V(t−a)dadt=β11¯T1∫∞0q1(a)σ1(a)da⋅∫∞0V(t)dt=β11N1¯T1cV0=R11V0, |
then t01t10=R11. Similarly, t02t20=R21.
We consider t11 and t22. We denote by ij(t,0) the age density at a=0 of infected cells created directly from an infected cell in compartment j. By the boundary condition, it holds
ij(t,0)=βj2¯Tj∫∞0pj(a)ij(t,a)da. |
Then
ij(t,0)=βj2¯Tj∫t0pj(a)ij(t,a)da+βj2¯Tj∫∞tpj(a)ij(t,a)da=∫t0βj2¯Tjpj(a)σj(a)ij(t−a,0)da+∫∞tβj2¯Tjpj(a)σj(a)σj(a−t)ij0(a−t,0)da. | (2.7) |
Put
ψ(t)=ij(t,0),K(a)=βj2¯Tjpj(a)σj(a),g(t)=∫∞tβj2¯Tjpj(a)σj(a)σj(a−t)ij0(a−t,0)da, |
then (2.7) is written as
ψ(t)=∫t0K(a)ψ(t−a)da+g(t). | (2.8) |
This is a renewal equation with respect to ψ(t), and the basic reproductive number R10 of (2.8) is calculated as
R10=∫∞0K(a)da. |
Since t11 is equal to R10,
t11=β12¯T1M1=R12. |
Similarly, t22=R22.
The quantity Rm is a type reproduction number defined in [11].
Lemma 2.6. We assume that R12<1 and R22<1. Then the type reproduction number TV of the class of pathogens is well defined, and is equal to Rm.
Proof. We assume that R12<1 and R22<1. Then the average number of pathogens which are newly infected from V class at DFE is as follows:
t10∞∑p=0tp11t01+t20∞∑p=0tp22t02=t10t01∞∑p=0Rp22+t20t02∞∑p=0Rp33=R111−R12+R211−R22. |
Then it holds TV=Rm.
We note that the characteristic equation of K is as follows:
Λ3−(R12+R22)Λ2−(R21+R11−R12R22)Λ+R12R21+R22R11=0. | (2.9) |
The quantity R0 is the largest real solution of (2.9). Since the equation is cubic, it is not easy to calculate R0. But, it is possible to describe the following threshold condition R0>1 (Cheng et al. [9]).
Lemma 2.7. R0>1 is equivalent to the following:
R1>1orR2>1or(R1≤1,R2≤1,andRm>1) |
Proof. Suppose R0>1. First, assume R1≤1, R2≤1. Since R12<1, R22<1, Rm=TV. By [11], Rm>1.
Conversely, assume Rj>1. Let Kj be the NGM of the j-compartment model. Kj is nonnegative and Kj≤K. Then the Perron-Frobenius eigenvalue of K is equal or greater than that of Kj, then R0>1. If R1≤1, R2≤1, Rm>1, by [11] R0>1.
The following proposition is proved in Theorem 6 of [9].
Proposition 2.8. (Cheng et al. [9]) If R0>1, then an interior equilibrium exists.
We note that if T∗j>0 (j=1,2) are specified for the interior equilibrium, then V∗ and i∗j are uniquely determined as:
V∗=1c(f1(T∗1)N1+f2(T∗2)N2),i∗j(a)=fj(T∗j)σj(a). |
Lemma 2.9. If R0≤1, interior equilibria do not exist.
Proof. By Theorem 4.4 in Section 4.2, if R0≤1, then DFE is GAS and interior equilibria do not exist.
Theorem 2.10. The semiflow {St}t≥0 on X is point dissipative.
Proof. It is shown using differential inequalities as in [13].
Proposition 2.11. The semiflow {St}t≥0 on X is asymptotically smooth.
Proof. It is proved by the method in Demasse et al. [15] and Kajiwara et al. [16]. Let B be a forward invariant bounded subset of X. Take an infinite sequence {up}p=1,2,…=((T1)p),(i1)p,(T2)p,(i2)p,Vp)p=1,2,… in B and an infinite sequence {tp}p=1,2,… in R+ with tp→∞. For t≥0, we put up(t)=S(t)up. We show that {up(tp)} contains a convergent sequence. Since the bounded subset B is positively invariant, we can assume that subsequences {(Ti)p(tp)}p=1,2,…, {Vp(tp)}p=1,2,… (i=0,1) of R are convergent. For t≥−tp we define
(Ti)p(t)=(Ti)p(t+tp),Vp(t)=Vp(t+tp). |
We extend (Ti)p(t) and Vp(t) (i=0,1) for t≤−tp continuously such that their maximums do not exceed those in t≥−tp, their Lipschitz norms are not greater than 1 and their values are zero for sufficiently small t. Since B is forward invariant, they are uniformly bounded. Moreover their Lipschitz norms are also uniformly bounded because they are elements of the solution of (2.6). Using Ascolli-Arzella Theorem for R, we can take a subsequence which is convergent locally uniformly from {(Tj)p}p=1,2,… and {Vp}p=1,2…. Using the Cantor diagonal process, they contain subsequences of the same indices which converge uniformly on each compact interval, Using the Volterra expression (2.4) for ij(t,a) (j=1,2), and by the standard arguments, we can show that {(ij)p(tp,a)}p=1,2,… has convergent subsequence with respect to L1 topology for j=1, 2. The semiflow {St}t≥0 is asymptotically compact on each forward invariant bounded subset, then {St}t≥0 is asymptotically smooth.
