Travelling Waves of a Diffusive Epidemic Model with Latency and Relapse

An SEIR epidemic model with relapse and spatial diffusion is studied. By analyzing the corresponding characteristic equations, the local stability of each of the feasible steady states to this model is discussed.The existence of a travelling wave solution is established by using the technique of upper and lower solutions and Schauder’s fixed point theorem. Numerical simulations are carried out to illustrate the main results.


Introduction
In most epidemic models, individuals are often divided into several classes such as susceptible, infective, and recovered classes.Some diseases such as human or bovine tuberculosis have a latency (see, e.g., [1,2] and references therein): infected individuals do not infect other susceptible individuals until sometime later.Hence, it is reasonable to include a latent class in such disease models.Also, for some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class.This recurrence of disease is an important feature of some animal and human diseases.For human tuberculosis, incomplete treatment can lead to relapse, but relapse can also occur in patients who took a full course of treatment and were declared cured [3][4][5].Most tuberculosis in human adults (caused by Mycobacterium tuberculosis) in the USA results from reactivation of latent infection [3].
In order for a model to be more realistic, taking into consideration a general exposed distribution and the possibility of relapse, van den Driessche et al. [6] where  is the size of the population; , , , and  are the population sizes of susceptible, latent, infective, and recovered classes, respectively;  > 0 is a constant rate at which an individual in the recovered class reverts to the infective class;  > 0 is the recruitment rate and the death rate (this guarantees that  can be assumed as a constant);  > 0 is the transmission coefficient;  ⩾ 0 is the recovery rate; () denotes the probability that a latent individual still remains in the latent class  time units after entering the latent class.
As in [6], when () is a step function, that is, model (1) reduces to the following delay differential equation system: () =  −  () −  ()  () ,   () =  −  ( − )  ( − ) +  () − ( + )  () ,   () =  () − ( + )  () , where  > 0 can be explained as the average latent period of the disease.The global stability of the steady states for system (1) and ( 4) was analyzed in [6,7].Note that it is implicitly assumed that the population is well mixed, and the spatial mobility of individuals has been ignored in model (1) as well as most other epidemic models.In reality, the environment in which an individual lives is often heterogeneous making it necessary to distinguish the locations, and due to the large mobility of individuals within an area or even worldwide, spatially uniform models are not sufficient to give a realistic picture of a disease's transmission.For this reason, the spatial effects cannot be neglected in studying the spread of epidemics.If the environment is spatially continuous, random diffusion is often used to describe the mobility of the population, leading to models in the form of reaction-diffusion equations (see, e.g., [8][9][10]).Noble [9] applied reaction-diffusion theory to describe the spread of plague through Europe in the mid-14th century.Maidana and Yang [10] proposed a spatial model to analyze the West Nile Virus propagation across the USA, from east to west.
Motivated by the work mentioned above, the main purpose of this paper is to study the effect of the spatial diffusion of the individuals on the dynamics of an epidemic model with latency and relapse.If we further consider the spatial diffusion, then the epidemic model corresponding to (4) should be described by the following delayed reaction diffusion system: with initial conditions In problem ( 5) and ( 6),  1 , with the inner product ⟨⋅, ⋅⟩, where / denotes the outward normal derivative on Ω.The organization of this paper is as follows.In the next section, by analyzing the corresponding characteristic equations, the local stability of an endemic steady state and a disease-free steady state of system ( 5) is discussed.In Section 3, by constructing a pair of upper-lower solutions, we use the cross-iteration method and the Schauder's fixed point theorem to prove the existence of travelling wave solutions to system (5).In Section 4, numerical simulations are carried out to illustrate the main theoretical results, and by combining with numeric simulations, we discuss the effects of some parameters on the dynamics of system (5).A brief conclusion is given in the last section.

