An Introduction to Mathematical Epidemiology pp 249279  Cite as
Ecological Context of Epidemiology
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Abstract
This chapter is focused on ecoepidemiology. It introduces and studies a number of models related to infectious diseases in animal populations. Animals are typically subject to ecological interactions. The chapter first introduces SI and SIR models of species subject to a generalist predator and studies the impact of selective and indiscriminate predation. The classical Lotka–Volterra predator–prey and competition models are reviewed together with their basic mathematical properties. Furthermore, the chapter includes and discusses a Lotka–Volterra predator–prey model with disease in prey and a Lotka–Volterra competition model with disease in one of the species. Hopf bifurcation and chaos are found in some of the ecoepidemiological models.
Keywords
Reproduction Number Prey Population Generalist Predator Basic Reproduction Number Interior Equilibrium10.1 Infectious Diseases in Animal Populations
Infectious disease pathogens affect numerous animal populations. An animal population, by definition, is a collection of individuals of the same species occupying the same habitat. A species is a group of individuals who generally breed among themselves and do not naturally interbreed with members of other groups. Fortunately, familiar baseline models of infectious diseases in humans such as the SI, SIS, SIR, SIRS models also can be used to model diseases in animal populations. There are some very important distinctions, however, that we will discuss below.

The simplest and most obvious reason is that a nonhuman species, such as species of fish, birds, and other mammals, represents a much simpler biological system for scientific study than the human species. These species can often be manipulated for better understanding of its properties and dynamics. One such example, in which population dynamics principles were validated through experiments with a beetle population is given in [47].

There is also a practical reason to study animal diseases. Historically, human diseases have been inextricably linked to epidemics in animal populations. Rapid expansion of civilization in the last few millennia has increased the contact at the human–animal interface, through urban and agricultural expansion that encroaches on wildlife habitats, domestication of cattle and other livestock, or simply by keeping pets. Because of such close intimacy between humans and nonhuman animals, viruses and bacteria that cause various animal diseases continuously “jump across the species barrier” and infect humans. Recent examples include HIV, whict came from monkeys; SARS, which came from bats; and avian flu H5N1, which was first found in wild birds. In fact, about 60% of all human infectious diseases have their origin in animal species, and about threequarters of all emerging infectious diseases of humans—those that are occurring for the first time in humans—have been traced back to nonhuman species. Thus, understanding disease dynamics and ways to control disease in animal populations has tremendous human health implications.

Another practical reason to study disease in natural populations is that the disease can be a regulator of a population, leading to control of dangerous pests [11, 114]. Furthermore, parasites can have substantial effect on community composition [76].
Epidemic models of human populations assume that the population being considered is a closed population, completely separate from the rest of the world. This is rarely true for animal populations. Each animal species is a part of an intricate web of ecological interactions. Interacting populations that share the same habitat form a community. Two fundamental community interactions are competition for resources and predation. Such interspecies interactions impart complex feedback on the dynamics of the diseases in the population that is studied. This necessitates the integration of the principles of infectious disease epidemiology and community ecology.

Often, a single pathogen simultaneously infects many different species in the community. Even if the pathogen is completely removed from a given host, it may survive in the other hosts and can reinfect the original host. This is of particular concern if one of the species being infected is humans, while another species is an animal species.

Intense competition of the focal host species with the other species in the community can bring down the host population size to threateningly small levels, and a pathogen attack can eventually drive it to extinction.

