Disease Spread among Hunted and Retaliating Herding Prey

Abstract

Two new ecoepidemic models of predator–prey type are introduced. They feature prey that gather in herds. The specific novelty consists of the fact that the prey also has the ability to defend themselves if they are in large numbers. The two deterministic models differ in the way a disease spreading among the ecosystem is transmitted, either by direct contact among infected and susceptible animals or by the intake of a virus present in the environment. Only the disease-free and the endemic equilibrium are allowed, and they are analyzed for feasibility and stability. The boundedness results allow us to gather some results regarding global stability. Persistent oscillations can be triggered when some relevant model parameters cross specific thresholds, causing repeated epidemic outbreaks. Furthermore, the environmental contamination through a free viruses destabilizes the endemic equilibrium and may lead to large amplitude oscillations, which are dangerous because they are potentially harmful to ecosystems. The bifurcation parameters leading to the limit cycle onset are related to the epidemics. For instance, they could be the disease-related mortality and the transmission rates, whether by direct contact among individuals or through the environment. The results of this investigation may provide insights to theoretical ecologists and may provide useful indications for epidemic spread containment.

1. Introduction

In mathematical biology, a prominent place is taken by population theory. This area of research is not confined to the investigation of the behavior of only one population but extends to account for population interactions, trophic chains and food webs, which involve many species living in the same environment. These investigations date back almost a century.

A particular feature of animal behavior has also recently been introduced, observing that herbivores generally gather in herds. Models accounting for predators hunting on prey living together have been studied in the past decade, see e.g., [1,2,3,4] and even earlier [5]. The concept is also common in other sciences, such as economics [6,7,8], decision making [9] and social sciences [10,11,12]. The main feature is that predation in such situations most affects the prey that occupy the outermost positions of the herd. Mathematically, this translates to the fact that the predator–prey interactions are not one-to-one but involve only these prey.

Their number is shown [1,13] to be a power of the population size. The generally considered models are based on simplicity with the square root response, which adapts itself well for populations living in two-dimensional environments. However, variants of this assumption have been already studied [13,14], possibly accounting for fish schools and insect swarms living in a three-dimensional environment or for planar shapes that are not smooth or even become fractals. Further, the response function has also been modified to render it more realistic at low population values or to allow generic formulations that are not necessarily tied to a specific mathematical form [15,16].

Among the most interesting results of these investigations is the fact that novel features arise that distinguish them from the models of Lotka–Volterra type—for instance, tristability in the context of competing populations [17], whose impossibility is shown instead for the classical quadratic interactions case. The latter case gives the principle of exclusion among competitors. Extensions of these ideas to spatial models are also present in the current literature, [18,19,20]. These investigations have been further extended to incorporate the possible retaliating behavior of prey under predator pressure [21,22,23].

Other models accounting for the spread of diseases have been investigated. These partition the population among different classes. Among them, at least the susceptible and the diseased individuals should be accounted for. Many other variations are possible, depending on the infection considered, its mode of transmission and other features of the population being affected.

In more recent years, the two fields, mathematical population theory and mathematical epidemiology, have merged, and thus models of the spread of transmissible diseases in interacting populations have been constructed and investigated. Such models have originated the field of ecoepidemiology.

The presence of epidemics in the context of herding populations has been formerly introduced and studied [24,25,26,27,28]. In this paper, we continue these investigations formulating a model that introduces a disease affecting prey gathering in herds. In particular, we consider two alternatives: one in which the disease is transmitted directly upon contact among the infected and susceptible prey, and the second contemplates a pathogenic agent possibly present in the environment.

2. Materials and Methods

The main feature of the models that are considered in the following is that the prey gather together to form a herd. Individualistic predators hunt them. However, coming close to the prey herd, as prospective animals to be attacked, predators mainly spot individuals on the herd’s boundary. The number of the latter is therefore proportional to the herd perimeter length, which is a one-dimensional manifold.

