Discrete Dynamics in Nature and Society

Volume 2017, Article ID 5310837, 13 pages

https://doi.org/10.1155/2017/5310837

## Discrete Models of Disease and Competition

^{1}U.D. Matemáticas, Universidad de Alcalá, 28871 Alcalá de Henares, Spain^{2}Departamento de Matemática Aplicada, ETSI Industriales, Universidad Politécnica de Madrid, 28006 Madrid, Spain

Correspondence should be addressed to R. Bravo de la Parra; se.hau@ovarb.leafar

Received 9 June 2017; Revised 25 July 2017; Accepted 10 August 2017; Published 20 September 2017

Academic Editor: Vicenç Méndez

Copyright © 2017 R. Bravo de la Parra 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

The aim of this work is to analyze the influence of the fast development of a disease on competition dynamics. To this end we present two discrete time ecoepidemic models. The first one corresponds to the case of one parasite affecting demography and intraspecific competition in a single host, whereas the second one contemplates the more complex case of competition between two different species, one of which is infected by the parasite. We carry out a complete mathematical analysis of the asymptotic behavior of the solutions of the corresponding systems of difference equations and derive interesting ecological information about the influence of a disease in competition dynamics. This includes an assessment of the impact of the disease on the equilibrium population of both species as well as some counterintuitive behaviors in which although we would expect the outbreak of the disease to negatively affect the infected species, the contrary happens.

#### 1. Introduction

Species interactions are a central issue in ecology in order to explain community structure and its dynamics. Among the frameworks that try to explain ecosystems dynamics, ecoepidemiology has become a proper discipline in its own right [1, 2]. Loosely speaking, its main objective is to understand the effect of diseases or parasites on species interactions and vice versa. The empirical evidence of such an influence has motivated a great deal of mathematical modeling in order to reveal the underlying mechanisms.

In this work we focus on ecoepidemic competition models that have attracted great attention [3]. Empirical observations have revealed that disease/parasites can affect the outcome of species competition in several different ways. For instance, a shared parasite between the competing species can change the fate of competition, allowing the otherwise inferior competitor to rule out the superior competitor, as observed by Park [4] with* Tribolium castaneum* (the stronger competitor),* Tribolium confusum*, and the parasitoid* Adelina tribolii* that lowers the fecundity of* T. castaneum*. There is also evidence of species coexistence induced by specialist (i.e., affecting only one competitor) parasite. This is the case of* Anolis* lizards in the Caribbean area [5].* Anolis gingivinus* is a stronger competitor than* A. wattsi* and coexisting populations are found only in the presence of the parasite* Plasmodium azurophilum*. This parasite that reduces the competitive abilities of the host rarely affects* A. wattsi* but can be common in* A. gingivinus*.

It is usually assumed that disease/parasites reduce either the growth or competitive abilities of infected individuals or both of them. Nevertheless, this is not always the case. Regarding their effect on growth, some parasites spread through host offspring, so that the strategy consists on enhancing host fecundity to improve their spreading. For instance, this is the case of* Wolbachia* and its host* Bemisia tabaci* [6, 7].* Bemisia t.* competes with* Trialeurodes vaporariorum* (Westwood) [8] and it has been reported that there are no evidences of Westwood infected by* Wolbachia* [9]. As far as competitive abilities are concerned, a laboratory study [10] dealt with the effect of the tapeworm* Hymenolepis diminuta* on competition between* T. castaneum* and* T. confusum*.* T. castaneum* (as we have said, the superior competitor) has higher infection and fitness costs from parasitism, so that a parasite-mediated reversal of the outcome of competition was expected. Nevertheless,* T. confusum* was excluded even faster than what had happened in the absence of infection; it was suggested that intraguild predation is behind this outcome [10].

From the mathematical point of view, complete ecoepidemic competition models are known for the difficulty of their analytical study (see [3] box 2.3 and references therein). There exist a number of studies in continuous time [3] but, to our knowledge, our work is the first attempt in discrete time.