Predation of the host by other species in the community can regulate pathogen outbreaks in the host species.
Conversely, host–pathogen interactions can feed back and impact the community by weakening the competitive abilities of the affected species.
In this chapter, we will consider the interrelation between infectious diseases and the two principal community interactions: predation and competition. Predation, or predator–prey interactions, refers to the feeding on the individuals of a prey species by the individuals of a predator species for the survival and growth of the predator. Some common examples of predator–prey interactions are big fish eating small fish, birds feeding on insects, and cats eating rats and mice. The predator itself can be of two types—a specialist predator, whose entire feeding choice is restricted to a single prey species, and a generalist predator that feeds on many different prey species. The dynamics of a specialist are strongly coupled to those of the prey, so that any rise or fall of the population size of the prey triggers a consequent rise or fall in the population size of its predator. The interaction between a specialist predator and its prey is modeled by the familiar Lotka–Volterra predator–prey model. By contrast, the dynamics of a generalist predator are only weakly coupled to any particular focal prey species and may not be influenced by the dynamics of the prey. The predation of a focal species by a generalist predator can be modeled by a predatoradded mortality on the prey.
10.2 Generalist Predator and SIType Disease in Prey
10.2.1 Indiscriminate Predation
While the direct effect of the predator is to harm the prey by increasing its mortality rate, predation can indirectly benefit the prey by keeping its population size low and thereby ruling out epidemic outbreaks.
10.2.2 Selective Predation
In summary, with an SI model, we see that predation reduces disease load and prevalence in the prey population, irrespective of whether the predator selectively attacks either susceptible or infected prey, or indiscriminately preys on both prey types. The main reason is that the incidence rate β S I, which gives the rate at which new infections appear in the host, depends bilinearly on both the susceptible and infected prey. Therefore, whether the predator eats selectively or indiscriminately, it always reduces the incidence β S I, and hence the disease level in the prey population. In the next section, we discuss disease in prey with permanent recovery, and investigate the impact of the predator on the disease load and prevalence of the disease in the prey population.
10.3 Generalist Predator and SIRType Disease in Prey
10.3.1 Selective Predation
Therefore, once the pathogen is established in the prey, \(\mathcal{R}_{0}> 1\), the prevalence decreases with increasing predation level P when the predator selectively attacks susceptible or infected prey only. In contrast, prevalence increases with increasing predation level P when the predator attacks preferentially the recovered prey. This last result, although counterintuitive, can be explained by the fact that by attacking recovered prey alone, the predator decreases total population size without impacting the incidence of the disease β S I.
10.3.2 Indiscriminate Predation
Thus, we are faced with an intriguing situation. The basic reproduction number \(\mathcal{R}_{0}\) and the disease load I^{∗} decrease with increasing predation pressure P, which tends to give the impression that the epidemic in the prey is weakened in the presence of predation. However, prevalence p^{∗}, which gives another measure of the disease in the population, increases, at least for low predation level P. The reason is that even though both I^{∗} and N^{∗} both decrease with increasing P, N^{∗} falls faster than I^{∗} for low values of P, and therefore the ratio I^{∗}∕N^{∗}, which gives the prevalence, increases for those values of P. In other words, while both the population size of the prey and the number of infective hosts are depressed under indiscriminate predation, the proportion of infective individuals in the population increases for low levels of P. The epidemiological significance of this result is that the overall disease burden in the population is reduced, but the risk that a randomly chosen individual is infected can increase with predation, at least at small predation levels.
10.4 Specialist Predator and SI Disease in Prey
Specialist predators prey on a given species, which is our focal species. The dynamics of a specialist predator are closely linked to those of the prey. These dynamics are those described by the Lotka–Volterra predator–prey model. In the next subsection, we discuss various types of predator–prey models.
10.4.1 Lotka–Volterra Predator–Prey Models

The prey population finds ample food at all times.

The food supply of the predator population depends entirely on the prey population.

The rate of change of population is proportional to its size.

During the process, the environment does not change in favor of one species, and genetic adaptation is sufficiently slow.
The Lotka–Volterra model represents one of the early triumphs of mathematical modeling, because it captures the oscillatory behavior observed in natural predator–prey systems with a specialist predator. Unfortunately, the model cannot explain these oscillations, because the oscillations in the model are structurally unstable, that is, small changes to the model can significantly change the qualitative behavior of the model, e.g., it can stabilize the oscillations. Ideally, we would like the oscillations in the model to be structurally stable, that is, if we make small changes to the model to better reflect reality, the qualitative predictions of the model remain the same, and in particular, the model continues to exhibit oscillations.
10.4.2 Lotka–Volterra Model with SI Disease in Prey
The predator–prey dynamics described by the above models can be impacted by the presence of a disease. The disease may affect the prey, it may affect the predator, or it may affect both the predator and the prey if it is caused by a pathogen that can jump the species barrier. We will consider here a prey population that is subject to predation and impacted by a disease. As a baseline model of the predator–prey dynamics we use the Lotka–Volterra model with selflimiting prey population size and linear functional response. This model exhibits simple dynamics—global convergence to a preyonly equilibrium or to a predator–prey coexistence equilibrium. We will see that the introduction of disease in the prey can lead to much more complex dynamics, namely oscillation and even chaos.

The disease is transmitted only in the prey and does not affect the predator.

Infected prey do not recover from the disease—the disease is of SI type for the prey.

Attack rates of the predator for healthy and infected prey may be different.

