Dynamics of a Host-pathogen System on a Bounded Spatial Domain

We study a host-pathogen system in a bounded spatial habitat where the environment is closed. Extinction and persistence of the disease are investigated by appealing to theories of monotone dynamical systems and uniform persistence. We also carry out a bifurcation analysis for steady state solutions, and the results suggest that a backward bifurcation may occur when the parameters in the system are spatially dependent. 1. Introduction. Mathematical disease models play an important role in studying the mechanism of infectious disease. Recent evidences have shown that diseases can affect the dynamics of animal populations and communities. In classical epidemiological models, the host population is divided into infected and susceptible classes, with one differential equation representing each class. Anderson and May [3] introduced an additional class representing the population of infectious pathogen particles. These particles are found in invertebrate pathogens and they allow pathogens to survive in the environment for several decades. The following host-pathogen system was proposed by Anderson and May [3]:

1. Introduction.Mathematical disease models play an important role in studying the mechanism of infectious disease.Recent evidences have shown that diseases can affect the dynamics of animal populations and communities.In classical epidemiological models, the host population is divided into infected and susceptible classes, with one differential equation representing each class.Anderson and May [3] introduced an additional class representing the population of infectious pathogen particles.These particles are found in invertebrate pathogens and they allow pathogens to survive in the environment for several decades.The following host-pathogen system was proposed by Anderson and May [3]: where u 1 (t) is the density of susceptible hosts, u 2 (t) is the density of infected hosts, u 3 (t) is the density of pathogen particles, r is the reproductive rate of the host, β is the transmission coefficient, α is the rate of disease-induced mortality, λ is the rate of production of pathogen particles by infected hosts, and δ is the decay rate of the pathogens.Dwyer [9] revised the system (1.1) and obtained a mathematical disease model that includes density-dependent host population dynamics: where the new parameter K is the carrying capacity for the hosts.In order to simplify the system (1.2),Dwyer [9] further ignored the consumption of the pathogen by the hosts and investigated the following system: ( In many situations ordinary differential equations are appropriate mathematical models for the progress of infectious diseases.However, it has been recognized that spatial structure is also a central factor that affects the spatial spreading of a disease.Taking into consideration the host movement, the author in [9] modified (1.3) to a reaction-diffusion model in a spatial environment: (1.4) Here the host movement is described by the diffusion terms d∆u 1 and d∆u 2 where ∆ is the usual Laplacian operator; d > 0 denotes the diffusion coefficient, which represents the rate of host movement; x and t represent location and time, respectively.All coefficients in (1.4) are positive constants.Dwyer [9] assumed that the habitat in (1.4) is one-dimensional and unbounded, and accordingly, investigated the existence of travelling wave and spreading speed.
In the real world, a habitat in which a host population settles is typically bounded, and this motivates us to consider a mathematical system modelling the dynamics of a disease in a bounded spatial domain (not necessarily one-dimensional).Also, the parameters in a model involving space are typically space dependent due to the spatial heterogeneity.Based on these basic facts, we will, in this paper, further modify the more general (than (1.4)) model system (1.2) by replacing the constant parameters with spatial dependent parameters, and consider the solution dynamics on a general bounded domain with zero-flux boundary condition.In other words, we consider the following problem: x ∈ Ω, t > 0, x ∈ Ω, i = 1, 2, 3. (1.5) Here Ω ⊂ R m is a bounded open set with smooth boundary ∂Ω; ∂ ∂ν denotes the differentiation along the unit outward normal ν to ∂Ω; K(x) is the carrying capacity.The spatially dependent functions β(x), λ(x), K(x) are assumed to be continuous and positive in Ω.