An ecoepidemic competition model combines three different processes: each species demography, intra- and/or interspecies competition, and epidemics. To our knowledge, all the ecoepidemic competition models in the literature consider that the three processes evolve within the same time scale. In other words, it is assumed that the effects of demography, competition, and infection are accounted for at the population level. This assumption is fairly true for certain long-term infections (as AIDS, tuberculosis, etc.) but is not for some others. Indeed, there are diseases such that a number of infection/recovery episodes take place within each demography period. This assumption is equivalent to that of the existence of different time scales associated with each process. An example of this scenario is the* human trachoma*, an infectious disease that is the leading cause of blindness worldwide and is caused by a microorganism called* Chlamydia trachomatis*. The immune system of human beings can recover but reinfections happen often [11]. After repeated episodes, successive scars cause the edge of the eyelid to fold inward, causing pain and even permanent damage to the cornea which may lead to irreversible blindness. In fact, the assumption of an endemic background scenario can be understood as the effect on the population dynamics of the disease having attained one of its equilibrium states.

In this work we consider two models of competition and disease. The first one is a single species model in which there is intraspecific competition and the population is affected by a disease-parasite. Demography-competition is assumed to be governed by an adaptation of the Beverton-Holt model and the disease corresponds to a discrete SIS epidemic model with frequency-dependent transmission [12]. The second model corresponds to two competing species, one of them affected by specialist disease/parasite. Competition is built up from the classical discrete time competition Leslie-Gower model [13], and the disease is again introduced by means of a discrete SIS epidemic model. This ecoepidemic model happens to be rich enough to exhibit the complex behaviors described above but simple enough to be analytically tractable.

One of the differences between continuous and discrete models is that in the former all processes involved in the model (demography, competition, and infection/recovery) occur instantaneously at the same time, whereas in the later it is usual to consider that processes take place sequentially [14]. Keeping this idea as well as the separation of time scales in mind, in the construction of both models we assume that a large number of epidemic-related events take place within each demographic/competition step. Concentrating our attention on the case of the two competing populations, the resulting model is a three-dimensional system. Making use of the existence of different time scales, it is possible to build up a reduced two-dimensional system that approximates the behavior of the original system and that simplifies the analysis of the model (see [15] for a recent review in the field of discrete approximate reduction techniques). The resulting reduced system can be regarded as a generalization of the Leslie-Gower competition model and retains some of its properties, as the fact that the unique attractor states are equilibrium points. However, the reduced model allows for disease-induced multistability scenarios.

This work is organized as follows: in Section 2 we consider the interplay between intraspecific competition and a disease on a single species model. In Section 3 we introduce a competing species and carry on with the mathematical analysis of the resulting system and its ecological interpretation. Subsequently, we analyze separately the scenarios of a disease affecting either species growth or species competitive abilities. The conclusions in Section 4 and Appendix with the proof of the mathematical results complete the manuscript.

#### 2. Disease and Intraspecific Competition

We consider a host population with density dependent regulation that is affected by a disease which acts on a shorter time scale than the demographic dynamics. The time unit of the discrete model is the one associated with its demographic part. We consider that, in this time unit, called slow, a single episode of demographic change following a number of disease infection-recovery cycles happens. In this way we intend to describe the fact that pathogens exhibit outbreaks on short time scales, on the order of days or weeks, whereas demographic changes might be considered annual.

Time in the slow time unit is denoted by . and represent the susceptible and the infected individuals, respectively, at time . Between time and time we consider the disease dynamics acting sequentially times followed by one demographic episode.

The disease dynamics is defined by means of the discrete time SIS epidemic model studied in [12] whose associated map is so that the total effect of disease during one slow time unit is represented by its th iterate that we denote as .

We assume that transmission is frequency-dependent, with being the transmission coefficient. The recovery coefficient represents the fraction of infected individuals that recover in a unit of time.

The disease dynamics keeps constant the population size

Henceforth we assume that the following inequalities hold: so that if we denote and , one has [12]. Note that condition amounts to requiring that the number of recovered individuals in one time step can not be greater than the number of infected individuals, which clearly is necessary for the number of infected individuals to remain positive. The second condition in (3) says that the transmission coefficient can not be too high with respect to the recovery coefficient because, if this were not so, the number of susceptibles might become negative.

The asymptotic behavior of the solutions of the discrete system defined by map (1) is studied in [12]. The basic reproduction number for this model is . If then for any positive initial condition the number of infected individuals monotonically decreases to 0; that is, positive solutions converge to the disease-free equilibrium , where is the constant population size. On the other hand, if the disease becomes endemic. Assuming, in addition to inequalities (3), that positive solutions converge to an asymptotically stable endemic equilibrium

If then the monotonic convergence to an endemic equilibrium disappears giving rise to period-doubling and chaotic behavior.