Lemma 2.12. The semiflow {St}t≥0 has a compact attractor A for bounded subsets in X.
Proof. The semiflow {St}t≥0 is asymptotically smooth and each positive orbit of a bounded subset is bounded under {St}t≥0. Then by Theorem 2.33 in [12], {St}t≥0 has a compact attractor in X.
The following proposition is used to show that if a compact attractor contains only one point, it attracts all points and is locally stable. The proof is contained in [13].
Proposition 2.13. (Simplified version of Lemma 23.7 in Sell and You [17]) Let X be a compact metric space and {St}t≥0 be acontinuous semiflow on X. Assume that a compact attractor of {St}t≥0 consists of one equilibrium x∗. Then the equilibrium x∗ is locally stable and is globally asymptotically stable. If there exists a persistence attractor for {St}t≥0, the unique equilibrium in the persistence attractor is globally asymptotically stable by a similar argument.
In this section, we always assume that R0>1, and present results of the semiflow {St}t≥0.
The persistence result of the semiflow {St}1≥t is necessary for the definition of the Lyapunov functional that will be defined in Section 4.1. We use the method of Smith et al. [12] for persistence.
Define a persistence function ρ for u=(T1,i1(⋅),T2,i2(⋅),V)∈X by
ρ(u)=J1[i1]+J2[i2]+V. |
Since {u∈X|ρ(x)=0} is not forward invariant in general, put X0 by
X0={u∈X|ρ(St(u))=0,foreacht≥0}. |
Theorem 3.1. The disease free equilibrium (DFE) is globally asymptotically stable in X0, and the attractor of the semiflow {St}t≥0 restricted to X0 consists of DFE.
Proof. For u∈X0, put St(u)=(T1(t),i1(t,⋅),T2(t),i2(t,⋅),V(t)). The equation of Tj is
dTjdt=fj(Tj), |
then Tj(t)→¯Tj. On the other hand, since V(t)=0 and Jj(t)=Jj[ij(t,⋅)]=0 for each t≥0 in (2.4), it holds
∫∞0ij(t,a)da=∫∞tσj(a)σj(a−t)ij0(a−t)da,≤e−mjt∫∞0ij0(a)da≤e−mjt‖ij0‖1. |
Then ij(t,⋅) tends to 0 in L1 topology.
Lemma 3.2. We assume ρ(u)>0. Then V(t)>0 for some t>0.
Proof. We note that Tj(t)>0 for each t∈R. We assume ρ(u)>0. If V>0, V(t)>0 for each t>0, and there is nothing to prove. We assume Jj[ij]=Jj(0)>0. Then Jj(t)>0 for some neighborhood of 0. Then by the V(t) equation of (2.6), V(t)>0 for some t>0.
We show that DFE is uniformly weakly ρ-repelling by contradiction.
Lemma 3.3. If we take sufficiently small ε>0, then for a solution u(t) with initial value (T01,i01,T02,i02,V0) such that
¯T1−ε<T01<¯T1+ε,¯T2−ε<T02<¯T2+ε,0<V0<ε, |
there exists t1>0 such that u(t1)=(T1(t1),i1(t1,⋅),T2(t1),i2(t1,⋅),V(t1)) does not satisfy at least one inequality of
¯T1−ε<T1(t1)<¯T1+ε,¯T2−ε<T2(t1)<¯T2+ε,0<V(t1)<ε. | (3.1) |
Proof. We assume that (3.1) holds for all t≥0. Then it holds
Jj(t)=∫∞0pj(a)ij(t,a)da≥∫t0pj(a)ij(t,a)da=∫t0pj(a)ij(t−a,0)σj(a)da=∫t0βj1pj(a)Tj(t−a)V(t−a)σj(a)da+∫t0βj2pj(a)σj(a)Tj(t−a)Jj(t−a)da≥∫t0βj1(¯Tj−ε)pj(a)σj(a)V(t−a)da+∫t0βj2(¯Tj−ε)pj(a)σj(a)Jj(t−a)da. | (3.2) |
On the other hand, it holds
dVdt≥∫t0β11q1(a)σ1(a)T1(t−a)V(t−a)da+∫t0β12q1(a)σ1(a)T1(t−a)J1(t−a)da+∫t0β21q2(a)σ2(a)T2(t−a)V(t−a)da+∫t0β22q2(a)σ2(a)T2(t−a)J2(t−a)da−cV(t)≥∫t0β11q1(a)σ1(a)(¯T1−ε)V(t−a)da+∫t0β12q1(a)σ1(a)(¯T1−ε)J1(t−a)da+∫t0β21q2(a)σ2(a)(¯T2−ε)V(t−a)da+∫t0β22q2(a)σ2(a)(¯T2−ε)J2(t−a)da−cV(t). | (3.3) |
Note that Jj(t) and V(t) are bounded continuous functions. We assume λ>0. Let ˆJj(λ), ˆV(λ), Mj(λ) and Ni(λ) denote the Laplace transformations of Jj(t), V(t), pj(a)σj(a) and qj(a)σj(a). We note that the limits limλ→+0Mj(λ) and limλ→+0Nj(λ). Let Mj and Nj denote these positive values.
