Advanced Nonlinear Dynamics of Population Biology and EpidemiologyView this Special Issue
Traveling Wave Solutions for a Delayed SIRS Infectious Disease Model with Nonlocal Diffusion and Nonlinear Incidence
A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder's fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.
Mathematical modeling has been proven to be valuable in studying the transmission dynamics of infectious diseases in a host population. We note that in disease progression, the spatial content of the environment plays a crucial role; the spread of germs, bacteria, and pathogen in the area is the main reason which leads to the spread of infectious disease. Thus, due to the large mobility of people within a country or even worldwide, spatially uniform models are not sufficient to give a realistic picture of a disease’s diffusion. Considering the spatial effects, Gan et al.  considered the following SIRS epidemic model with spatial diffusion and time delay: where represents the number of individuals who are susceptible to the disease, represents the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and represents the number of individuals who have been removed from the possibility of infection through full immunity. The parameters , , , , , are positive constants in which is the recruitment rate of the population, is the death rate due to disease, is the natural death rate of the population, is the transmission rate, is the recovery rate of the infective individuals, and is the rate at which recovered individuals lose immunity and return to the susceptible class. is a fixed time during which the infectious agents develop in the vector and it is only after that time that the infected vector can infect a susceptible human. , , and denote the corresponding diffusion rates for the susceptible, infected, and removed populations, respectively. In , by constructing a pair of upper-lower solutions, the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state was given. In recent years, there has been a fair amount of work on epidemiological models with spatial diffusion (see, e.g., [2–6]).
In system (1), the Laplacian operator has been used to model the diffusion of the species, which suggests that the population at the location can only be influenced by the variation of the population near the location . However, in dynamics of infectious diseases, dispersal is better described as a long range process rather than as a local one. At the same time, studies of disease infections have also shown that reaction-diffusion equation does not accurately describe the spatial and temporal behavior of some diseases, for example, in the incubation period of SARS patients, who can move freely and the movement may transmit the disease to other people. Since the long range effect is taken into account, nonlocal diffusion equations have received great interest and have been recently intensively studied to analyze the long range effects of the dispersal (see, e.g., [7–12]). A basic nonlocal diffusion equation is of the form  where the kernel of the convolution is a nonnegative function of mass one and a given nonlinearity. As stated in , if represents the density of a species at the point and time and is regarded as the probability distribution of jumping from location to location , then is the rate at which individuals arrive at position from all other places and is the rate at which they leave location to travel to all other sites. The diffusion is modeled by a convolution operator which looks to be biologically reasonable.
We note that in system (1), Gan et al. used a bilinear incidence rate based on the law of mass action. If the number of susceptible individuals is very large, it is unreasonable to consider the bilinear incidence within a certain limited time, because the number of effective contacts between infective individuals and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the protection measures by the susceptible individuals. After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio  introduced a saturated incidence rate into epidemic models, where tends to a saturation level when gets large; that is, ; here measures the force of infection of the disease, and measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the susceptible individuals. This incidence rate seems more reasonable than the bilinear incidence rate, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters .
Motivated by the works of Capasso and Serio , Gan et al. , and Li et al. , in this paper, we study the following delayed SIRS infectious disease model with nonlocal diffusion: where the parameter denotes the corresponding diffusion rates for the three populations, respectively. Here, for simplicity, we assume . is a kernel function which is continuous satisfying(A1), and , for . For any fixed , and
The initial conditions for system (3) take the form
In the biological context, it is important to analyse the epidemic waves which are described by traveling wave solutions propagating with a certain speed. In this paper, our focus is on the existence of traveling wave solutions to the SIRS infectious disease model (3).
The rest of this paper is organized as follows. In Section 2, by constructing a pair of upper-lower solutions and using Schauder’s fixed point theorem, the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state of system (3) is established. In Section 3, a brief concluding remark is given to end this work.
2. Existence of Traveling Waves
In this section, we apply Schauder’s fixed point theorem, the method of cross-iteration scheme associated with upper-lower solutions, to study the existence of traveling wave solutions of system (3) connecting the disease-free steady state and the endemic steady state.
Denote is called the basic reproduction ratio of system (3), which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. This quantity determines the thresholds for disease transmissions. It is easy to show that system (3) always has a disease-free steady state . Further, if , system (3) has a unique endemic steady state , where
Denoting , then system (3) is equivalent to the following system:
A traveling wave solution of (9) is a special translation invariant solution of the form , where is the profile of the wave that propagates through one-dimensional spatial domain at a constant speed . On substituting , , into (9) and denoting the traveling wave coordinate still by , we derive from (9) that where Equation (10) will be solved subject to the following boundary value conditions:
Now, we give the definition of upper and lower solutions of system (10) as follows.
