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International Journal of Differential Equations
Volume 2017, Article ID 5796958, 14 pages
https://doi.org/10.1155/2017/5796958
Research Article

Global Dynamics of a Periodic SEIRS Model with General Incidence Rate

Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje 13615, 97119 Mérida, YUC, Mexico

Correspondence should be addressed to Eric Ávila-Vales; xm.ydau.oerroc@aliva

Received 25 July 2017; Revised 3 October 2017; Accepted 10 October 2017; Published 9 November 2017

Academic Editor: Khalid Hattaf

Copyright © 2017 Eric Ávila-Vales et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form , and it is shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease. Numerical simulations are performed using a nonlinear incidence rate to estimate the basic reproduction number and illustrate our analytical findings.

1. Introduction

Epidemiological models in mathematics have been recognized as valuable tools in analyzing the dynamics of an infectious disease nowadays. They are used to describe the spread of disease and also to make control measures known to avoid its persistence, for example, via vaccination terms or treatment terms. These models consider the total population divided into compartments, given by the biological assumptions on the model and represented by functions depending on time . The most common categories used are susceptible (), infected (), recovered (), exposed (), quarantined (), and vaccinated (), and the dynamics of model is given by transmission rates from a compartment to another. We have then indicated that the models could be of type , , , , , , and so forth.

To ensure that the model can give a justified qualitative description of the disease, the choice of the incidence rate plays an important role. An incidence rate is defined as the number of new health related events or cases of a disease in a population exposed to the risk in a given time period. Some examples are the bilinear incidence rate, the saturated incidence rate, or a general incidence rate. The bilinear incidence rate has been repeatedly used by several authors. It is given by , where is the transmission rate and the product represents the contact between infected and susceptible individuals (based on the law of mass action). It was introduced by Kermack and McKendrick [1] in 1927, and even when it is mathematically simple to use, it faces multiple problems and challenges when it is used to describe disease propagation among gregarious animals or persons [2], because it goes to infinity when becomes larger. In order to improve the modelling process to study the dynamics of infection among a large population, Capasso and Serio [3] in 1978 introduced a saturated incidence rate by studying the Cholera epidemic spread in Bari, given by , where is the transmission rate and the saturation constant. Unlike the bilinear incidence, saturated incidence does not grow up without a limit, but it goes to a saturation limit as goes to infinity. Multiple types of saturated incidence have been used in the literature; see, for example, [2] for a list of them. To avoid the use of a single incidence function, the use of a general incidence rate that includes a family of particular functions with similar properties has become a topic of interest by several authors (see, e.g., [48]).

The basic reproduction (represented by ) is defined as “the average number of secondary cases produced by a single infected case when it is introduced in a susceptible population” and it has an important role in the study of disease transmission. In biological terms, usually when this number is less than one, the disease is eradicated from population, but when it is greater than one, the infection persists. Mathematically, it is of interest to compute a threshold parameter with the properties of the basic reproduction number. A method to compute this number for certain compartmental disease models is via the next-generation matrix method developed in [9]; however, it is not useful when the model presents time periodic seasonal terms. Authors like [10, 11] have defined its basic reproduction number for periodic models as an average, to give some results about extinction or persistence of infection. However Bacaër and Guernaouni in [12] introduced the definition of basic reproduction number for periodic environments, and, later, Wang and Zhao [13] made a formal definition of it, via the monodromy matrix.

In the present work, we focus on a family of SEIRS epidemic models with a time periodic seasonal term, improving the model of Moneim and Greenhalgh in [14], by introducing an incidence rate with a general function taken from [4] and the references therein.

We propose the following SEIRS model:where is the total population size, with denoting the fractions of population that are susceptible, exposed, infected, and recovered, respectively. is the transmission rate and it is a continuous, positive -periodic function. () is the vaccination rate of all newborn children. is the vaccination rate of all susceptibles in the population and it is a continuous, positive periodic function with period , where is an integer. is the common per capita birth and death rate. , and are the per capita rates of leaving the latent stage, infected stage, and recovered stage, respectively. It is assumed that all parameters are positive constants.

Bai and Zhou in [5] answered some open problems stated in [14], they also showed that their condition is a threshold between persistence and extinction of the disease via the framework established in [13]. They assumed that the incidence was bilinear. In our study, the nonlinear assumptions on function are listed below (see [4]).(A1) is continuously differentiable.(A2), and for all .(A3).

