The periodic traveling waves in a diffusive periodic SIR epidemic model with nonlinear incidence.

In this paper, a reaction-diffusion SIR epidemic model is proposed. It takes into account the individuals mobility, the time periodicity of the infection rate and recovery rate, and the general nonlinear incidence function, which contains a number of classical incidence functions. In our model, due to the introduction of the general nonlinear incidence function, the boundedness of the infected individuals can not be obtained, so we study the existence and nonexistence of periodic traveling wave solutions of original system with the aid of auxiliary system. The basic reproduction number R 0 and the critical wave speed c * are given. We obtained the existence and uniqueness of periodic traveling waves for each wave speed c > c * using the Schauder's fixed points theorem when R 0 > 1 . The nonexistence of periodic traveling waves for two cases (i) R 0 > 1 and 0 < c < c * , (ii) R 0 ≤ 1 and c ≥ 0 are also obtained. These results generalize and improve the existing conclusions. Finally, the numerical experiments support the theoretical results. The differences of traveling wave solution between periodic system and general aperiodic coefficient system are analyzed by numerical simulations.

Introduction

Consider the several outbreaks of infectious disease in human history, such as the Black Death in the late Middle Ages, from the 16th to the 18th century, smallpox, measles, typhoid fever, plague, in the middle of the 20th century, Ebola, SARS and bird flu, these infectious diseases have brought huge impacts on human society. Most recently, in 2020, the COVID-19 outbreak has spread globally in just a few months and has infected more than 50 million people, killed more than one million two hundred as of mid-November 2020, and had an incalculable impact on human lives and the global economy. In the history of mankind, the researches on this kind of terrible infectious disease have never stopped, and the prediction, prevention and control of the disease spread are the research focus. It is a straightforward and effective way to study the spread and control of disease applying appropriate mathematical models.

Besides, to make the model better reflects or describes the real environment, traveling wave solutions, as a special form solution, have been studied more and more widely, the traveling wave solutions cover a wide range of fields, such as in chemical reactions, biological population, genetic engineering, fluid mechanics, mathematical ecology, etc. (See [1], [2], [3], [4], [5], [6]). Particularly, in the epidemic models, traveling wave solutions can be understood as a state of disease propagation in space at a specific speed from a certain area to the surroundings. The existence of traveling waves and the minimum wave speed have important significance for the prediction and control of diseases (See [7], [8], [9], [10], [11], [12], [13], [14], [15]). At present, the traveling waves for autonomous reaction-diffusion and nonlocal diffusion epidemic models have been studied extensively. Some important results can be found in the literatures [7], [8], [9], [10], [13], [16], [17], [18], [19], [20], [21], [22], [23] and their references.

On the other hand, we know that the disease transmission coefficient and recovery rate are not always constant, due to the influence of factors such as weather, temperature, season and so on (See [24], [25], [26], [27], [28], [29], [30] and references therein). Consequently, the nonautonomous (especially, time periodic) systems have received widespread attention from scholars [31], [32], [33], [35]. In [31], Zhang et al. proposed the following reaction-diffusion SIR model with time-dependent

It is worth noting that, different from the previous model with constant transmission rate and recovery rate in [34], this model presents the traveling waves of periodic parabolic systems, which render the previous methods in [34] unusable. In order to effectively break through these problems, the authors turn the problem into a truncated problem, and then investigate the periodic traveling waves by applying some limiting arguments. They established the existence of periodic traveling waves when wave speed is large than critical wave speed and the basic reproduction number . They also obtained the nonexistence for either or and .

Recently, Wang et al. in [35] investigated the following reaction-diffusion time periodic SIR modelThe authors have used fixed-point theories to investigate the periodic traveling waves. Theoretical results show that the system has periodic traveling waves for each wave speed large than the critical value when the basic reproduction number and there is no periodic traveling wave when or and .