The area that the herd occupies is a two-dimensional manifold, and it is proportional to the prey population size. The relationship between the perimeter $\pi$ and the area $\alpha$ of a square is expressed by a square root term, $\pi =\sqrt{\alpha }$. For more complicated planar figures, this equation slightly changes. A similar relationship therefore exists among the corresponding population occupying the boundary and the whole population in the herd. This remark is used in the models to modify the classical one-to-one predation term. If A denotes the predators and B, the prey, this interaction is thus of the form $AB$, a mass-action term. However, in view of the previous considerations, this becomes $A\sqrt{B}$.

2.1. The Model for Direct Disease Transmission among Individuals

Denoting the predators by P, we partition the prey among the susceptible S and infected I and consider the following system

$$\begin{array}{ccc}\hfill \frac{dS}{dt}& =& rS-mS-c_{SS}{S}^2-c_{SI}SI-\frac{a\sqrt{S}P}{1+bS}-\beta SI,\hfill \\ \hfill \frac{dI}{dt}& =& \beta SI-\mu I-\frac{aIP}{1+bI},\hfill \\ \hfill \frac{dP}{dt}& =& sP\left(1-\frac{P}{H}\right)+\frac{ea\sqrt{S}P}{1+bS}+\frac{eaIP}{1+bI}.\hfill \end{array}$$

(1)

The first equation considers reproduction, that is allowed only for the susceptible. This is modeled by the first term. Here, we assume implicitly that either the disease renders mating impossible or that the induced disease mortality kills the individuals before they can reproduce. This occurs, for example, in the EHBS (European Brown Hare Syndrome) affecting the European hare [29]. The effect of natural mortality represents the second term in the equation. Susceptible prey experience intraspecific competition from other susceptible prey as well as from the infected ones—the third and fourth terms.

The next term accounts for the damage caused by the hunting predators. Note that the numerator of the fraction represents the prey captured by predators on the edge of the herd. This is explained in detail in the preamble of this section. Instead, the denominator models the fact that a large number of prey may induce the predator to hesitate in attacking because the prey could retaliate. For instance, a large herd of buffalo can dissuade an individual predator from even approaching. In sum, this term indicates reduced predator pressure. The last term denotes disease transmission at the rate $\beta$. Upon contact among an infected and a susceptible, the latter leaves their class and enters into the infected set.

The second equation shows the infected dynamics. The newly infected enter into it as stated above. They die by natural plus disease-related mortality at rate $\mu$—the second term in the equation. They are also hunted by the predators as expressed by the fractional term. Note that the last term differs from the corresponding one in the first equation. The reason is that the infected individuals are assumed to be left behind by the wandering herd in search of new pastures and are thus subject to predator attacks on a one-to-one basis. This is expressed by the mass action term in the numerator. The denominator also has a different meaning than in the first equation. It denotes predator satiation. Indeed, when a large number of (infected) prey are available, predators reach a point at which they do not feed any longer; a maximal daily food intake exists for each living organism.

The third equation models the predators. They are assumed to feed also on other food sources, not explicitly modeled in (1). This explains the logistic term. The remaining ones correspond to the hunting of susceptible and infected prey, as explained above, both scaled via a conversion factor e.

2.2. Analysis of the Model for Direct Disease Transmission among Individuals

A dynamical system is analyzed by establishing its stationary points. Their interpretation in ecological terms is shown in

Table 1. Equilibria $E_k=(S_k,I_k,P_k)$ of (1) and their ecological interpretation.

Equilibrium	Interpretation
$E_1=(0,0,0)$	system collapse
$E_2=(0,0,H)$	prey-and-disease-free environment
$E_3=((r-m)c_{SS}^{-1},0,0)$	disease-and-predator-free environment
$E_4=(S_4,0,P_4)$	disease-free environment
$E_7=\left(\frac{\mu }{\beta },\frac{\beta (r-m)-c_{SS}\mu }{\beta (\beta +c_{SI})},0\right)$	predator-free environment
$E_{*}=(S_{*},I_{*},P_{*})$	population coexistence with endemic disease

. The components of the equilibria are populations and thus must be nonnegative. This leads to the feasibility conditions that are summarized in

Table 2. Equilibria $E_k=(S_k,I_k,P_k)$ of (1) and their feasibility.