Under these assumptions, function includes various types of incidence rate; in particular, when , we are on the bilinear case considered in [14].

In addition, we assume the following extra conditions (see [15]).(A4).(A5)There exists such that when ,

This set of assumptions on the function allows for more general incidence functions than the bilinear one, like saturated incidence functions and functions of the form ; in particular, in the case when , they represent psychological or media effects depending on the infected population. In this last case the incidence function is nonmonotone on . (A3) regulates the value of comparing it with the value at of a line containing the origin of slope (note that this line varies as increases), (A4) requires a concave at the origin, and (A5) imposes the geometrical condition that in a small neighborhood of the origin must lie between the tangent line of at and a concave parabola tangent to at .

We consider a family of epidemic models with periodic coefficients and general incidence rate in epidemiology. Then we show that the global dynamics of solutions is determined by the basic reproduction number , generalizing the results in [5]. The layout of this paper is as follows: In Section 2, we prove the existence of a disease-free periodic solution and we introduce the basic reproduction number via the theory developed in [12, 13]. In Section 3, we adapt the arguments given in [5] to prove that the disease-free periodic solution of system (1) is globally asymptotically stable if and it is persistent when . Finally, in Section 4, we give some numerical simulations of our results, making a comparison between our basic reproduction number and the average reproduction number used by several authors (see, e.g., [10, 11]).

2. The Basic Reproduction Number

First of all, we prove nonnegativity of the solutions under nonnegative initial conditions.

Theorem 1. Let . The solution of (1) with is nonnegative in the sense that , , and satisfies , with constant.

Proof. Let ; then, adding all equations of system (1), we can see that , so the value of is constant. Now, set as the solution of system (1) under initial conditions . By the continuity of solutions, for all of and that have a positive initial value at , we have the existence of an interval such that for . We will prove that .
If for a and other components of remain nonnegative at , then implying that whenever the solution touches the -axis, the derivative of is nondecreasing and the function does not cross to negative values. Similarly, when for a and other components remain nonnegative, we have When for a and other components remain nonnegative, Finally, when for a and other components remain nonnegative, Therefore, whenever touches any of the axes , , , or , it never crosses them.

In order to make the analysis of the model in a simpler way from now on, we make a reduction of dimension in system (1) making , obtaining the following:

The dynamics of system (1) is equivalent to that of (7); moreover, due to positivity of solutions, we have , so we study the dynamics of system (7) in the region

A disease-free periodic solution can be found for (7). To find it, set ; then, from the first equation of (7) we can obtain the following initial value problem:

From [5, 14], the equation above admits a unique positive -periodic solution given bywhere

Therefore, is a disease-free periodic solution of (7); moreover, from [5] we have that ; therefore, lives in .

Using the notation of [9], we sort the compartments so that the first two compartments correspond to infected individuals. Let and define(i): the rate of new infection in compartment ,(ii): the rate of individuals into compartment by other means,(iii): the rate of individuals transfer out of compartment .

System can be written aswhere ,

Linearizing system (12) around the disease-free solution, we obtain the matrix of partial derivatives , where

Using Lemma of [9], we part and and set

For a compartmental epidemiological model based on an autonomous system, the basic reproduction number is determined by the spectral radius of the next-generation matrix (which is independent of time) [9]. The definition of basic reproduction number for nonautonomous systems has been studied for multiple authors; see, for example, [12, 13]. Particularly, Wang and Zhao in [13] extended the work of [9] to include epidemiological models in periodic environments. They introduced the next infection operator given bywhere is the ordered Banach space of all periodic functions from to , which is equipped with the maximum norm. is the initial distribution of infectious individuals in this periodic environment, and is the evolution operator of the linear periodic system:meaning that, for each , the matrix satisfies

is the distribution of accumulative new infections at time produced by all those infected individuals introduced before , with kernel . The coefficient in row and column represents the expected number of individuals in compartment that one individual in compartment generates at the beginning of an epidemic per unit time at time if it has been in compartment for units of time, with [16].

Let is an eigenvalue of if there is a nonnegative eigenfunction such that

Therefore, the basic reproduction number is defined asthe spectral radius of . The basic reproduction number can be evaluated by several numerical methods and approximations [1517]; in Section 4 we discuss this topic.