We take Laplace transformations of both sides of (3.2).
For j=1, 2, we have
ˆJj(λ)≥βj1Mj(λ)(¯Tj−ε)ˆV(λ)+βj2Mj(λ)(¯Tj−ε)ˆJj(λ). | (3.4) |
First consider the case of R12>1 or R22>1. If R12=β12M1¯T1>1, for sufficiently small ε>0 it holds
β12M1(¯T1−ε)>1. | (3.5) |
By (3.4), we have
ˆJ1(λ)≥β12M1(λ)(¯T1−ε)ˆJ1(λ). |
If the initial value of V(t) is positive, V(t)>0 for each t>0. Then for some t>0, J1(t)>0, then ˆJ1(λ) is also positive for λ>0. Then it holds
1≥β12M1(λ)(¯T1−ε). | (3.6) |
Then (3.6) contradicts to (3.5). For the case of R22>1, a contradiction holds using a similar argument.
We consider the case R12≤1 and R22≤1, and R1>1 or R2>1. We assume that R12≤1 and R1>1. The assumption contains the case R12=1. If ε>0, then 1>β12M1(¯T1−ε). If we take sufficiently small λ>0, it holds
1−β12M1(λ)(¯T1−ε)>0. |
By (3.4), we have
ˆJ1(λ){1−β12M1(λ)(¯T1−ε)}≥β11M1(λ)(¯T1−ε)ˆV(λ), |
and hence we obtain
ˆJ1(λ)≥β11M1(λ)(¯T1−ε)1−β12M1(λ)(¯T1−ε)ˆV(λ). | (3.7) |
We use the following equation obtained by the Laplace transformation of both sides of (3.3):
λˆV(λ)−V(0)≥β11N1(λ)(¯T1−ε)ˆV(λ)+β12N1(λ)(¯T1−ε)ˆJ1(λ)+β21N2(λ)(¯T2−ε)ˆV(λ)+β22N2(λ)(¯T2−ε)ˆJ2(λ)−cˆV(λ). | (3.8) |
Thus if we drop the last two nonnegative terms, by (3.7), we have
λˆV(λ)−V(0)≥β11N1(λ)(¯T1−ε)ˆV(λ)+β11M1(λ)β12N1(λ)(¯T1−ε)21−β12M1(λ)(¯T1−ε)ˆV(λ)−cˆV(λ). |
Then it holds
{λ−β11N1(λ)(¯T1−ε)−β11M1(λ)β12N1(λ)(¯T1−ε)21−β12M1(λ)(¯T1−ε)+c}ˆV(λ)≥V(0). | (3.9) |
Here the coefficient of ^V(λ) is
λ−β11N1(λ)(¯T1−ε)−λβ12M1(λ)(¯T1−ε))}+c−cβ12M1(λ)(¯T1−ε)1−β12M1(λ)(¯T1−ε). |
Since R1=R11+R12>1, by taking sufficiently small ε>0, λ>0, we can take the left hand side of (3.9) to be negative. Since the right hand side is positive, a contradiction holds. The case R2>1 is similar.
Last consider the case of R12<1, R22<1, R1≤1 and R2≤1. We note that Rm>1, since R0>1 is assumed. By taking sufficiently small ε>0, it holds
β12M1(¯T1−ε)<1,β22M2(¯T2−ε)<1. |
By taking sufficiently small λ>0, it holds
β12M1(λ)(¯T1−ε)<1,β22M2(λ)(¯T2−ε)<1. | (3.10) |
For sufficiently small ε>0, λ>0, by (3.4), it holds
ˆJ1(λ)≥β11M1(λ)(¯T1−ε)1−β12M1(λ)(¯T1−ε)ˆV(λ),ˆJ2(λ)≥β21M2(λ)(¯T2−ε)1−β22M2(λ)(¯T2−ε)ˆV(λ). |
Substituting these to (3.8),
λˆV(λ)−V(0)≥{β11N1(λ)(¯T1−ε)+β12N1(λ)(¯T1−ε)β11M1(λ)(¯T1−ε))1−β12M1(λ)(¯T1−ε)+β21N2(λ)(¯T2−ε)+β22N2(λ)(¯T2−ε)β21M2(λ)(¯T2−ε)1−β22M2(λ)(¯T2−ε)−c}ˆV(λ)=c{β11N1(λ)(¯T1−ε)c(1−β12M1(λ)(¯T1−ε))+β21N2(λ)(¯T2−ε)c(1−β22M2(λ)(¯T2−ε))−1}ˆV(λ). |
Then it holds
[λ−c{β11N1(λ)(¯T1−ε)c(1−β12M1(λ)(¯T1−ε))+β21N2(λ)(¯T2−ε)c(1−β22M2(λ)(¯T2−ε))−1}]ˆV(λ)≥V(0). |
Since Rm>1, the coefficient of ˆV(λ) tends to −c(Rm−1)<0 as ε→+0 and λ→+0. Hence the coefficient is negative for sufficiently small λ>0 and ε>0. A contradiction occurs because ˆV(λ)>0 and V(0)≥0.
Lemma 3.4. If R0>1, the equilibrium DFE is uniformly weakly ρ-repelling in X.