Equilibrium	Feasibility Conditions
$E_1=(0,0,0)$	-
$E_2=(0,0,H)$	-
$E_3=((r-m)c_{SS}^{-1},0,0)$	$r\ge m$
$E_4=(S_4,0,P_4)$	sufficient: (6)
$E_7=\left(\frac{\mu }{\beta },\frac{\beta (r-m)-c_{SS}\mu }{\beta (\beta +c_{SI})},0\right)$	$\beta r\ge \beta m+c_{SS}\mu$
$E_{*}=(S_{*},I_{*},P_{*})$	numerical

. In addition, even if feasible, in order to be attained, an equilibrium must be stable. These issues are now addressed.

2.2.1. Boundedness

From the susceptible equation, as long as $S>(r-m)C_{SS}^{-1}+1$, adding the first two equations of (1) and dropping all negative terms that do not involve $S^2$, it follows that $S^{\prime }\left(t\right)+I^{\prime }\left(t\right)<0$, and $S+I$ decreases steadily. Hence, there is a time $\widehat{t}$ beyond which $S\left(t\right)+I\left(t\right)<(r-m)C_{SS}^{-1}+1$ for all $t>\widehat{t}$. It follows that the prey population $S+I$ is bounded by some positive quantity M (hence, $S+I<M$) and by the nonnegativity of the populations, $S,I<M$. Dropping the quadratic term in the predator equation and using the above estimate, we find

$$P^{\prime }\left(t\right)\le (s+ea\sqrt{M}+eaM)P-\frac{s}{H}{P}^2.$$

Again, for $P>H+1$, the predators decrease because $P^{\prime }<0$. There is a time $\tilde{t}$ such that, for all $t>\tilde{t}$, we then have $P\left(t\right)<H+1$, showing the boundedness of the predators as well.

2.2.2. The System’s Equilibria and Their Feasibility

The following are easily established unconditionally feasible stationary points of (1): the origin, $E_1=(0,0,0)$ and the predator-only equilibrium $E_2(0,0,H)$. The healthy-prey-only point $E_3=((r-m)c_{SS}^{-1},0,0)$ is conditionally feasible, namely, for

$$r\ge m.$$

(2)

The predator-free equilibrium is

$$E_7=\left(\frac{\mu }{\beta },\frac{\beta (r-m)-c_{SS}\mu }{\beta (\beta +c_{SI})},0\right)$$

with feasibility condition

$$\beta r\ge \beta m+c_{SS}\mu .$$

(3)

For the disease-free point, we can solve the first and third equilibrium equations of (1) to obtain

$$P=\psi \left(S\right)=\sqrt{S}\frac{(1+bS)(r-m-c_{SS}S)}{a},P=\varphi \left(S\right)=H+\frac{eaH\sqrt{S}}{s(1+bS)}.$$

The coordinates of $E_4=(S_4,0,\psi \left(S_4\right))$ are obtained from the intersection of these two curves. We turn to the study of these two functions, mimicking what is reported in [23].

The function $\psi \left(S\right)$ has a negative zero at $b^{-1}$; however, in the domain of interest $S\ge 0$, it has a zero at the origin, another one at $S_3=(r-m)c_{SS}^{-1}$, which is positive only if (2) holds, and

$$\underset{S\to \infty }{\lim }\psi \left(S\right)=-\infty .$$

Thus, its feasible branch lies in the domain $0<S<S_3$. We differentiate,

$$\frac{d\psi \left(S\right)}{dS}=\frac{\mathsf{\Pi }\left(S\right)}{2a\sqrt{S}}=\frac{r-m+3[b(r-m)-c_{SS}]S-5c_{SS}b{S}^2}{2a\sqrt{S}}.$$