3. The Threshold Dynamics of

3.1. Disease Extinction

Theorem 2. Let be defined as (20); then the disease-free periodic solution is asymptotically stable if and unstable if .

Proof. We use Theorem of [13] and check conditions (A1)–(A7). Conditions (A1)–(A5) are clearly satisfied from the definitions of and given in Section 2. We prove only conditions (A6) and (A7). Define and let be the monodromy matrix of system(A6). Let be a fundamental matrix for system , with defined as before and periodic; the monodromy matrix is given by . The general solution of (22) is so and . Note that , so andDue to the fact that is a constant, its eigenvalue is itself and for .(A7). Solving the system , we arrive at the general solution so  Computing , we haveClearly, for .

Note 1. Due to the fact that is a fundamental solution of a periodic system, we can always choose it such that , so the monodromy matrix satisfies . This property is used in further analysis.

In order to prove the global stability of the disease-free periodic solution, we enunciate some useful definitions and some lemmas.

Let be continuous, cooperative, irreducible, and -periodic matrix function, and the fundamental matrix of system . Denote by the spectral radius of .

Lemma 3. Let . Then there exists a positive, -periodic function such that is a solution of (see proof in Lemma  2.1 of [18]).

Lemma 4. Function of model (1) satisfies , .

Proof. Using assumptions on function , we haveso function decreases and then .

Lemma 5. Let be a solution of system (7) with initial conditions , and the disease-free periodic solution of (7); then

Proof. Proof is similar to Lemma of [14]. satisfies the first equation of system (7); then Let ; then Using Gronwall’s inequality , Taking limits in both sides, we obtain that .

Now, we are able to enunciate our theorem for global stability of disease-free periodic solution.

Theorem 6. The disease-free periodic solution of system (7) is globally asymptotically stable if .

Proof. From Theorem 2 we have that is unstable for and asymptotically stable for , so it is sufficient to prove that any solution with nonnegative initial conditions approaches as tends to infinity.
Let ; from Lemma 5 we have so there exists a such that for all which implies that . Then, from the definition of supremum, we have that for all Then, we have proved that for all we can find a such that for all .
Now, using Lemma 4, for we can find a such that for We consider the following perturbed subsystem:which can be rewritten as with defined in (15) andMatrix is -periodic, cooperative, irreducible, and continuous. Using Lemma 3, if , then there exists a positive and -periodic function such that is solution of system (38). Note that for all , function is also a solution of system (38) with initial condition at .
Choose a and such that ; then from (37) we have that and using a comparison principle (see, e.g., [19] Theorem ), we have for all .
From Theorem of [13], iff . By the continuity of the spectrum for matrices (see [20], Section  II.5.8), we can choose small enough so that and then (see Note 1). Thus, using positivity of solutions and comparison, And similarly for , we obtain thatWe need only to prove that approaches . At disease-free periodic solution , where satisfies Thus, satisfiesLet be arbitrary and . Due to (43) we can find a such that for ; moreover, we can find a such that for . Then, let ; we have for Multiplying in both sides by and integrating from to , we obtainSo, , where is arbitrarily small. Then , and using similar arguments for and , we can find a with for . Also, from (43), we can find with for , so, for , we haveOr, equivalently, , with being arbitrarily small, and this implies that . We conclude by comparison and using Lemma 5 that , completing the proof.

Theorem 6 shows that disease will completely disappear as long as . Thus, reducing and keeping below the unity would be sufficient to eradicate infection, even in a periodic environment and a general incidence rate.

3.2. Disease Persistence

Uniform persistence is an important concept in population dynamics, since it characterizes the long-term survival of some or all interacting species in an ecosystem [21].

In this section we consider the dynamics of the periodic model when . We will show that actually is a threshold parameter for the extinction and the uniform persistence of the disease. Our results are inspired by [5, 15, 18, 22].

Let be the Poincaré map associated with system (7); that is, where is defined in (8) and is the unique solution of system (7) with . We define the following sets:

Note that is not the boundary of , but it is a standard notation of persistence theory.

Lemma 7. Set is positively invariant under system (7).