Infected prey does not reproduce but participates in the competition for resources, so it participates in selflimitation.
 1.
Extinction or trivial equilibrium: \(\mathcal{E}_{0} = (0,0,0)\). The trivial equilibrium always exists, but it is always unstable, since the community matrix has an eigenvalue r > 0.
 2.Diseasefree and predatorfree equilibrium: \(\mathcal{E}_{1} = (K,0,0)\). This equilibrium also always exists. If we define a disease reproduction number in the absence of predatorand predator invasion number in the absence of disease$$\displaystyle{\mathcal{R}_{0} = \frac{\beta K} {\mu _{0}} }$$$$\displaystyle{\mathcal{R}_{P}^{0} = \frac{\varepsilon \gamma _{S}K} {d},}$$then the equilibrium \(\mathcal{E}_{1}\) is locally asymptotically stable ifand unstable if either inequality is reversed.$$\displaystyle{\mathcal{R}_{0} <1\qquad \mathrm{and}\qquad \mathcal{R}_{P}^{0} <1}$$
 3.Predatorfree endemic equilibrium in which the disease persists in the prey but the predator dies out, \(\mathcal{E}_{2} = (S_{2},I_{2},0)\), whereThe equilibrium \(\mathcal{E}_{2}\) exists if and only if \(\mathcal{R}_{0}> 1\). The equilibrium \(\mathcal{E}_{2}\) is locally asymptotically stable if and only if the invasion number of the predator in the presence of disease is less than one, \(\mathcal{R}_{p} <1\), where the invasion number of the predator in the presence of the disease is$$\displaystyle{S_{2} = \frac{\mu _{0}} {\beta } \qquad I_{2} = \frac{r\left (1  \frac{1} {\mathcal{R}_{0}} \right )} { \frac{r} {K}+\beta }.}$$$$\displaystyle{\mathcal{R}_{P} = \frac{\varepsilon } {d}\left (\gamma _{S}\frac{\mu _{0}} {\beta } +\gamma _{I}\frac{r\left (1  \frac{1} {\mathcal{R}_{0}} \right )} { \frac{r} {K}+\beta } \right ).}$$
 4.Predator–prey diseasefree equilibrium, where the disease dies out and the predator and the prey coexist diseasefree: \(\mathcal{E}_{3} = (S_{3},0,P_{3})\), whereThe diseasefree predator–prey coexistence equilibrium \(\mathcal{E}_{3}\) exists if and only if the predator invasion number in the absence of disease satisfies \(\mathcal{R}_{P}^{0}> 1\). This equilibrium is locally asymptotically stable if and only if the disease reproduction number in the presence of the predator is less than one: \(\mathcal{R}_{1} <1\), where$$\displaystyle{S_{3} = \frac{d} {\varepsilon \gamma _{S}}\qquad P_{3} = \frac{r} {\gamma _{S}}\left (1  \frac{1} {\mathcal{R}_{P}^{0}}\right ).}$$$$\displaystyle{\mathcal{R}_{1} = \frac{\beta d} {\varepsilon \gamma _{S}(\gamma _{I}P_{3} +\mu _{0})}.}$$
 5.Predator–prey–disease coexistence equilibrium: \(\mathcal{E}^{{\ast}} = (S^{{\ast}},I^{{\ast}},P^{{\ast}})\). The system for the coexistence equilibrium is given byThe system for the interior equilibrium is a linear system, and if it has a nonnegative solution, that solution is unique under the assumption that the determinant is not zero. The system can be solved, but the expressions obtained do not offer much insight. So we take a different approach. The conditions for existence of a positive equilibrium are stated in the following theorem:$$\displaystyle\begin{array}{rcl} r\left (1 \frac{N} {K}\right ) \gamma _{S}P \beta I& =& 0, \\ \beta S \gamma _{I}P \mu _{0}& =& 0, \\ \varepsilon (\gamma _{S}S +\gamma _{I}I)  d& =& 0. {}\end{array}$$(10.21)
Theorem 10.1.
Proof.
Theorem 10.2.
The interior equilibrium is locally stable if γ _{I} > γ _{S} . If γ _{I} < γ _{S} , the interior equilibrium is unstable.
We highlight the main conclusion:
The presence of a disease in the prey and a preferential predation of susceptible individuals can destabilize otherwise stable predator–prey dynamics.
Fixed parameter values used in simulations
Parameter  Interpretation  Value  
r  Prey growth rate  2  
K  Prey carrying capacity  1,000  
μ _{0}  Diseaseinduced death rate  0.001  
ε  Predator conversion efficiency  0.2  
γ _{ I }  Attack rate on infectious prey  1.0  
γ _{ S }  Attack rate on susceptible prey  9.1  
d  Predator death rate  0.2 
10.5 Competition of Species and Disease
Predation, which we considered in the previous sections, is one of the interactions in the ecological community, and it is certainly the most dynamic interaction. For many years, however, competition has been thought to be the main mode of interaction. There is no question that competition is a very important community interaction.
10.5.1 Lotka–Volterra Interspecific Competition Models
Lotka and Volterra also developed competition models. Lotka–Volterra competition models describe the competition between two or more species for limited resources. Such competition is called interspecific competition, which is contrasted with intraspecific competition, which is competition among individuals of one species for limited resources. Lotka–Volterra models are representatives of the interference competition models, whereby the increase in the size of one species is assumed to decrease the other species per capita growth rate [90].
 Case 1.