Local Stability
In this section, by analyzing the corresponding characteristic equations, we investigate the local stability of steady states to system (5) with the initial conditions (6) respectively.The homogeneous Neumann boundary conditions imply that the populations do not move across the boundary Ω.Let R 0 is called the basic reproductive number of system (5), which describes the average number of newly infected individuals at the beginning of the infectious process.It is easy to show that system (5) always has a disease-free uniform steady state  0 (1, 0, 0).Further, if R 0 > Let Ê( 0 ,  0 ,  0 ) represent any feasible uniform steady state of system (5).If we set  =  −  0 ,  =  −  0 ,  =  −  0 , and drop the bars for simplicity of notations, then (5) Denote  = ((⋅, ), (⋅, ), (⋅, ))  .Then the linearization of (11) can be rewritten as an abstract differential equation in the phase space C = ([−, 0], ), where  = diag( 1 ,  2 ,  3 ), and  is defined by for  = ( 1 ,  2 ,  3 )  ∈ C, and its characteristic equation is where  ∈ dom(Δ) and  ̸ = 0, dom(Δ) ⊂ .From the properties of the Laplacian operator defined on the bounded domain, the operator Δ on  has the eigenvalues − 2 with the relative eigenfunctions where   = cos(),  ∈  0 = {0, 1, 2, . ..}.Clearly, ( From ( 16) and ( 17), ( 14) is equivalent to where If  0 =  0 , (18) is equivalent to Clearly, for any  ⩾ 1, (20) has a negative root − 2  1 − .Its other roots satisfy If R 0 > 1, then for  = 1, we get that, for  real, Hence, when  = 1, (21) has a positive real root, and accordingly,  0 is unstable.If R 0 < 1 and  = 0, we have which means that the roots of (21) are negative constants, then  0 is locally asymptotically stable.For  > 0, if ( > 0) is a solution of (21), separating the real and imaginary parts, we derive that where Squaring and adding the two equations of (24), it follows that Note that if  0 < 1, we obtain Hence, (26) has no positive roots for all  ⩾ 1.Therefore, if R 0 < 1,  0 is locally asymptotically stable for all  > 0.
If  0 =  * , (18) is equivalent to where When  = 0, (28) becomes It is not difficult to verify that which indicates that all roots  ,1 ,  ,2 , and  ,3 of (30) have negative real parts.
Next we can conclude that there exists a positive constant  such that In fact, let  =   V, then It is easy to see that these three roots V 1 , V 2 , and V 3 of ℎ(V) have negative real parts from the Routh-Hurwitz criterion again, and thus, there exists a positive constant  such that By continuity, we see that there exists  0 such that the three roots V ,1 , V ,2 , and and thus, Let then ξ, and (32) holds for  ∈ ( ξ,   0 /2).Consequently, the spectrum of L, consisting only of eigenvalues, lies in Re  ⩽ −.Hence, the infected steady state is locally asymptotically stable when  = 0.
For  > 0, if ( > 0) is a solution of (28), separating the real and imaginary parts, we derive that Squaring and adding the two equations of (39), it follows that By calculation, it is not difficult to verify that, for all  ⩾ 1, Hence, (40) has no positive roots.Therefore, if R 0 > 1,  * is locally asymptotically stable for all  > 0.
From the discussion above, we can obtain the following results.

Existence of Travelling Waves
In this section, we investigate the existence of travelling wave solutions to system (5).The technique of the proofs is to use the Schauder's fixed point theorem, the method of upperlower solutions, and its associated cross iteration scheme.
From Lemmas 4 and 5, we see that the existence of travelling wave solutions for system (5) follows from the existence of a pair of upper and lower solutions (, , ) and (, , ) of (43) satisfying the following conditions.