It is well known that the incidence function in many infectious disease models are bilinear or standard incidence. From the perspective of the development mechanism of infectious diseases, the bilinear incidence is generally used for small-scale susceptible group and exposure. As the population size increases, the bilinear incidence will become infinite and lose its practical significance. At this time, the standard incidence is adopted, which is applicable to a large number of people. However, more and more nonlinear incidence have been mentioned many times (See [36], [37], [38]). Korobeinikov and Maini in [39] investigated the the nonlinear incidence in a variety of epidemic models. So here, we wonder if the system has a variety of incidence functions, can we come to the same conclusions? Incorporating these above factors, it is then very interesting and natural to investigate the periodic traveling wave solution of periodic epidemic model with general nonlinear incidence.

Motivated by the articles in [31], [35], we investigate the following reaction-diffusion time periodic SIR epidemic model with general nonlinear incidence:where and denote the sizes of susceptible, infective and removed individuals in location and at time respectively; stand for the diffusion coefficients for the three individuals, respectively; and are considered as strictly positive and T-periodic functions; the term is the nonlinear incidence.

It is worth mentioning that our model makes the boundedness of infected individuals difficult to obtain due to the introduction of general response function, which will lead to the Schauder’s fixed point theorem cannot be used like in [31] and [35]. To overcome these difficulties, we have to establish an auxiliary system to further construct a bounded closed convex set, and use the fixed point theorem to obtain the traveling wave solution of the auxiliary system, and use the Arzel-Ascoli theorem to extract diagonal elements to obtain the existence of the traveling wave solution of the system (3).

This paper is organized as follows. Next Section, some important lemmas are introduced and the critical wave speed and the basic reproduction number are given. Furthermore, the auxiliary system is introduced. In Section 3, a pair of sub- and super-solutions are constructed for the auxiliary system. In Section 4, the convex and closed set is constructed. The fixed point problems derivation from model (3) are discussed. In Section 5, the existence of periodic travelling wave solutions will be firstly proved for the auxiliary system. Then, the corresponding conclusions are further proved for model (3). In Section 6, we study the nonexistence of periodic traveling wave solutions. Section 7, the numerical examples are given to verify our conclusions and a simple comparison is made with the general aperiodic model. Lastly, in Section 8, a brief conclusion is given.

Preliminary

We first introduce some notations, important lemmas and definitions to be used in the next sections.

Denote and be the sets of natural numbers and positive natural numbers, respectively. Denote be a Banach space of bounded and uniformly continuous functions with the supremum norm. Let From Theorem 1.5 in [40] we know that the -realization of generates a strongly continuous analytic semigroup on and for any . Additionally, from (Section 2, [31]) or (Chapter I, [40]) we obtain

For any given constants and let be a space of uniformly bounded and uniformly continuous functions . Definewith the norm

Next, we need to introduce the following important lemmas on the parabolic initial value problems and Harnack inequalities (See [41], [42]), which will be used in the proofs of Lemma 10 and Theorem 3, respectively.

(See ) Let initial value . Set Then, for each and . Furthermore, the following inequalities hold for some constants and and where

(See ) (Harnack inequalities) Let the operators be uniformly parabolic in an open domain of that is, there exists such that the inequality always holds for every where and is a bounded open set. Assume that and satisfy for some . Suppose that satisfies where and if and there exists a constant such that Assume that and there exist some positive constants and such that dist and . Let and then there exist positive constants and dependent on diam and satisfying Where particularly, . Furthermore, if all inequalities in (8) become equalities, then the lemma is also valid for and independent of .

In this paper, for the convenience, let with be a continuously differentiable function, we denote .

For nonlinear incidence in model (3) in this paper we always assume that the following assumptions hold.

(A1) is twice continuously differentiable for and for all .

(A2) and for all and . For any fixed is positive and bounded for all .

There are some concrete incidence functions satisfying assumptions (A1) and (A2), such as (See [31]), (See [35]), with constant (See [43]) and with constants (See [44]).

Because the first two equations of (3) are relatively independent, it suffices to consider the reduced system

The T-periodic traveling wave of system (9) is defined as a special solution with satisfying

Obviously, such solutions with must satisfy

In this paper, we investigate the traveling wave solutions of system (9) satisfying the following asymptotic boundary conditionsHere is thought to be the sizes of susceptible individuals in the initial state. The constant stands for the sizes of susceptible individuals after the epidemic. Define the basic reproduction number of periodic system (9) as belowIn fact, reflects the average number of newly infected individuals generated by a single infected individual at the beginning of the disease infection process.