Proof. Let , that is, , and letbe the solution of (7) with . Due to nonnegativity of solutions and assumptions on function and , we have Using a comparison theorem (see, e.g., [19] Appendix B.1), we have for all Similarly, so, Therefore, remains on for all .

To use persistence theory developed in [21], we show thatwhere

Let and be the solution that passes through that initial condition. We have that , with being a solution of (9) and being a solution that satisfies the initial condition. By uniqueness of solutions we have , so lives on .

Now, if , we want . We prove an equivalent sentence: if , then it does not belong to . Consider an initial point ; then , or . Suppose that and ; then holds

By continuity of and sign of derivative of , we have that, for small , , so, for , . Using invariance of (Lemma 7) we have for all . Finally, for a such that , we have and this implies (56). By the existence of a disease-free periodic solution (proved in Section 2), it is clear that there is one fixed point of in given by ([23]).

Now, we are in a position to introduce the following result of uniform persistence of the disease.

Theorem 8. Let ; then there exists an such that any solution of (7) with initial values satisfies

Proof. We first prove that is uniformly persistent (see Definition from [21]) with respect to , because this implies that the solution of (7) is uniformly persistent with respect to (see [21], Theorem ). Clearly, is relatively open in , so is relatively closed.
Define we show that
By Theorem of [13], if and only if . Choose an small enough with the property (see Appendix A). For , let us consider the following perturbed equation:System above admits a unique positive -periodic solution of the formwhit , which is globally attractive for all solutions of (61) (see Appendix B for proof), and withSince is continuous in , for all there is a such that for we have . Moreover, by continuity of solutions with respect to initial values we can find for all an such that if , then Therefore, for established before, we can find small enough such that ,  .
Again, by continuity of solutions with respect to initial values, for this small , there exists a such that for all with we have ,  .
We now claim thatBy contradiction, suppose thatWithout loss of generality, we can assume that for all (see Appendix C). From the discussion above, , and .
For any , let , where and is the greatest integer less than or equal to . Then, we get If we set , then we have , , , and from the first equation of (7) and Lemma 4 we arrive atwhich is exactly the equation in (61). Since the unique periodic solution of (61) is globally attractive, we have for solution of (61) that . So for given before, there exists such that for all or equivalently . Moreover, from previous analysis, ; therefore, using comparison principle on (68) we arrive atfor .
We have , and is fixed small, so we can take and use assumption (A5) in Introduction (see Appendix D) to obtainwhere are defined in (15), is defined in (40), and By Theorem of [13], we have iff . By continuity of spectrum (see [20] Section II), we can find such that Consider the auxiliary system then, using Lemma 3 there exists a solution of (71) with the form , with . Choose a and a small number such that . Using comparison principle we get , which implies and . This leads to a contradiction.
The claim above shows that is weakly uniformly persistent with respect to Note that has a global attractor (see Lemma 5). It follows that is an isolated invariant set in , . Every orbit in converges to and is acyclic. By the acyclicity theorem on uniform persistence for maps ([21] Theorem and Remark ), it follows that is uniformly persistent with respect to ; that is, there exists such that any solution of (7) satisfies ,

4. Numerical Simulations

In this section we provide some numerical simulations to illustrate the results obtained in our theorems and compare them with previous results.

To improve previous models used in references, we use a particular functionwhich includes the case used in [5]. One can check that function (75) satisfies conditions (A1)–(A5). Using this function, system (7) is rewritten as

Set an initial population and take time in years. Suppose per year, corresponding to an average human life time of 50 years. Following [5] take the parameters as follows: per year, per year, , and . Choose the periodic transmission as , with being the transmission parameter, and the periodic vaccination rate . Both functions have period .

There exists multiple methods for computing the basic reproduction number, via numerical approximations, or finding a positive solution of the equation (see Theorem of [13]). In order to compare our work with previous works, we approximate the basic reproduction number with its average value , used by several authors as the reproduction number (for example [10, 11]), so definewhere is given by (15) and with being the average of functions defined as . Computing each average, we obtain so for

Following Theorem of [13], to compute , let , be the evolution operator of the systemthat is, for each , , , and . With this operator, is the unique solution of

Example 1. To illustrate our results, fix . Computing , we have , which is a first approximation of . To solve system (80) numerically, we substitute the terms of expression of in (10): The previous integral cannot be computed analytically, so we approach using Taylor expansion around 0 (remember that we want so solve