Numerical Simulations
In this section, focusing on the travelling wave solutions of system (5), we perform some numerical simulations.Firstly, we illustrate the results stated in Theorems 1 and 8.As in [6], we take parameter values motivated by bovine tuberculosis in a cattle herd, with time unit of one year.It is assumed that the exposed period is 9 months (i.e.,  = 0.75).Taking  = 0.1,  =  = 0.5 (equal periods of infection and recovery before relapse),  = 0.6, and  1 =  2 =  3 = 0.01, system (5) with above coefficients has a unique endemic equilibrium  * (0.3294, 0.3394, 0.2828).For convenience, we truncate the spatial domain R by [−10, 10].It is easy to show that R 0 = 3.0363 > 1.If we use the following trivial functions as initial conditions: ( (, ) ,  (, ) ,  (, )) = (0.9, 0.1, 0) , then the numerical simulations shown in Figures 1(a)-1(c) indicate that the endemic steady state  * is stable.If we change the initial conditions,  * keeps stable (see Figure 1(d); for simplicity, we show only the solutions of (, )), which indicates that  * is in fact globally asymptotically stable, although we fail to give a proof theoretically at present.Now, to illustrate the existence of travelling wave solutions, we use the following piecewise functions as initial conditions: ( (, ) ,  (, ) ,  (, )) The numerical simulations shown in Figure 2 indicate that system (5) has a travelling wave solution with speed  ≈ 0.07 connecting  0 and  * .
Secondly, we address the effects of parameters on the dynamics of system (5).For this part, the following parameter values will be fixed: and the initial conditions are the same as in (68).We first investigate the impact of the diffusion rates.To this end, we fix  = 0.5,  = 0.6,  = 0.75, and let the diffusivity  1 ,  2 , and  3 vary.In this case, the values of R 0 and  * are the same as those in Figure 2 (Note that R 0 and  * are independent of the diffusion rates).Figure 3 is the simulation results of (5).In Figure 3(a),  1 =  2 =  3 = 0.01, but in Figures 3(b)-3(d), we increase the values of  1 ,  2 , and  3 to 0.04, respectively.By contrasting these figures, we can see that the travelling waves in Figures 3(c) and 3(d) are much fast than those in Figure 3(a), but the wave speed in Figure 3(b) is just the same as that in Figure 3(a), which indicates that both the diffusion rates of infective and recovered individuals can advance the time to arrive at the infection steady state, but the diffusion rate of susceptible individuals cannot.Even so, there is an obvious difference between the wave profiles in Figure 3(b) and the other three ones: the former has a hump in wave profile for , but the latter does not seem to show that any more; that is, the diffusion rate of susceptible individuals could lead to nonmonotone travelling waves.In biological meaning, large diffusion of susceptible individuals may enhance the accumulative effect of infection.
Next, we study the impact of , that is, the average latent period of the disease.To this end, we fix  = 0.5,  = 0.6,  1 = 0.02, and  2 =  3 = 0.01, and let  vary.In this case, the values of R 0 and  * are different.(note that both R 0 and  * are decreasing in ). Figure 4 is the simulation results of (5).In Figure 4(a),  = 0.75, but in Figure 4(b), we increase  to 7.5.By contrasting the two figures, we can see that the travelling wave speed in Figure 4(b) is less than that in Figure 4(a), and the humps in Figure 4(a) disappear in Figure 4(b), which indicates that the average latent period of the disease can impact not only the wave speed but also the monotonicity of the wave profile, as well as the level of the positive steady state.In biological meaning, large latent period of the disease could lead to low and slow infection, and may weaken the accumulative effect of infection.Now we investigate the impact of transmission coefficient .To this end, we fix  = 0.5,  = 0.75,  1 =  2 =  3 = 0.01, and let  vary.In this case, the values of R 0 and  * are different (note that both R 0 and  * are increasing in ). Figure 5 is the simulation results of (5).In Figure 5(a),  = 0.3, but in Figure 5(b), we increase  to 1.2.By contrasting the two figures, we can see that the travelling wave in Figure 5(b) is rather fast than that in Figure 5(a), but the travelling waves keep monotone as  varies, which indicates that the trans-mission coefficient can obviously impact the wave speed, but it cannot change the monotonicity of the wave profile.In biological meaning, large transmission coefficient could lead to high and fast infection.
At last, we study the impact of an important parameter , that is, the rate at which an individual in the recovered class reverts to the infective class.In the following, for convenience, we call  the relapse rate.Now we fix  = 0.6,  = 0.75, and  1 =  2 =  3 = 0.01, and let  vary.In this case, the values of R 0 and  * are also different (note that both R 0 and  * are increasing in ). Figure 6 is the simulation results of (5).In Figure 6(a),  = 0.1, but in Figure 6(b), we increase  to 2. By contrasting the two figures, we can see that the travelling wave in Figure 6(b) is rather fast than that in Figure 6(a), but the travelling waves keep monotone as  varies, which indicates that the relapse rate can obviously impact the wave speed, but it cannot change the monotonicity of the wave profile.In biological meaning, large transmission coefficient could lead to high and fast infection, which is similar to the case of transmission coefficient.
To end this section, we point out that for the parameters given above, the following properties of the spread speed  can be obtained from the simulations and discussions above:  is increasing in  2 ,  3 , , and  but decreasing in .In addition, small  and large  1 may lead to nonmonotone travelling waves.

Conclusions
In this paper, we formulated an SEIR model with latent period, relapse and spatial diffusion.The dynamics of problem ( 5) and ( 6) was addressed.It was shown that the basic reproductive number of system ( 5) is given by R 0 =  − ( + )/ (( +  + )), which describes the average number of newly infected individuals at the beginning of the infectious process.We have shown when the average number of newly infected individuals is less than unity, that is, R 0 < 1; system (5) has a unique disease-free steady state  0 (1, 0, 0), which is asymptotically stable; when the average number of newly infected individuals is greater than unity, that is, R 0 > 1, system (5) admits two steady states,  0 (1, 0, 0) and  * ( * ,  * ,  * ), where  * ,  * , and  * are defined in (10).In this case, the endemic steady state  * is always stable, while the disease-free steady state  0 is unstable.Clearly, the spatial diffusion cannot destabilize the spatially homogenous steady state.Then by using the technique of upper and lower    solutions and Schauder's fixed point theorem, we derived the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state.Furthermore, we gave some numeric simulations to illustrate the main results, and combining with numeric simulations, we discussed the effects of some parameters on the dynamics of system (5).
From the discussion in Section 4, we could conclude that the travelling wave speed is increasing in the diffusion rates of infective and recovered individuals, the transmission coefficient, and the relapse rate, but decreasing in the average latent period of the disease.In addition, small latent period or large diffusion rate of susceptible individuals may lead to nonmonotone travelling waves.In other words, to slow down or weaken the infection, we can reduce the diffusion rates, the transmission coefficient, or the relapse rate.