In the following we will restrict to the case where (4) holds and therefore the endemic equilibrium is stable. The fact that the fast disease process attains an equilibrium point in the long term allows one to reduce the dimension of the system that we are proposing. This equilibrium is expressed as the limit, , of the iterates of map where and is defined by Note that implies that the disease is eradicated.

To build up our demographic model where we have to take into account susceptible and infected individuals, we start from the Beverton-Holt model [16] where the positive parameters and are the intrinsic growth rate and the intraspecific competition coefficient. There are two possible asymptotic behaviors of the solutions of equation (8) with positive initial condition:(1)If then .(2)If then .

Note that this corresponds to a logistic behavior with carrying capacity .

Now we adapt the Beverton-Holt model to consider individuals classified into susceptible and infected. The trait-mediated indirect effects of parasites on hosts are considered both in the growth and in the intraspecific competition. We assume different intrinsic growth rates for susceptible, , and infected, , individuals. We also distinguish four intraspecific competition coefficients: , , , and . The resulting model is

To build up the complete model combining the demographic and the disease processes we compose the th iterate of map (1) and the map defining system (9). Using the notation the complete model has the form

System (10) can be reduced to a scalar equation from which we can approximate the asymptotic behavior of its solutions. To do so we make use of discrete approximate reduction techniques and follow the reduction procedure presented in [15, 17]. Assuming that the disease process has attained its equilibrium (6), the dynamics of the total number of individuals in the population, , can be approximated by the following equation:obtained by adding up the two equations in (10) and using the following approximations: and [18].

We can write (11) in the next simplified form with four parameters defined as , , , and . The asymptotic behavior of its solutions is stated in the next theorem.

Theorem 1. *One considers equation (12) with , , and being positive and nonnegative. Let be any solution of (12) with positive initial condition:*(1)*If then *(2)*If then , where *

*Proof. *See Th. A.4 in [18].

*Making use of the last theorem and the results on approximate aggregation in [18], we can decide the asymptotic behavior of solutions of system (10). We assume that the disease becomes endemic (, i.e., ) because otherwise system (10) behaves as the scalar Beverton-Holt model. Therefore we have that, for any positive initial condition, the solution of system (10) verifies the following:(1)If then it approximately tends to .(2)If then it approximately tends to where is the transformation associated with the competition process (9); that is, *

*Note that if the disease is established, either the population tends to extinction or it tends to an endemic equilibrium. To analyze the influence of the parasite on the population dynamics we compare the asymptotic behaviors of the solutions of (8), representing the dynamics of the disease-free population, and (11), which yields the approximate long-term behavior of system (10). For the comparison we assume that parameters affecting susceptible individuals in system (10) coincide with the corresponding ones in (8), that is, and .*

*It is straightforward to check that implies extinction and endemic stabilization. On the other hand, if , that is, if the population without disease tends to , then the disease leads the population to extinction for small enough, specifically when . Conversely, if the population is not viable without disease, , a large enough infected growth rate, , entails the population attaining equilibrium .*

*If the population attains a positive equilibrium in both cases, with and without disease, we can explore the influence of the disease by comparing the two final population sizes. Without disease it is and with disease . We do the comparison in the relevant particular case in which the disease does not affect the competitive abilities of individuals, that is, . In this case we obtain the ratio As a function of the growth rate of infected individuals, , this ratio is increasing and equals 1 for . More interesting is its dependence on If then it increases with ; that is, the larger , the smaller the ratio. Conversely, if the larger , the larger the ratio.*

*3. Disease and Interspecific Competition*

*3.1. Model Construction and Reduction*

*In this section we generalize the setting of the previous one and introduce a second species, which we consider disease free, which competes, according to the well-known Leslie-Gower model [13], with the species which is affected by the disease. Specifically, let us consider two species that we denote by 1 and 2, in competition and a disease defined by system (1) and which affects only species 1. Let and represent the number of susceptible and infected individuals of species 1 and let be the amount of individuals of species 2. Then, the slow dynamics is described bywhere all growth and competition coefficients are assumed to be positive.*

*As in Section 2, we consider that in the time unit of the model a single episode of demographic change happens, defined by (17), followed by a number of disease infection-recovery cycles corresponding to the SIS model (1) introduced in Section 2. Then, the complete system that takes into account the joint effect of the slow and fast processes takes the form If we follow the reduction procedure presented in [15, 17, 18], which essentially consists in adding up the first two equations in (18) and using the approximations and , we obtain the reduced systemwhere , , , and .*