Linearizing the second equation of system (11) at yieldsDefinewhere Clearly, if and only if . LetIf then we have and for all

In this paper, we will consider the periodic travelling wave solutions for model (3) by using the auxiliary system method. We introduce the auxiliary systemwhere is a constant. Our idea is that firstly we will discuss the existence of travelling wave solution which is dependent of parameter for auxiliary system (14) with the asymptotic boundary conditions (12) when and by using the sub- and super-solutions method and the fixed point theorems. Then taking we will further prove that there exists the limit function of which is the travelling wave solution of model (3) with the asymptotic boundary conditions (12) when and .

Sub- and super-solutions

In what follows, we always assume that and . Define function We easily verify for all . Letwhere . Parameters and are positive constants and will be given later. Now, we will prove that and are the upper-lower solutions of system (14).

Function satisfies the inequality

The proof of Lemma 3 is simple since .

Function satisfies the inequality

By a simple calculation, we getAssumption (A2) implies that for . ThusThis is end of the proof. □

Suppose and large enough, then function satisfies the inequality for any

If then and (17) holds. If then and . The inequality (17) is equivalent tothat is,From assumptions (A1) and (A2) one can see that andTo obtain inequality (18), it is sufficient to verifyNoting the periodicity and positivity of and keeping and taking sufficiently large, then inequality (19) holds. This proof is completed. □

Assume that small enough and large enough such that . Then function satisfies the inequality for any

It follows from that . If then which implies that (20) holds. If then and . Rewriting inequality (20), we haveBy direct calculations, we can obtainThen inequality (21) is equivalent toAssumption (A2) implies that is a decreasing function on . Thus, for any given we can find a number such that as and Since and as for given above we can choose a such that and when . Let then we getDue to inequality (23) is valid for any which implies that That is,Noting (24), to obtain (22), it suffices to verifyIn view of to obtain (25) it suffices to verifyThat is,Since and we have . Hence, for it is sufficient to verifyIn fact, it is easy to see that and . The above inequality is true for large enough. The proof of this lemma is shown. □

Fixed-point problems

Now, we construct a set as followswhere is sufficiently large constant satisfying for all . In fact, we can choose . For any letwhere and are positive constants satisfying with and . For a given we investigate the following parabolic initial value problem:Rewrite (28) as an integral system:where and is the analytic semigroup (See [41]) generated by operator defined by andwhere the space can be found in [40]. Furthermore, from (4) we haveThe solution of integral system (29) is the mild solution of problem (28). For a given defineequipped with the normLetObviously, is convex and closed set. Next, we show that the system (29) has a unique solution for any .

Now, we show that the invariance of integral system (29). To obtain this, we introduce the following Lemma.

The functions and satisfy the following inequalities: and

Since and we haveSince the positivity of semigroup we know thatwhich implies that inequality (31) holds.

From the Eq. (15) in Lemma 4 and we know thatSince the positivity of semigroup one hasHence, inequality (32) holds.

Next, we consider and . Wherein we let and . It follows from Lemma 5 thatfor any By a direct computation, yieldsLetandCalculating the partial derivative of one hasNoting for and using the integration by parts formula, one also hasRecalling we obtainSinceis integrable on noting the continuity of in from (Chapter I, [40]) we knowTherefore,Due to it follows from above equality thatBy the definition of and and applying an integral transformation, we obtainIn the above inequality, let we obtainwhich implies that inequality (33) holds. By using the same scheme as above, we can also prove that inequality (34) holds. The proof is completed. □

With the aids of the discussions of above lemma, we obtain the following invariance of integral system (29).

Let be the solution of integral system (29) with initial value . Then for .

Since and then andRecalling we have . Since the analytic semigroup generated by operator we haveCombining the positivity of semigroup and (35), one hasConsequently thus for any .