Proof. Let u∈X satisfy ρ(u)>0. It means that pathogens or forces of infection are present. Since V(t2)>0 for some t2>0 by Proposition 3.2, the solution u(t) with u(0)=u, if it enters a neighborhood of the DFE, escapes from the neighborhood by Lemma 3.3, provided that the neighborhood is taken sufficiently small. Then the conclusion holds.
Lemma 3.5. If R0>1, the equilibrium DFE is isolated in X.
Proof. By Proposition 3.1, DFE is globally asymptotically stable in X0, that is DFE has an isolated neighborhood in X0. Moreover, DFE is uniformly weakly ρ-repelling in X by Lemma 3.4. Then DFE is shown to be isolated in X using Lemma 8.18. of Smith and Thieme [12].
Lemma 3.6. There exists no cycle in X0 connecting the sets of attractor.
Proof. It is shown from the fact that DFE is globally asymptotically stable in X0.
Proposition 3.7. The semiflow {St}t≥0 on X is uniformly weakly ρ-persistent.
Proof. It follows from Theorem 8.17 in Smith and Thieme [12].
It is necessary to exclude total solutions u(t) which is ρ(u(t0))>0, ρ(u(t1))=0 and ρ(u(t2))>0 for some t0<t1<t2 to get uniformly ρ-persistence from uniformly weakly ρ-persistence by using the method of Section 5 in Smith and Thieme [12].
Let u(t)=(T1(t),i1(t,a),T2(t),i2(t,a),V(t)) a total solution. Then the following equations are satisfied:
V(t)=e−ctV(0)+e−ct∫t0ecs(∫∞0q1(a)σ1(a)(β11T1(s−a)V(s−a)+β12T1(s−a)J1(s−a))da+∫∞0q2(a)σ2(a)(β21T2(s−a)V(s−a)+β22T2(s−a)J2(s−a))da)ds, | (3.11) |
Jj(t)=∫∞0pj(a)σj(a)(βj1Tj(t−a)V(t−a)+βj2Tj(t−a)Jj(t−a))da.(j=1,2) | (3.12) |
Lemma 3.8. Let u(t) be a total solution. Then if V(t1)>0, thenit holds V(t)>0 for t>t1.
Proof. By (3.11), if V(t1)>0 then V(t)>0 for t>t1.
Lemma 3.9. There exists no total solution such that ρ(u(r))>0, ρ(u(t0))=0, ρ(u(s))>0 (r<t0<s).
Proof. Without loss of generality, we can set t0=0. We assume the existence of such a total solution u(t). Since ρ(u(0))=0 implies V(0)=0, V(t)=0 for t≤0 by Lemma 3.8. We will show that Jj(t)=0 for t<0 (j=1,2). We note that q1(a) is continuous, and that q1(a0)>0 for some a0>0. Suppose J1(t1)>0 for some t1<0, then the integrand q1(a)σ1(a)β11T1(s−a)J1(s−a) is positive at s=a0+t1. Then it holds
e−ct∫t0ecs(∫∞0q1(a)(σ1(a)β11T1(s−a)J1(s−a)da)ds>0. |
Since the other integrands are nonnegative, V(t1)>0. Thus V(t)>0 for t>t1. By Lemma 3.8, it contradicts to V(t)=0 for t≤0. Then for t<0, Jj(t)=0 (j=1,2). By shifting time, we consider the total solution as the solution with initial value at t=t2<0. Since initial value of V(t) and Jj(t) are zero, Tj(t) and V(t)=0, Jj(t)=0 is a solution. By the uniqueness of the integral equation, V(t)=0, Jj(t)=0 for t≥t2. Then there exists no total solution such that ρ(u(r))>0, ρ(u(0))=0, ρ(u(s))>0.
From the proof above, we also obtain the following.
Corollary 3.10. Let u(t) be a total solution. Then V(t) element of u(t) is always positive or always 0. If V(t)=0 identically, Jj(t) is also identically 0.
Lemma 3.11. The semiflow {S(t)}t≥0 on X is uniform ρ-persistence.
Proof. We verify that two conditions (H0), (H1) in Chapter 5 of Smith and Thieme [12] are satisfied. (H0) follows from the existence of compact attractor in Section 2. The condition (H1) follows from Lemma 3.9. Then semiflow {S(t)}t≥0 is uniform ρ-persistence by Theorem 5.2 in [12].
Proposition 3.12. (Theorem 5.7 in Smith and Thieme [12]) If the semiflow {St} is uniformly ρ-persistent, a compact attractor A is decomposed as A=A0∪A1∪C with invariant sets A0, A1 and C. A0 and A1 are compact, and they satisfy (a), (b) and (c) in [12].
The set A1 is called the persistence attractor. We note that the persistence attractor A1 is a union of total trajectories.
Lemma 3.13. The V-element and Tj-elements (j=1,2) of A1 have positive minimum values.
Proof. Since persistence attractor is a compact set, V-element and Tj-elements have minimum values. Since Tj can not be 0, the minimum value is positive. We assume that the minimum value of V is zero, and denote by u such element in the phase space. Let u(t) be a total solution with u(0)=u. By Corollary 3.10, it holds ρ(u(t))=0 for each t∈R, then u∈X0. It contradicts to u∈A1.
We denote by ˜Tj and ˜V the minimum values of Tj, V in A1 respectively.