We point out that although there have been numerous ODE models for disease and/or pathogen dynamics in literature, the studies of PDE models with spatial variables are much fewer, among which are Allen et.al. [2], Capasso and Maddalena [4], Capasso and Wilson [5], Guo et.al. [12], Li and Zou [18], Peng [24], Peng and Yi [25], Vaidya et.al. [34], Wang and Zhao [36,37].The main reason is that a PDE model with spatial variables (important in disease spread), such as (1.5), is infinite dimensional, and thus, is much harder to analyze.Taking basic reproduction number as an example, for an ODE model, following the "recipe" given in [33], the basic reproduction number can be easily identified as the spectral radius of the next generation that enjoys some nice properties, and can be conveniently calculated in most cases.However, for a PDE model, one needs to work on operators between function spaces in order to obtain the next generation operator.Moreover the spectral radius of a general operator is very hard, or even impossible in most cases, to calculate, and thus, one has to heavily depend on numerical simulations.See, e.g., [2,4,12,24,25,34,36,37] for a taste of what is stated above.Furthermore, unlike in [2,4,12,24,25,34,36,37] where all equations in the models have diffusion terms and hence the solution semiflows are alway compact, here in our model (1.5), the compactness is an issue because of the lack of diffusion term in the u 3 equation, and this makes analysis more challenging.In [5], a model on temporal and spatial evolution of orofecal transmitted disease with immobile human population and Dirichlet boundary condition was considered, but our approach is quite different.
In the rest of this paper, we will investigate the dynamics of this modified model.In Section 2, we explore the solution properties of the system (1.5) by appealing to the theories of monotone dynamical systems and uniform persistence.In Section 3, we utilize bifurcation theory to investigate the steady state solutions of the system (1.5).A brief discussion section concludes the paper.
2. Basic properties of solutions.This section is devoted to establishing some basic properties of (1.5), starting with the well-posedness.
For convenience of discussions later, we recall some well-known results for some auxiliary systems.First consider the following equation 3) It is easy to see that system (2.3) has a unique positive steady state λ(x) A(x) if A(x) > 0. By [38, Theorem 2.2.1], we have the following result: Lemma 2.2.Suppose that A(x) > 0 and λ(x) > 0 for x ∈ Ω, then the system (2.3) admits a unique positive steady state λ(x) A(x) which is globally asymptotically stable in C(Ω, R).
Secondly we consider the dynamics of the following diffusive logistic equation: (2.4) The dynamics of the system (2.4) is well-known (see for example, [38, Theorem 3.1.5and the proof of Theorem 3.1.6]): Lemma 2.3.For any d, r > 0 and W 0 (x) ≡ 0, the diffusive logistic equation (2.4) admits a unique positive steady state u * 1 (x) which is globally asymptotically stable in C(Ω, R) .Now we are in a position to show that solutions of the system (1.5) exist globally for t ∈ [0, ∞) in X + .Lemma 2.4.For every initial value function φ ∈ X + , the system (1.5) has a unique solution u(x, t; φ) defined on [0, ∞) with u(•, 0; φ) = φ and a semiflow Ψ(t) : X + → X + is generated by (1.5) which is defined by (2.5) (2.6) The system (2.6) is bounded from above by the system (2.More precisely, there exist t 1 > 0 and a > 0 such that This implies that U (x, t) is ultimately bounded.It follows from Lemma 2.1 that u i (x, t) is also ultimately bounded, i = 1, 2. Then there exists a positive number A such that the third equation of system (1.5) for u 3 satisfies By standard comparison theorem and Lemma 2.2, it follows that u 3 is also ultimately bounded.Thus, the solution exists globally, i.e., for all t ∈ [0, ∞), and moreover, the solution semiflow generated by (1.5) is point dissipative.