*Note that system (19) generalizes the classical Leslie-Gower competition model [13] in the sense that at (i.e., the disease-free scenario) system (19) becomes the usual Leslie-Gower competition model.*

*3.2. Model Analysis*

*In order to carry out the mathematical treatment of system (19) we express it in the form where is the map defined byNote that all the parameters of the model are positive except possibly that is nonnegative. As we will see, the numbers will play an important role in the dynamics of the system.*

*We note that the functional form of system (20) is a particular case of a model obtained in [19] which studies the dynamics of two spatially distributed populations in competition in which there is fast migration of individuals between the different patches. In that work an analytical study of the system was carried out in the case in which for at least one value of , which essentially corresponds to our Proposition 3 and Theorem 4. In Theorem 6 we provide a full analytical study of the behavior of the orbits of the system in the case for , which was only addressed numerically in [19].*

*Let and be the positive axes. It is immediate to realize that the sets , , and , are forward invariant by . In the sequel, unless otherwise stated we will always assume that we are working on .*

*Let us study the isoclines of the system, that is, the sets defined by and . Clearly, besides the positive axes (resp., ) in which the variable (resp., ) is constant, the isoclines are the curves defined by , for .*

*Lemma 2. The isoclines and are, respectively, a hyperbola and a straight line. For each , intersects if and only if and in that case intersects both the positive axes and . Moreover can be written in the form , for a certain , where is a strictly decreasing convex function.Except in the degenerate case in which and are in fact the same straight line, there can exist at most two positive equilibriums.*

*Proof. *See Appendix.

*Let us now consider the existence of equilibriums for system (20). Note that is an equilibrium point for all values of the parameters. Using Lemma 2 we conclude the follwoing:(i)For each , there exists a semitrivial equilibrium point on the positive axis if and only if . In that case the semitrivial equilibrium is unique.(ii)A necessary condition for the existence of a positive equilibrium is that for both , and in that case there can exist at most two such equilibriums.*

*In what follows we will write (resp., ) to denote that and (resp., and ). Similarly, we will write (resp., ) to denote that and (resp., and ).*

*Proposition 3. Let us consider system (20).(a)All solutions in are forward bounded. More specifically, (b) is strongly competitive in ; that is, if are distinct points with it follows that [20].(c)All orbits in are eventually componentwise monotone; that is, for each , the corresponding solution verifies that is eventually monotone for each . Moreover, all orbits tend to an equilibrium as .*

*Proof. *See Appendix.

*The next result analyzes the behavior of solutions of system (20) except in the case in which , .*

*Theorem 4. Let us consider system (20).(a)For each , if then for any initial value on species tends to extinction as .(b)If for , all orbits in tend to , and if the inequalities are strict then is hyperbolic. If for , then is a repeller and therefore no orbit can converge to .(c)If and then (1) all orbits with tend to and (2) all orbits with tend to .(d)If and then (1) all orbits with tend to and (2) all orbits with tend to .*

*Proof. *See Appendix.

*Let us now consider the case in which for . In the first place we study the local stability of the semitrivial equilibriums and the behavior of orbits on the positive axes.*

*Proposition 5. Let us consider system (20) and let us assume that for .(a)All orbits starting on the positive -axis converge to , .(b)Let be fixed. If , , then is hyperbolic and attracting. If , , then is hyperbolic and unstable.*

*Proof. *(a) Let be fixed. We know that is invariant for and then any orbit corresponding to an initial condition on must converge to either or . From Theorem 4 part (b) is a repeller and therefore the result follows.

(b) It follows from the usual analysis of the eigenvalues of the corresponding Jacobian matrix. Standard calculations lead to the results bearing in mind that .

*In order to study the behavior of solutions for positive initial conditions, we will consider different cases based on the relative position of the intercepts of the isoclines. To this end, we define as the -intercept of , . It is immediate to check that is given bywhere , .*

*Now we distinguish the following scenarios:(i)Case : and .(ii)Case : and .(iii)Case : and .(iv)Case : and .*

*Taking into account Lemma 2, the isoclines divide in a finite number of open connected and disjoint sets , , in which is either strictly increasing or strictly decreasing for each . To describe the kind of monotonicity in each region we will use arrows, in such a way that, for example, the situation in which decreases and increases corresponding to a horizontal arrow pointing to the left and a vertical arrow pointing up. Figure 1 shows the different configurations. Note that in case we can distinguish two generic cases depending on whether the two isoclines do not intersect (case ) or intersect at two points (case ).*