Lemma 3.14. Let (T∗1,i∗i(⋅),T∗2,i∗2(⋅),V∗) be an interior equilibrium. We assume that a total solution u(t) is contained in the persistence attractor A1. Then there exist M and M′ such that 0<M≤ij(t,a)/i∗j(a)≤M′ for t∈R, a∈R+.
Proof. Let u(t)∈A1, and ij(t,a) be an element of u(t). Then it holds
ij(t,a)i∗j(a)=ij(t−a,0)i∗j(0)=βj1V(t−a)Tj(t−a)+βj2Tj(t−a)Jj(t−a)βj1V∗T∗j+βj2T∗j+Jj[ij]≥βj1V(t−a)Tj(t−a)βj1V∗T∗j+βj2T∗j+Jj[ij]≥βj1˜V~Tjβj1V∗T∗j+βj2T∗j+Jj[ij], |
where ˜V and ˜Tj are minimum values. We note that
Jj[ij]=∫∞0p(a)i∗j(a)da |
is also a positive value. Then we can take such M. The existence of M′ follows from that u(t) is contained in the compact set A1.
Remark 3.1. It is not necessary to assume that ij(t,a) is differentiable with respect to t for the use of Lemma 9.18 in Smith and Thieme [12],
For R0>1, we construct a Lyapunov functional, which is defined in A1, for the system (2.1). By Proposition 2.8, there exists an interior equilibrium. In Section 4.1, we fix one interior equilibrium u∗=x(T∗1,i∗1(⋅),T∗2,i∗2(⋅),V∗). By the V-equation, it holds
∫∞0q1(a)i∗1(a)da+∫∞0q2(a)i∗2(a)da−cV∗=0. |
Then if we define c1, c2 by
c1=∫∞0q1(a)i∗1(a)daV∗,c2=∫∞0q2(a)i∗2(a)daV∗, |
then it holds c1V∗+c2V∗=cV∗, c=c1+c2, c1>0, c2>0. Moreover it holds
∫∞0q1(a)i∗1(a)da−c1V∗=0,∫∞0q2(a)i∗2(a)da−c2V∗=0. | (4.1) |
As in Kajiwara et al. [3], we rewrite the equation of V as
dVdt=(∫∞0q1(a)i1(t,a)da−c1V)+(∫∞0q2(a)i2(t,a)da−c2V). |
Put, for j=1 or 2,
Aj=∫∞0βj1T∗jqj(b)σj(b)cjdb,Bj=∫∞0βj2T∗jpj(b)σj(b)db. |
By the boundary condition of (2.1), it holds
i∗j(0)=βj1T∗jV∗+βj2T∗j∫∞0pj(a)i∗j(a)da. | (4.2) |
By substituting
i∗j(a)=i∗j(0)σj(a),V∗=(1/cj)∫∞0qj(a)i∗j(a)da | (4.3) |
obtained from (4.1), we get Aj+Bj=1. Put
ψj1(a)=∫∞aβj1T∗jqj(b)cjσj(b)σj(a)−1db,ψj2(a)=∫∞aβj2T∗jpj(b)σj(b)σj(a)−1db. |
We note that ψj1(a) and ψj2(a) are integrable on (0,∞) by the assumption (2.3). It holds ψj1(0)=Aj, ψj2(0)=Bj. Define functionals Wj1, Wj2,1, and Wj2,2 on the persistence attractor A1 by
Wj1(Tj)=Tj−T∗j−T∗jlogTjT∗j,Wj2,1(ij(⋅))=∫∞0ψj1(a)(ij(a)−i∗j(a)−i∗j(a)logij(a)i∗j(a))da, | (4.4) |
Wj2,2(ij(⋅))=∫∞0ψj2(a)(ij(a)−i∗j(a)−i∗j(a)logij(a)i∗j(a))da. | (4.5) |
By Proposition 3.14, the right hand sides of (4.4) and (4.5) are well defined. Then
Wj2,1(ij(⋅))=∫∞0ψj1(a)σj(a)(ij(a)−i∗j(a)−i∗j(a)logij(a)i∗j(a))σj(a)−1da,Wj2,2(ij(⋅))=∫∞0ψj2(a)σj(a)(ij(a)−i∗j(a)−i∗j(a)logij(a)i∗j(a))σj(a)−1da, |
and
ψj1(a)σj(a)=∫∞aβj1T∗jqj(b)cjσj(b)db,ψj2(a)σj(a)=∫∞aβj2T∗jpj(b)σj(b)db. |
Then it holds
dda(ψj1(a)σj(a))=−βj1T∗jqj(a)cjσj(a),dda(ψj2(a)σj(a))=−βj2T∗jpj(a)σj(a). |
On the other hand, if u(t) is a total solution and ij(t,a) is an element of u(t),
(ij(t,a)−i∗j(a)−i∗j(a)logij(t,a)i∗j(a))σj(a)−1=ij(t−a,0)−i∗j(0)−i∗j(0)logij(t−a,0)i∗j(0) |
is a function of t−a.