Now we prove . Let from inequality (33), we obtainSince and we have therefore, Next, we consider . Recalling we know thatfor . Recalling (32), we haveAccording to for all yieldsSince satisfies we haveCombining and a similar arguments as above, one hasAdditionally, from the positiveness of and (34), we obtaintherefore, for Since for any we get This completes the proof. □

Define time-T map of system (29) : as follows.

For any given map has a unique fixed point .

Lemma 8 implies . Let then and are uniformly bounded with respect to . Using Lemma 1 we know that defined by system (29) belongs to for arbitrary and moreover, there exist dependent on satisfyingandCombining the estimates (36) and (37), we know that is compact on . Next, we need to prove the compactness and continuity of . For any given sequences let . Due to and satisfy the (36) and (37), therefore, we can choose a subsequence of for presentation convenience, still labeled by satisfying as in in other words, for any

Next, we will prove thatThe boundedness of meansalso uniformly bounded for . For any given we can choose constant satisfyingfor and . Additionally, it follows from (38) that there exists such thatfor and . Thus, it follows from the inequalities (39) and (40) that as with respect to the norm . Therefore, the Shauder’s fixed point theorem ensures that admits a fixed point satisfying and for any . Finally, we show that the uniqueness of . Suppose that there exists satisfying for any . We haveHowever, due to thus, one haswhich contradicts with . Consequently, . can be obtained similarly. □

It is clear that the fixed point of map is the solution of system (29) defined on .

From Theorem 1 we define an operator as follows

To obtain the fixed point of operator we need to prove the following conclusions.

The operator is continuous in .

For any given let . Using (30) and the first equation of system (29), we deduce thatLet . Taking small enough such that yieldsLet Noting we obtain thatSince satisfiesTherefore, the continuity of and in can be obtained similarly. □

The operator is compact in .

For any given let where is the solution of system (29). From the estimates (36) and (37), and the uniform boundedness of and for any we know that and for some constant . Thanks to Lemma 1, we know that and there exist some and satisfying andFor any given sequences let . Due to and satisfying the (41) and (42) respectively, we can choose a subsequence of without loss of generality, still labeled by satisfyingBy a similar method as in Theorem 1, we can obtain as therefore, the operator is compact in . □

Applying Lemma 9, Lemma 10 and the Schauder’s fixed point theorem, we further obtain the following theorem.

The operator has a fixed point .

It is clear that the fixed point of operator is the solution of system (14) defined on .

Existence of periodic traveling waves

Now, on the basis of the above discussion we show that the existence of periodic solution for auxiliary system (14).

Suppose and . Then auxiliary system (14) admits a nontrival time periodic solution satisfying asymptotic boundary conditions (12) . Moreover, and

By the above discussion, for every operator admits a unique fixed point satisfying moreoverfor . Denote where integers can be chosen to satisfy . We can verify that and are T-periodic function for . From we know that . It is clear that satisfies following systemfor . Without loss of generality, still labeled by . Thanks to ([41], Theorem 5.1.2, 5.1.3, 5.1.4), it follows that satisfiesfor . Moreover,It is clear that and uniformly for . Now, we intend to show that the second equality of (43). To obtain this, we need to investigate the asymptotic behavior of and as . Using the estimate (47) and Landau type inequalities (See [45]), for any given we haveandConsequently,Taking the partial derivatives with respect to on both sides of the first equation of (46), yieldsApplying ([41], Theorem 5.1.3, 5.1.4), we know that and Repeating the above procedure and Landau type inequality, one can see thatConsider the system (46), combining the (48) and (50), we haveSetting . It follows from (48) that as . From the first equation of (46), we haveBy integrating (51) from to we obtainIt follows from uniform boundedness of and that and are bounded. Therefore, from (52) we knowCombining (51), it follows thatIntegrating above equality from to we findHence, for all which indicates that exists and let . Clearly, . On the other hand, applying the Barblat’s lemma (See [46]), it follows that as . Taking in (52), we find that