K_{1} > α_{12}K_{2} and K_{2} < α_{21}K_{1}. In this case, there is no interior equilibrium, since the numerators of N_{1}^{∗} and N_{2}^{∗} have opposite signs. The boundary equilibrium \(\mathcal{E}_{1}\) is a stable node, while the boundary equilibrium \(\mathcal{E}_{2}\) is a saddle (unstable). All orbits tend to (K_{1}, 0) as t → ∞. Thus, species one persists at carrying capacity, while species two becomes extinct.
 Case 2.

K_{1} < α_{12}K_{2} and K_{2} > α_{21}K_{1}. This is a symmetric case to Case 1. In this case, again there is no interior equilibrium. The boundary equilibrium \(\mathcal{E}_{1}\) is a saddle (unstable), while the boundary equilibrium \(\mathcal{E}_{2}\) is a stable node. All orbits tend to (0, K_{2}) as t → ∞. Thus, species two persists at carrying capacity, while species one becomes extinct.
 Case 3.
 K_{1} < α_{12}K_{2} and K_{2} < α_{21}K_{1}. These two inequalities imply that 1 < α_{12}α_{21}. Thus the interior equilibrium exists. However, since the determinant of the community matrix evaluated at the interior equilibrium is negative,the community matrix has two real eigenvalues of opposite sign (q < 0). Therefore, the coexistence equilibrium is a saddle. At the same time, both semitrivial equilibria \(\mathcal{E}_{1}\) and \(\mathcal{E}_{2}\) are stable nodes. In this case, the coexistence of the two species is again impossible. One of the species always outcompetes and eliminates the other. However, the winner of the competition is determined by the initial conditions. We recall that this dependence on the initial conditions is called the founder effect. This is another example of bistability. Solution orbits that start from the upper part of the plane converge to the equilibrium \(\mathcal{E}_{2}\), while those that start from the lower part converge to the equilibrium \(\mathcal{E}_{2}\) (see Fig. 10.7).$$\displaystyle{{\mathrm{Det}}J(N_{1}^{{\ast}},N_{ 2}^{{\ast}}) <0,}$$
 Case 4.
 K_{1} > α_{12}K_{2} and K_{2} > α_{21}K_{1}. In this case, 1 > α_{12}α_{21}, and the interior equilibrium also exists. The community matrix at the interior equilibrium has negative trace andTherefore, the coexistence equilibrium is locally asymptotically stable. It can be further shown that it is a stable node. The community matrices of the two semitrivial equilibria have one positive eigenvalue and one negative eigenvalue. Hence, the two semitrivial equilibria are saddle points. In this case, every orbit that starts from the interior tends to the coexistence equilibrium as t → ∞.$$\displaystyle{{\mathrm{Det}} J(N_{1}^{{\ast}},N_{ 2}^{{\ast}})> 0.}$$
In Cases 1 and 2, we say that competitive exclusion occurs. That means that one of the species excludes the other and dominates by itself. We recall that the principle of competitive exclusion was first formulated by Gause in 1934 [64] on the basis of experimental evidence.
10.5.2 Disease in One of the Competing Species
 Equilibrium \(\mathcal{E}_{12}\) is an equilibrium in which the disease is present in species one but species two is absent:This equilibrium exists when the following inequalities are satisfied:$$\displaystyle{\mathcal{E}_{12} = \frac{1} {\beta ^{2}} ((r_{1}  r_{I})a_{11}  r_{I}\beta,(r_{I}  r_{1})a_{11} + r_{1}\beta,0).}$$See Problem 10.4 for further details.$$\displaystyle{r_{I}\beta <(r_{1}  r_{I})a_{11} <r_{1}\beta.}$$
 Equilibrium \(\mathcal{E}_{13}\) is an equilibrium in which species one is present with susceptible individuals only and species two is also present:where \(\varDelta = a_{11}a_{22}  a_{12}a_{21}\) is the determinant of the matrix of coefficients. Hence, the existence and stability of equilibrium \(\mathcal{E}_{13}\) is exactly the same as the existence and stability of the coexistence equilibrium of the two species.$$\displaystyle{\mathcal{E}_{13} = \frac{1} {\varDelta } (r_{1}a_{22}  r_{2}a_{12},0,r_{2}a_{11}  r_{1}a_{21}) = \left (\frac{K_{1} \alpha _{12}K_{2}} {1 \alpha _{12}\alpha _{21}},0, \frac{K_{2} \alpha _{21}K_{1}} {1 \alpha _{12}\alpha _{21}} \right ),}$$