Since the third equation in (1.5) has no diffusion term, the solution map Ψ(t) is not compact.In order to overcome this problem, we introduce the Kuratowski measure of noncompactness κ (see [7]), which is defined by κ(B) := inf{r : B has a finite cover of diameter < r}, (2.9) for any bounded set B. We set κ(B) = ∞ whenever B is unbounded.It is easy to see that B is precompact (i.e.B is compact) if and only if κ(B) = 0. Then the solution map Ψ(t) has some partial compactness in the following sense.Proof.Let G(u 1 , u 2 , u 3 ) = −δu 3 + λ(x)u 2 − β(x)(u 1 + u 2 )u 3 represent the reaction term for the third equation of (1.5).Then With this inequality, the rest of the proof is similar to the one in [15,Lemma 4.1] (see also [15,Lemma 3.2]).Now we are ready to show that solutions of system (1.5) converge to a compact attractor in X + .Theorem 2.1.Ψ(t) admits a connected global attractor on X + .Proof.By Lemma 2.4 and Lemma 2.5, it follows that Ψ(t) is point dissipative and κ-contracting on X + .From the proof of Lemma 2.4 (see (2.6) and (2.8)), we also know that the positive orbits of bounded subsets of X + for Ψ(t) are (uniformly) bounded.By [20, Theorem 2.6], Ψ(t) has a global attractor that attracts every bounded set in X + .2.2.Extinction.In this subsection, an extinction result for large α is established.We first consider the following linear system: (2.10) Denote by Σ(t) the solution semiflow Σ(t) of (2.10), that is, Σ(t) : , and its generator B can be written as Further, B is a closed and resolvent positive operator (see, e.g., [32,Theorem 3.12]).Substituting u i (x, t) = e µt ψ i (x), i = 2, 3, into (2.10),we get the following associated eigenvalue problem: (2.11) We point out that Σ(t) is not compact since the second equation in (2.10) has no diffusion term and its sign.
The following lemma concerns with the existence of the principal eigenvalue of (2.11).
Lemma 2.6.Let s(B) be the spectral bound of B. Then (i) s(B) is the principal eigenvalue of the eigenvalue problem (2.11) which has a strongly positive eigenfunction; (ii) s(B) has the same sign as ξ 0 , where ξ 0 is the principal eigenvalue of the eigenvalue problem (2.12) Proof.In order to make use of the results in [37, Theorem 2.3 (i)], we define an one-parameter family of linear operators on C( Ω, R): admits a principal eigenvalue, denoted by η0 = −α, with an associated eigenvector ϕ 0 0. Let µ * be the larger root of the algebraic equation By [37, Theorem 2.3 (i)], we complete the proof of (i).
Since −δ < 0, it follows from [37, Theorem 2.3 (ii)] that s(B) has the same sign as s(L 0 ), where Now we are ready to show that s(B) is an index for disease extinction.
By Theorem 2.2 and (2.15), it follows that the disease will become extinct if αδ is large or max x∈ Ω [β(x)λ(x)] is small.

2.3.
Persistence.Next we show the persistence for small α in the system (1.5).We establish some lemmas for that purpose.First we show the strict positiveness of solutions of (1.5).
(ii) From Lemma 2.1, it follows that u 2 (x, t) satisfies where it follows from the strong maximum principle and the Hopf boundary lemma that (2.16) For fixed x ∈ Ω, u 3 (•, t) satisfies an ordinary differential equation, then u 3 (x, t) > 0 for x ∈ Ω, t > t 0 from u 2 > 0 and the equation of u 3 .This completes the proof of part (ii).
It is easy to see that system (1.5) has a trivial equilibrium at M 1 = (0, 0, 0) and a disease-free equilibrium at M 2 = (u * 1 (•), 0, 0), where u * 1 (x) is the unique positive steady state of (2.4) which is globally asymptotically stable in C(Ω, R) for the dynamics of (2.4).Linearizing system (1.5) at the disease-free equilibrium (u * 1 (x), 0, 0), we get the following linear system: (2.17) Note that the equations for the infected host (w 2 ) and pathogen populations (w 3 ) decouple from the that for uninfected host population (w 1 ) in (2.17), forming the following subsystem which is cooperative: ( The perturbation of the eigenvalue problem (2.18) will play a central role in proving the persistence of the system (1.5).To proceed further, we first consider the following more general linear parabolic system: where h 1 (x) > 0 and h 2 (x) > 0 for x ∈ Ω. Denote by Π t the solution semiflow of (2.17) on C. The it is easy to see that Π t is a positive C 0 -semigroup on C, and its generator B h1,h2 can be written as .
Furthermore B h1,h2 is a closed and resolvent positive operator (see, e.g., [32, Theorem 3.12]).Again, Π t is not compact since the second equation in (2.19) has no diffusion term.