Next, we consider the limitation of as . Let and . Combining the second equation of (46), one hasThe two roots of the characteristic equation are denoted byLet Then, it follows from (54) thatIntegrating from to on both sides of above equation, we haveFrom (53) we know that is integrable on . The Fubinis theorem (See [47]) ensures that is integrable on andBy a similar argument as the boundedness of we can obtain that is bounded on . Therefore, the Barbrlat’s lemma guarantees as . We consider the second equation of system (46), taking and with . Then, the Lemma 2 means that there is a constant independent of such thatNow, we show that . Suppose that this conclusion is false, then there exists a constant and sequence satisfying as such that . The continuity of with respect to implies that there exists constant such that for all . We deduce thatwhich is a contradiction with . Thus combining the periodicity of we can obtain uniformly for as .

Next, letThanks to the T-periodicity of we need to show From (55), we know that there exist sequences and with and as satisfying Letting and . The estimation (47) implies that we can choose a subsequence of without loss of generality, still denoted by satisfying as . Consequently, is a T-periodic function, and for all

Due to there exists a such that as . Let then satisfiesFurther, we know . Due to and then the maximum principle guarantees that is for all . Next, let then . Furthermore, one can seewhich indicates . Thus . Similarly, we can show that . Therefore, uniformly for as .

Finally, by using Landau type inequalities to and we can deduceFrom the (54), we haveMaking an integration to Eq. (57) on combining the (53) and (56), we obtainDefineIt is clear that function satisfiesCombining (56), (58) and L’Hpital’s rule, we haveand Note that . Let Combining (57) and (60), we findMultiplying by to two sides of the (62) and integrating from to yieldswhich implies that is nondecreasing in . On the other hand, it follows from (62) and (63) that . Therefore, that is for all . □

Now, to return system (11), we obtain the following conclusion.

Suppose and . Then system (11) admits a time periodic solution satisfying asymptotic boundary conditions (12) . Moreover, and

Let is a decreasing sequence and as . Theorem 3 guarantees that system (14) admits a solution sequence with satisfyingandMoreover,Furthermore, we know thatIt follows from that as .

For there exists such that for all and . Therefore, solution sequences are equi-continuous and uniformly bounded on . The Arzel-Ascoli theorem means that there exists a uniformly convergent subsequences of as .

It’s not hard to see that there is a subsequence which uniformly converges on every interval . Let . For every we knowLet we know that . Thus, satisfies system (11). Combining Theorem 3 and (65), we know that and (63) holds. Finally, we need to prove . It follows from and thatandTherefore, we haveFrom (67), we have for . By the maximum principle (see, [48]), we know that on and . This completes the proof. □

It is clear that when incidence functions and then assumptions and are satisfied. Therefore, the main results on the existence of periodic travelling wave solutions for models (1) and (2) obtained in [31], [35] are the direct corollaries of Theorem 4.

Nonexistence of periodic traveling waves

In what follows, we shall establish the non-existence of periodic traveling wave solutions for or and .

Assume that and then there does not exist nonnegative and periodic traveling wave solution satisfying asymptotic boundary conditions (12) .

Suppose that there exists a periodic traveling wave solution for system (11) satisfying (12). Then, we havefor . Let . Integrating (68) with respect to on and combining asymptotic boundary conditions (12) and (15), yieldswhere Due to and are T-periodic, we haveIntegrating above equality from 0 to combining the periodicity of yieldsNote that which implies . Combining the yieldswhich is a contradiction. This is the end of the proof. □

Assume then system (11) does not have a periodic traveling wave solution satisfying asymptotic boundary conditions (12) for any .

Assume that the solution is a traveling wave solution satisfying conditions (12) for some . Since means which implies that there exists a sufficiently small such that . LetWe can choose a satisfying .

Fixed a constant and let . DefineObviously, Define a function . We can deduce for any . Then, by a direct calculation, yieldsWe choose such that . Further, we can choose . Then, we have and for all . Since is strictly positive on interval thus we can find a constant satisfying for all . Define Since is a solution of system (11), we haveDue to and uniformly for combining we can deduce that there exists a satisfying for all uniformly for . Since and we have for all . Therefore, from (70) we knowfor Let for all . It follows from (69) and (71) thatApplying the maximum principle, we can derive that for all and that is for all and which contradict with when since . This is the end of the proof. □

We easily see that the main results on the non-existence of periodic travelling wave solutions for models (1) and (2) established in [31], [35] can be directly obtained form Theorems 5 and 6.