 Equilibrium \(\mathcal{E}_{23}\) is an equilibrium in which species one is present with infected individuals only and species two is also present:where \(\varDelta = a_{11}a_{22}  a_{12}a_{21}\) has the same meaning as above. Hence, the existence and stability of equilibrium \(\mathcal{E}_{23}\) can be derived in a similar way to that of the existence and stability of the coexistence equilibrium of the two species. See Problem 10.5.$$\displaystyle{\mathcal{E}_{23} = \frac{1} {\varDelta } (0,r_{I}a_{22}r_{2}a_{12},0,r_{2}a_{11}r_{I}a_{21}) = \left (0, \frac{\frac{r_{I}} {r_{1}} K_{1} \alpha _{12}K_{2}} {1 \alpha _{12}\alpha _{21}}, \frac{K_{2} \frac{r_{I}} {r_{1}} \alpha _{21}K_{1}} {1 \alpha _{12}\alpha _{21}} \right ),}$$
Animal populations are subject to a number of ecological interactions. Introducing disease in one or more of the interacting populations is an interesting area of exploration often referred to as ecoepidemiology.
Problems
10.1. Competition of Strains under Predation
 (a)
Is coexistence of the strains possible in this model? Show competitive exclusion.
 (b)
Determine which strain dominates depending on the predation level P.
10.2. Competition of Strains under Predation
 (a)
Is coexistence of the strains possible in this model? Determine the coexistence equilibrium and the conditions under which it exists.
 (b)
Use a computer algebra system to simulate the coexistence of the strains. How does changing the predator’s predation rates γ_{1} and γ_{2} affect the competition of the strains?
10.3. Specialist Predator with Disease in Predator
 (a)
Find the equilibria of the system.
 (b)
Compute the reproduction number of the disease in the predator. Determine the stability of the semitrivial equilibria.
 (c)
Compute the interior equilibrium. When is the interior equilibrium locally asymptotically stable? Does Hopf bifurcation occur?
10.4. Epidemic Model with Vertical Transmission
 (a)
Find the equilibria of model (10.40). Under what conditions does each equilibrium exist?
 (b)
Determine the local stabilities of each equilibrium.
 (c)
Use a computer algebra system to draw the phase portrait in each of the cases above.
 (d)
Explain how the vertical transmission is incorporated in the model.
10.5. Equilibria of Lotka–Volterra Competition Model with Disease
 (a)
Find the trivial and semitrivial equilibria of model (10.32). Under what conditions does each equilibrium exist?
 (b)
Determine the local stabilities of each equilibrium. Determine the corresponding conditions for stability/instability.
 (c)
Use a computer algebra system to simulate model (10.32). Set β = 0. Determine parameter values such that \(\mathcal{E}_{13}\) is locally stable. Start increasing β. How does the increase in the prevalence of the disease affect the competitive ability of species one?
10.6. Mutualism
 (a)
Find the equilibria of model (10.41). Under what conditions does each an equilibrium exist?
 (b)
Show that in the coexistence equilibrium, species persist at densities larger than their respective carrying capacities. What does that mean biologically?
 (c)
Determine the local stabilities of each equilibrium. Determine the corresponding conditions for stability/instability.
 (d)
Draw a phase portrait in each of the cases α_{12}α_{21} < 1 and α_{12}α_{21} > 1. Show that in the case α_{12}α_{21} > 1, orbits may become unbounded.
10.7. Mutualism with Disease
 (a)
Find the trivial and semitrivial equilibria of model (10.42). Under what conditions does each an equilibrium exist?
 (b)
Determine the local stabilities of each equilibrium. Determine the corresponding conditions for stability/instability.
 (c)
Determine the interior equilibrium of the system.
 (d)
Determine the stability of the interior equilibrium.
10.8. Specialist Predator with a TwoStrain Disease in Predator
 (a)
Find the semitrivial equilibria of the system above.
 (b)
Compute the reproduction numbers of the two strains in the predator. Determine the stability of the semitrivial equilibria.
 (c)
Is there a coexistence equilibrium for that system?
Notes
Acknowledgements
The first part of this chapter is based on lecture notes that Manojit Roy developed and delivered in the Biomathematics Seminar.
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