Definition 1. A pair of continuous functions and are called a pair of upper-lower solutions of system (10), if there exist constants such that and are twice differential on and satisfy
In what follows, we assume that there exist an upper solution and a lower solution of system (10) satisfying (P1)-(P2):(P1);(P2), .
Let where satisfy We look for traveling wave solutions to system (10) in the following profile set: Obviously, is nonempty, convex, closed, and bounded.
Furthermore, corresponding to (10), we make the following hypotheses.(A2)There exist three positive constants such that for and with , , , , are positive constants.
For , we define two operators and from to by Letting , and , then system (10) can be rewritten as and then is well defined such that Hence, a fixed point of is a solution of (10), which is a traveling wave solution of (9) connecting with if it satisfies (P2).
In the following, we introduce some lemmas to support our main results.
For , define Then it is easy to check that is a Banach space.
In view of the definition of and , we can easily see that they admit the following properties.
Lemma 2. Let hold. One has
(ii) for with , , .
By using a similar argument as in the proof of Lemmas 3.3–3.6 in , one can show the following lemmas.
Lemma 3. Assume that (A2) holds. is continuous with respect to the norm in .
Lemma 4. , where .
Lemma 5. is compact.
We now consider the following equations: Since and (15) hold, direct calculations show that Therefore, we obtain that there exist and such that and . Further, if , there exist and satisfying Similarly, we can show that there exist such that . If , there exist satisfying and .
Lemma 6. Let . Assume that ; then one has and .
It is easy to show that
If , then by (9), we see that and . Note that for all ; hence, we have and .
Suppose that and ; we can choose , and , , satisfying
In fact, noting that , for , there exist and such that which yield
Since , for , we can find and such that
If , then we can choose suitable values of such that Furthermore, we can choose , , satisfying Accordingly, there exist suitable constants such that By the second equation of system (10), we have . It then follows from (35) that
Now, we define the continuous functions and as follows: where , and is a constant sufficiently small to be chosen later. Then we can choose to be sufficiently small such that , , e satisfying where are defined in (15). Furthermore, we can choose such that . If , it is easy to show that . Clearly, and satisfy (P1) and (P2).
Lemma 7. is an upper solution of system (10).
If , and . By Lemma 6, it follows that
If , and . Then, we have Note that and . Hence, for sufficiently small, there exists such that for all .
If , and . We obtain that For sufficiently small, implies that , and there exists such that for all .
If , and . It follows that
If , , , , and . We obtain that where Then by (29), we have
Since , for sufficiently small, it is easy to show that and there exists such that for all .
If , , , , and . It follows that where For sufficiently small, by (29), we see that and there exists such that for all .
If , and . Then, by Lemma 6, we have
If , and . We derive that Note that ; then we have . By (29), for sufficiently small, it is easy to show that and there exists such that for all .
If , and . We obtain that For sufficiently small, by (29), we see that implies that and there exists such that for all .
Clearly, for all , . This completes the proof.
Lemma 8. is a lower solution of system (10).
If , . It is easy to see that .
If , and . Then, we have For sufficiently small, implies that and there exists such that for all .
If , and . Hence, .
If , . Noting that , and . We obtain that where By (29), we have . Accordingly, for sufficiently small, there exists such that for all .
If , . Then, we have .
If , and . We obtain that For sufficiently small, then, by (29), it is readily seen that and there exists such that for all .
Obviously, for all . This completes the proof.
Applying Lemmas 2–8 and Schauder’s fixed point theorem, we know that if and , system (9) has a traveling wave solution with speed connecting the steady states and . Accordingly, we have the following conclusion.
Theorem 9. Let . Assume that (A1) and hold. For every , system (3) always has a traveling wave solution with speed connecting the disease-free steady state and the endemic steady state .
3. Concluding Remark
In this paper, we have discussed a delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence. By constructing a pair of upper-lower solutions and using Schauder’s fixed point theorem, we investigated the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state . We now study the influence of the nonlocal diffusion terms and time delay describing the incubation period on the spreading speed . From the second equation of system (3), we have a linearized equation at that takes the form Letting yields the following characteristic equation: By direct calculations we have It is easy to show that the spreading speed is monotonically increasing for the nonlocal diffusion rate and is monotonically decreasing for the time delay .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (nos. 11371368, 11071254), the Natural Science Foundation of Hebei Province of China under Grant (no. A2014506015), the Natural Science Foundation for Young Scientists of Hebei Province (no. A2013506012), and the Science Research Foundation of Mechanical Engineering College (nos. YJJXM12010, YJJXM13008).