We define a functional W(u) for u=(T1,i1(⋅),T2,i2(⋅),V)∈A1 as follows:
W(u)=V−V∗−V∗logVV∗+2∑j=1cjβj1T∗j(Wj1(Tj)+Wj2,1(ij(⋅))+Wj2,2(ij(⋅))). | (4.6) |
We calculate the derivative of W(u) along solutions in A1:
dW(u(t))dt | (4.7) |
=2∑j=1{(1−V∗V)(∫∞0qj(a)ij(t,a)da−cjV)+cjβj1T∗j(dWj1(Tj(t))dt+dWj2,1(ij(t,⋅))dt+dWj2,2(ij(t,⋅))dt)}=2∑j=1cjβj1T∗j{dWj1(Tj(t))dt+dWj2,1(ij(t,⋅))dt+dWj2,2(ij(t,⋅))dt+βj1T∗jcj(1−V∗V)(∫∞0qj(a)ij(t,a)da−cjV)}. | (4.8) |
We calculated each term in (4.8). By (2.1), we have
dWj1(Tj(t))dt=1Tj(Tj−T∗j)(fj(Tj)−fj(T∗j))−ij(t,0)+i∗j(0)−βj1V∗(T∗j)2Tj+βj1VT∗j−βj2(T∗j)2Tj∫∞0pj(a)i∗j(a)da+βj2T∗j∫∞0pj(a)ij(t,a)da,dWj2,1(ij(t,⋅))dt=Aj(ij(t,0)−i∗j(0)−i∗j(0)logij(t,0)i∗j(0))−∫∞0βj1T∗jqj(a)cj(ij(t,a)−i∗j(a)−i∗j(a)logij(t,a)i∗j(a))da. |
Here we use Lemma 9.18 in Smith and Thieme [12]. We can use this Lemma in this case, because the solution u(t) is contained in the persistence attractor. In fact, ij(t,⋅) lies in a bounded set in L1([0,∞)), and we assume that qj is essentially bounded. Thus the sufficient condition for Fubini's theorem in the proof of Lemma 9.18 holds. Similarly it holds that
dWj2,2(ij(t,⋅))dt=Bj(ij(t,0)−i∗j(0)−i∗j(0)logij(t,0)i∗j(0))−∫∞0βj2T∗jpj(a)(ij(t,a)−i∗j(a)−i∗j(a)logij(t,a)i∗j(a))da, |
βj1T∗jcj(1−V∗V)(∫∞0qj(a)ij(t,a)da−cjV)=βj1T∗jcj∫∞0qj(a)ij(t,a)da−βj1T∗jV+βj1T∗jV∗−βj1T∗jV∗cjV∫∞0qj(a)ij(t,a)da. |
If we add these, ij(t,0)-terms and i∗j(0)-terms vanish by Aj+Bj=1.
We gather terms containing βj1. Noting that in (4.2), i∗j(0) contains both βj1 and βj2, we have by (4.3)
∫∞0βj1T∗jqj(a)i∗j(a)cj(−T∗jTj−logij(t,0)i∗j(0)+1+logij(t,a)i∗j(a)+1−V∗ij(t,a)Vi∗j(a))da. | (4.9) |
On the other hand, which are obtained by (4.3):
V∗=∫∞0qj(a)i∗j(a)cjda,∫∞0βj1T∗jqj(a)i∗j(a)cjda=∫∞0βj1T∗jqj(a)σj(a)cjda⋅i∗j(0)=Aji∗j(0). |
Using these, it holds
∫∞0βj1T∗jqj(a)i∗j(a)cj(TjVi∗j(0)T∗jV∗ij(t,0)−1)da=i∗j(0)ij(t,0)⋅βj1TjV∫∞0qj(a)i∗j(a)cjda⋅1V∗−Aji∗j(0)=i∗j(0)ij(t,0)⋅βj1TjV−Aji∗j(0), | (4.10) |
which will be used later.
Next we gather terms containing βj2:
∫∞0βj2T∗jpj(a)i∗j(a)(−T∗jTj−logij(t,0)i∗j(0)+1+logij(t,a)i∗j(a))da. | (4.11) |
We prepare
∫∞0βj2T∗jpj(a)i∗j(a)(Tji∗j(0)ij(t,a)T∗ji∗j(a)ij(t,0)−1)da=i∗j(0)ij(t,0)⋅βj2Tj∫∞0pj(a)ij(t,a)dt−Bji∗j(0). | (4.12) |
Adding (4.10) and (4.12), we have
∫∞0βj1T∗jqj(a)i∗j(a)c(TjVi∗j(0)T∗jV∗ij(t,0)−1)da+∫∞0βj2T∗jpj(a)i∗j(a)(Tji∗j(0)ij(t,a)T∗ji∗j(a)ij(t,0)−1)da=i∗j(0)ij(t,0)⋅βj1TjV−Aji∗j(0)+i∗j(0)ij(t,0)⋅βj2Tj∫∞0pj(a)ij(t,a)dt−Bji∗j(0)=i∗j(0)ij(t,0)(βj1TjV+βj2Tj∫∞0pj(a)ij(t,a)da)−(Aj+Bj)i∗j(0)=0. | (4.13) |
Then, by subtracting (4.13) from the sum, we have
dWj1(u(t))dt+dWj2,1(u(t))dt+dWj2,2(u(t))dt+βj1T∗jcj(1−V∗V)(∫∞0qj(a)ij(t,a)da−cjV)=1Tj(Tj−T∗j)(fj(Tj)−fj(T∗j))+∫∞0βj1T∗jqj(a)i∗j(a)cj(3−T∗jTj−V∗ij(t,a)Vi∗j(a)−TjVi∗j(0)T∗jV∗ij(t,0)+logij(t,a)i∗j(0)i∗j(a)ij(t,0))da+∫∞0βj2T∗jpj(a)i∗j(a)(2−T∗jTj−Tji∗j(0)ij(t,a)T∗ji∗j(a)ij(t,0)+logi∗j(0)ij(t,a)i∗j(a)ij0,t))da. |
Using these, it holds
dW(u(t))dt=2∑j=1[cjβj1T∗j{1Tj(Tj−T∗j)(fj(Tj)−fj(T∗j))+∫∞0βj1T∗jqj(a)i∗j(a)cj(3−T∗jTj−V∗ij(t,a)Vi∗j(a)−TjVi∗j(0)T∗jV∗ij(t,0)+logij(t,a)i∗j(0)i∗j(a)ij(t,0))da+∫∞0βj2T∗1pj(a)i∗j(a)(2−T∗jTj−Tji∗j(0)ij(t,a)T∗ji∗j(a)ij(t,0)+logi∗j(0)ij(t,a)i∗j(a)ij(t,0))da}]. | (4.14) |
Then the following theorem holds.