Substituting w i (x, t) = e Λt ψ i (x), i = 2, 3, into (2.19),we obtain the following eigenvalue problem: (2.20) The following lemma concerns with the existence of the principal eigenvalue of (2.20).
Proof.We first show that for each t > 0, Π t is an κ-contraction on C in the sense that for any bounded set B in C, where κ is the Kuratowski measure of noncompactness as defined in (2.9).Recall that T 2 (t) is the analytic semigroup on ) is a linear semigroup on C. Define a linear operator and a nonlinear operator where and hence From the boundedness of Π t and the compactness of U 2 (t) for t > 0, it follows that Q(t) : C → C is compact for each t > 0. For any bounded set B in C, there holds κ(Q(t)B) = 0 since Q(t)B is precompact, and consequently, In the above inequality, we have used the fact that κ(I(t)B) ≤ I(t) κ(B), t > 0, since I(t) is a linear operator.Thus, Π t is an κ-contraction on C with a contracting function e −δt .From (2.21), it follows that the essential spectral radius r e (Π t ) of Π t satisfies r e (Π t ) ≤ e −δt < 1, t > 0.
On the other hand, the spectral radius r(Π t ) of Π t satisfies This implies that r e (Π t ) < r(Π t ) for any t > 0. Since Π t is a strongly positive and bounded operator on C, it follows from a generalized Krein-Rutman Theorem (see, e.g., [21]) that the stated conclusion holds.Now we are ready to prove the main result of this section, which indicates that s(B u * 1 ,u * 1 ) is a crucial index for disease persistence.
Then, the infection is uniformly persistent in the sense that there exists an η > 0 such that for any φ ∈ X + with φ i ≡ / 0 for i = 1, 2, we have Moreover, System (1.5) admits at least one (componentwise) positive steady state û(x).
Since Λ > 0, it follows that u(x, t, φ 0 ) is unbounded.This is a contradiction.
From the above claims, it follows that any forward orbit of Ψ(t) in M ∂ converges to is the stable set of M i , i = 1, 2 (see [30]).It is obvious that there is no cycle in By [30,Theorem 3], it follows that there exists an η > 0 such that min Further, Lemma 2.7 implies that ũi (•) > 0, ∀ i = 1, 2. It remains to show that ũ3 (•) > 0. Indeed, from the third equation of (1.5), it follows that This implies that ũ(•) is a positive steady state of (1.5).The proof is completed.
Remark 2.2.We regret to point out that when s(B u * 1 ,u * 1 ) < 0, we are unable to determine the dynamics of the system (1.5) at the present.

2.4.
The basic reproduction number.In this subsection, we adopt the approach of next generation operators (see, e.g., [8,32], also see more recent work [12,34,36,37]) to define the basic reproduction number for the system (1.5).The cooperative system (2.18) is the linearized system of (1.5) at the disease-free equilibrium (u * 1 , 0, 0).Thus, the matrices F and V defined in [37,Eq. (3.4)] become ) be the C 0 -semigroup generated by the following system (2.27) In order to define the basic reproduction number for the system (1.5), we assume that the state variables are near the disease-free steady state (u * 1 (x), 0, 0).Then with an given initial distribution of infections described by (ϕ 2 (•), ϕ 3 (•)) ∈ C( Ω, R 2 ), solving (2.27) with this given initial distribution will give a distribution of total infections caused by (ϕ Let L : C( Ω, R 2 ) → C( Ω, R 2 ) be defined by the above integral, i.e., Then L is nothing but the next generation operator of the model system (see, e.g., [8,12,32,34,36,37]), the spectral radian of L we gives the basic reproduction number of the model, that is, R 0 := r(L).
(2.28) By [37, Theorem 3.1 (i) and Remark 3.1], we then have the following result.Lemma 2.9.R 0 − 1 and s(B u * 1 ,u * 1 ) have the same sign, where s(B u * 1 ,u * 1 ) is the generator associated with the linear system (2.18).By Lemma 2.9, we may restate Theorem 2.3 as follows: Theorem 2.4.Assume that R 0 > 1.Then, the infection is uniformly persistent in the sense that there exists an η > 0 such that such that for any φ ∈ X + with φ i ≡ / 0 for i = 1, 2, we have Moreover, System (1.5) admits at least one (componentwise) positive steady state û(x) .