Numerical examples

In this section, we present some numerical simulations to illustrate the theoretic results. Assume that the incidence function . We choose the following initial conditionsAll values of the parameters are given: and By simple calculation, we obtain and Theorem 4 means that there exists traveling wave solutions satisfying the asymptotical boundary conditions (12). That is, and The numerical simulations are given in Fig. 1 .

Fig. 1

Numerical simulation show that there exists traveling wave solution connecting the initial disease-free steady state and the steady state after the epidemic. Moreover, from the numerical stimulation we can obtain .

It is clear that there exists a constant which stands for the steady state of susceptible individuals after the epidemic, satisfying . However, we still don’t obtain the exact value of through theoretical analysis.

Now, we observe the effect of the critical wave speed by using the equipotential diagram. We choose all conditions and parameters are same as previous, that is and It follows from Theorem 3 and Theorem 6 that there exists traveling waves connecting and for every speed and there does not exist such traveling waves for any . The numerical simulations are given in Fig. 2 .

Fig. 2

Numerical simulation show that there exists traveling wave solution connecting the initial disease-free steady state and the steady state when and there does not exist such traveling wave solutions if .

In order to find out the difference of traveling wave solution when the infection coefficient and recovery rate are related to time or independent of time . We choose and assume that the rest of the parameters are same as above. We choose the following initial conditionsThe numerical simulations are given in Figs. 3 and 4 .

Fig. 3

Numerical simulation show that the traveling wave solution about susceptible individuals is more oscillatory or has a periodicity when the infection coefficient and recovery rate are related to time than usual.

Fig. 4

Numerical simulation show that the traveling wave solution about infected individuals is more oscillatory or has a periodicity when the infection coefficient and recovery rate are related to time than usual.

Conclusion

In this paper, we proposed a nonautonomous and diffusive SIR epidemic model with general nonlinear incidence given by system (3) assuming that the transmission rates and recovery rates are positive and T-periodic functions. We further investigated the existence and nonexistence of periodic traveling waves for system (3). It is important to mention that the classical methods such as shooting method [34], two-sided Laplace transform [18] and monotone iteration methods [49] can not be used directly, since the introduction of nonautonomous and nonlinear incidence. Consequently, we further refined and developed the previous methods in [31], [35]. More precisely, we constructed an auxiliary system (14) to obtain the boundedness of the infected individual, which guarantees the suitable bounded closed convex set between the upper and lower solutions. Finally, the existence of traveling wave solutions are obtained by the fixed-point theorem and some limiting arguments. Many of the methods and arguments in this investigation can be widely applied to the existence and nonexistence of periodic traveling wave solutions for other periodic reaction-diffusion systems.

The conclusions of this article show that there exists a critical wave speed such that the system (3) admits a T-periodic traveling wave solutions satisfying the asymptotic boundary conditions (12) when the reproductive number and there is no such solution for either or and . In fact, it is not difficult to find that the literatures [35] and [31] are the direct corollaries of our conclusions. Furthermore, we provided some numerical simulations to verify our theoretic results. It is not difficult to find that the traveling wave solution tends to a constant less than as when and and the exact value of can not be obtained by theoretical analysis, which will be our further research direction.

Finally, there are some very interesting problems that deserve further study. For example, on the one hand, we can consider the traveling wave solutions of the nonlocal diffusion time-periodic model. On the other hand, we can investigate the periodic model with susceptible input and natural death, because there are disease-free periodic solution and endemic periodic solution when the basic reproduction number is more than unit, so we can study the existence of traveling wave solutions connecting these two periodic solutions.

CRediT authorship contribution statement

Weixin Wu: Writing - review & editing, Visualization. Zhidong Teng: Funding acquisition, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.