Theorem 4.1. Let u(t) be a total solution in the persistence attractor A1. Then the time derivative of W(u(t)) is nonpositive. Moreover the maximum invariant subset of the set {u∈A1|˙W(u)=0} is the singleton set containing the interior equilibrium u∗.
Proof. By the property of fj,
1Tj(Tj−T∗j)(fj(Tj)−fj(T∗j))≤0 |
and the left hand side is 0 if and only if Tj=T∗j. It follows
3−T∗jTj−V∗ij(t,a)Vi∗j(a)−TjVi∗j(0)T∗jV∗ij(t,0)+logij(t,a)i∗j(0)i∗j(a)ij(t,0)≤0,2−T∗jTj−Tji∗j(0)ij(t,a)T∗ji∗j(a)ij(t,0)+logi∗j(0)ij(t,a)i∗j(a)ij(t,0)≤0, |
and the left hand sides are zero if and only if each term with minus sign equals 1 by [18].
We denote by μj the measure given by qj(a)da on [0,∞). Let u∈X be contained in the maximum invariant subset M of the set {u∈A1|˙W(u)=0}. Then ˙W(u(t))=0 for each t∈R, where u(t)=(T1(t),i1(t,a),T2(t),i2(t,a),V(t)) is the total solution such that u(0)=u. Then, using the measure μj, we have
3−T∗jTj−V∗ij(t,a)Vi∗j(a)−TjVi∗j(0)T∗jV∗ij(t,0)+logij(t,a)i∗j(0)i∗j(a)ij(t,0)=0,a.a.a∈[0,∞). |
Then, by Tj=T∗j, it holds
VV∗=ij(t,a)i∗j(a)=ij(t,0)i∗j(0),a.a.a∈[0,∞). |
Then for each t∈R
ij(t,a)=VV∗i∗j(a),a.a.a∈[0,∞). | (4.15) |
Substitute (4.15) to the equation of V,
dVdt=1V∗(∫∞0i∗1(a)q1(a)da+∫∞0i∗2(a)q2(a)da−cV∗)V=0. |
Then
Jj(t)=∫∞0pj(a)i∗j(a)da, |
is a constant, and we put J∗j=Jj(t). By the boundary condition,
ij(t,0)=βj1Tj(t)V(t)+βj2Tj(t)Jj(t)=βj1T∗jV∗+βj2T∗jJ∗j, |
and hence ij(t,0) does not depend on t. Then, by the equation that determines the equilibrium i∗j(0), we have ij(t−a,0)=ij(t,0)=i∗j(0). Then ij(t,a)=ij(t−a,0)σj(a)=i∗j(0)σj(a)=i∗j(a). It follows M={u∗}.
Since R0≤1, R12<1 and R22<1. Then Rm is well defined, and by [11], it holds that Rm≤1.
Lemma 4.2. We can take c1>0, c2>0, c1+c2=c with
β11¯T1N1c1+β12¯T1M1≤1,β21¯T1N2c2+β22¯T2N2≤1. |
Proof. Since Rm≤1, it holds
(β11¯T1N1)/c1−β12¯T1M1+(β21¯T2N2)/c1−β22¯T2M2≤1, |
or
β11¯T1N11−β12¯T1M1+β21¯T2N21−β22¯T2M2≤c. |
Then it is possible to find c1 and c2 with c1>0, c2>0, c1+c2=c, and
β11¯T1N11−β12¯T1M1≤c1,β21¯T2N21−β22¯T2M2≤c2. |
That is
β11¯T1N1c1+β12¯T1M1≤1,β21¯T1N2c2+β22¯T2M2≤1. |
We take and fix c1, c2 as in Lemma 4.2. For j=1, and 2, put
¯ψj1(a)=∫∞aβj1¯Tjqj(b)cjσj(b)σj(a)−1db,¯ψj2(a)=∫∞aβj2¯Tjpj(b)σj(b)σj(a)−1db¯ψj(a)=¯ψj1(a)+¯ψj2(a). |
Define a functional ¯W(u) on the compact attractor A by
¯W(u)=V+2∑j=1cjβj1¯Tj{Tj−¯Tj+¯TjlogTj¯Tj+∫∞0¯ψj(b)ij(b)db}. |
Theorem 4.3. We assume R0≤1. Let u(t) be a total solution contained in the compact attractor A. Then the derivative of ¯W(u(t)) is less or equal to 0. Moreover, the maximum invariant subset contained in the set {u∈A|˙¯W(u)=0} is the singleton set {DFE} that contains only the disease free equilibrium.