By the same arguments as in [37, Lemma 4.2 and Theorem 3.2 (ii)], we have the following observation.Lemma 2.10.Let η 0 be the principal eigenvalue of the following eigenvalue problem: Then R 0 = 1 η0 .Remark 2.3.When all parameters in (1.5) are constants, one can easily see that u * 1 (x) ≡ K, and one can actually calculate the spectral radius to obtain At the end of this section, we briefly mention a modified version of system (1.5).We may assume that susceptible and infected classes have different movement rates, and pathogen also adopts movement.Then system (1.5) can be modified as follows: ( We note that our arguments used in the analysis of (1.5) can be applied to system (2.31), except those in Lemma 2.4.Due to the fact that d 1 = d 2 , the arguments used in Lemma 2.4 do NOT work.Next, we sketch an approach in proving the boundedness of u i (x, t), i = 1, 2, 3. From the first equation of (2.31), it follows that Then it is easy to see that By Gronwall's inequality we get the L 1 estimates, With the L 1 estimates, one can show that u 2 (x, t) is uniformly bounded (see e. g. [1,17]).Since u 2 (x, t) is uniformly bounded, it follows from the third equation of (2.31) that u 3 (x, t) is uniformly bounded.Thus, the results in Lemma 2.4 can be obtained when system (1.5) is replaced by (2.31).We also note that d i > 0, i = 1, 2, 3, and hence, it follows that the solution maps generated by system (2.31) are compact, and hence, Lemma 2.5 is automatically valid.In other words, when we assume that susceptible and infected classes have different movement rates and pathogen also adopts movement, the mathematical analysis is similar to those in (1.5).
3. Bifurcation Analysis.In Theorem 2.4 we have proved that system (1.5) is uniformly persistent when R 0 > 1, thus (1.5) admits at least one positive steady state solution.In this section, we consider the steady state equation directly to obtain more information on the set of positive steady state solutions.The steady state solutions of (1.5) satisfy From the third equation of (3.1), it follows that u 3 satisfies It is easy to see that (u * 1 (x), 0) is a semi-trivial steady state solution of (3.3), where u * 1 (x) is the unique positive steady state solution of the diffusive logistic equation (2.4).
Next we apply [28,Theorem 4.4] to F (α, u, w) = 0 with V = R + × (X + ) 2 where X + = {u ∈ X : u > − } for some > 0. From the remarks after [28,Theorem 4.4] and discussions above, all conditions in [28,Theorem 4.4] are satisfied.Therefore there exists a connected component Σ 1 of Σ containing the curve C 1 , and Σ 1 satisfies either (i) it is not compact; or (ii) it contains a point (α * , u * 1 , 0) with α * = α 0 ; or (iii) it contains a point (α, u * 1 + U, W ), where (U, W ) = 0 and (U, W ) is in a compliment of span{(φ 0 , ψ 0 )}.Note that the equations of u and w are all in a form of d∆V + g(x)V = 0 for V = u or w thus a weak form of maximum principle holds, and all solutions on the connected component Σ 1 are necessarily positive.Since ψ 0 > 0, then W must be sign-changing and such (α, u * 1 + U, W ) cannot be on Σ 1 hence (iii) is not possible.Similarly (ii) is also impossible as the eigenvalue of (3.4) with positive eigenfunction is unique.Therefore Σ 1 is not compact.
Remark 2.3).In such a case, the impact of the model parameters on R 0 can only be explored numerically.To demonstrate this, we fix parameter values Ω = (0, 1), α = 0.02, d = 0.137, r = 1, δ = 0.0137, K(x) ≡ 3, β(x) ≡ 1.5, but let λ(x) = 0.9[1 + c cos(2πx + x 0 )], where c ∈ [0, 1].The spatial average of λ(x) on [0, 1] is always the constant 0.9 regardless of the values of c and x 0 , but the dependence of R 0 on c varies for different values of x 0 , as shown in Fig. 4.2.This indicates that the spatial variation can also affect the persistence/extinction of the disease.