Proof. It holds
d¯W(u(t))dt=dVdt+2∑j=1cjβj1¯Tj{(1−¯TjTj)dTjdt+ddt∫∞0¯ψj(b)ij(t,b)db}=2∑j=1∫∞0qj(a)ij(t,a)da−cjV+2∑j=1cjβj1¯Tj{(1−¯TjTj)dTjdt+ddt∫∞0¯ψj(b)ij(t,b)db}=2∑j=1cjβj1¯Tj{(1−¯TjTj)dTjdt+ddt∫∞0¯ψj(b)ij(t,b)db+βj1¯Tjcj(∫∞0qj(a)ij(t,a)da−cjV)}. |
We calculate each term in the summation. Then it holds
(1−¯TjTj)dTjdt=(1−¯TjTj)(fj(Tj)−fj(¯Tj)−βj1TjV−βj2Tj∫∞0pj(a)ij(t,a)da)=1Tj(Tj−¯Tj)(fj(Tj)−fj(¯Tj))−βj1TjV−βj2Tj∫∞0pj(a)ij(t,a)da+βj1¯TjV+βj2¯Tj∫∞0pj(a)ij(t,a)da=1Tj(Tj−¯Tj)(fj(Tj)−fj(¯Tj))−ij(t,0)+βj1¯TjV+βj2¯Tj∫∞0pj(a)ij(t,a)da |
Since ij(t,a)σj(a)−1 is a function of t−a, using Lemma 9.18 in Smith and Thieme [12], we have
ddt∫∞0¯ψj(a)ij(t,a)da=ddt∫∞0¯ψj(a)σj(a)ij(t,a)σj(a)−1da=¯ψj(0)ij(t,0)+∫∞0dda{¯ψj(a)σj(a)}ij(t,a)σj(a)−1da=¯ψj(0)ij(t,0)−∫∞0(βj1¯Tjcjqj(a)+βj2¯Tjpj(a))σj(a)ij(t,a)σj(a)−1da=¯ψj(0)ij(t,0)−∫∞0(βj1¯Tjcjqj(a)+βj2¯Tjpj(a))ij(t,a)da. |
The calculation for the term related with V is as follows:
βj1¯Tjcj(∫∞0qj(a)ij(t,a)da−cjV)=βj1¯Tjcj∫∞0qj(a)ij(t,a)da−βj1¯TjV. |
Then it holds
d¯W(u(t))dt=2∑j=1cjβj1¯Tj{1Tj(Tj−¯Tj)(fj(Tj)−fj(¯Tj))+(¯ψj(0)−1)ij(t,0)}. |
By Lemma 4.2, for j=1, 2, it holds
¯ψj(0)=βj1¯TjNjcj+βj2¯TjMj≤1, |
and hence
d¯W(u(t))dt=2∑j=1cjβj1¯Tj{1Tj(Tj−¯Tj)(fj(Tj)−fj(¯Tj))+(βj1¯TjNjcj+βj2¯TjMj−1)ij(t,0)}≤0. |
Then it holds
d¯W(u(t))dt≤0. |
Let u∈X be in the maximum invariant subset M of {u∈A|˙¯W(u)=0}. Then for each t∈R, u(t)∈M, where u(t)=(T1(t),i1(t,⋅),T2(t),i2(t,⋅),V(t)) is the total solution in A such that u(0)=u. Then Tj(t)=¯Tj. By the equation of Tj, it holds fj(¯Tj)−βj1¯TjV(t)−βj2¯TjJ(t)=0, and therefore βj1V(t)+βj2Jj(t)=0. Then V(t)=0 and Jj(t)=0 for t∈R. By the boundary condition, ij(t,a)=ij(t−a,0)σj(a)=0 for t∈R and a∈[0,∞). Then the maximum invariant subset of the set {u∈X|˙¯W(u)=0} is the singleton set {DFE}.
Theorem 4.4. If R0>1, the unique interior equilibrium exists and is globally asymptotically stable in X∖X0. The disease free equilibrium is globally asymptotically stable in X0. If R0≤1, the disease free equilibrium is globally asymptoticallystable in X.
Proof. We assume R0>1. By Proposition 3.1, DFE is globally asymptotically stable in X0. On the other hand, the alpha-limit set of each total solution in the persistence attractor A1 consists of an interior equilibrium u∗ used in Section 4.1 by Theorem 4.1, because A1 is compact. Then the persistence attractor A1 is the singleton set consists of the interior equilibrium u∗. Then by Proposition 2.13, the interior equilibrium u∗ is globally asymptotically stable in X∖X0, and then the interior equilibrium is unique.
We assume R0≤1. Then the alpha-limit set of each total solution of the compact attractor A is the singleton set which consists of the DFE by Theorem 4.3. Then also by Proposition 2.13, the DFE is globally asymptotically stable in X.
This work was supported by JSPS KAKENHI Grant Number JP17K05365.
The authors declare there is no conflict of interest.
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