The denominator is always positive, and thus it does not influence the sign of the derivative and also the slope of the function. The derivative annihilates at the zeros of the numerator $\mathsf{\Pi }\left(S\right)$, which is a concave parabola, with $\mathsf{\Pi }\left(0\right)=r-m$. Note that, if $\mathsf{\Pi }\left(0\right)<0$, i.e., (2) is not satisfied, $\mathsf{\Pi }\left(S\right)<0$ for all $S>0$. This entails ${\psi }^{\prime }\left(S\right)<0$ for all $S>0$, i.e., the function $\psi \left(S\right)$ would start from the origin and decrease continuously, thus, attaining always negative values. Therefore, this case is not further considered because no feasible intersections of $\psi$ with $\varphi$ exist. Instead, $\mathsf{\Pi }\left(0\right)$ is positive whenever (2) holds. In such a case, one root must be positive and the other one negative. The roots are easily obtained solving the corresponding quadratic equation:

$$S_{\pm }=\frac{3[b(r-m)-c_{SS}]\pm \sqrt{9{[b(r-m)-c_{SS}]}^2+20c_{SS}b(r-m)}}{10c_{SS}b}.$$

(4)

Note that the discriminant is always positive; thus, the above zeros are always real as should be expected. Hence, in the domain of interest $\psi \left(S\right)$ is increasing in $0<S<S_{+}$ and decreasing for $S>S_{+}$.

For the function $\varphi \left(S\right)$, we find $\psi \left(0\right)=H$ as well as

$$\underset{S\to \infty }{\lim }\varphi \left(S\right)=H.$$

For its derivative, we have

$$\frac{d\varphi \left(S\right)}{dS}=\frac{eaH(1-bS)}{2\sqrt{S}{(1+bS)}^2}.$$

Hence, $\varphi \left(S\right)$ increases in $0<S<b^{-1}$ and decreases for $S>b^{-1}$. Thus, it raises from the level H at the origin up to a maximum value attained in $b^{-1}$ to monotonically return to the asymptotic level H.

To have the equilibrium $E_4$, the functions $\psi$ and $\varphi$ must intersect. As Figure 1 shows, these are not guaranteed to always exist. The Figure is drawn with the set of parameters

$$r=4.0,m=3.0,c_{SS}=0.1,b=0.3,a=0.8,e=0.4,s=0.4$$

(5)

with two different values of the predators’ carrying capacity, $H=4.0$ and $H=10.0$. The figure also shows that possible intersections arise in pairs, through a saddle-node bifurcation.

Figure 2 shows that equilibrium $E_4$ can indeed be achieved at a stable level. A sufficient condition for the intersections among $\psi$ and $\varphi$ to occur is given by the maximum of the former having a height larger than the value of the latter for the same abscissa—namely, recalling (4)

$$\psi \left(S_{+}\right)>\varphi \left(S_{+}\right).$$

(6)

No equilibrium exists if instead

$$\psi \left(S_{+}\right)<H.$$

(7)

Coexistence is investigated numerically and is stably achievable as Figure 3 shows, while Figure 4 contains persistent oscillations around it, that will be investigated later in the paper.

2.2.3. Local Stability Analysis

The Jacobian of the system (1) is given by

$$J=\left[\begin{array}{ccc}{J}_{11}& -c_{SI}S-\beta S& -\frac{a\sqrt{S}}{1+bS}\\ \beta I& \beta S-\mu -\frac{aP}{{(1+bI)}^2}& -\frac{aI}{1+bI}\\ \frac{eaP(1-bS)}{2\sqrt{S}{(1+bS)}^2}& \frac{eaP}{{(1+bI)}^2}& J_{33}\end{array}\right]$$

where

$$\begin{array}{c}\hfill J_{11}=r-m-2c_{SS}S-c_{SI}I-\beta I-\frac{aP(1-bS)}{2\sqrt{S}{(1+bS)}^2},\\ \hfill J_{33}=s\left(1-2\frac{P}{H}\right)+\frac{ea\sqrt{S}}{1+bS}+\frac{eaI}{1+bI}.\end{array}$$

The equilibria with $S=0$ entail singularities in some of the above terms, for which they need to be analyzed with care.

Equilibria with vanishing prey

We first consider $E_1$. Here, from the first equation of (1), we look at the dominant terms for $S,I,P\to 0$:

$$\begin{array}{c}\hfill \frac{dS}{dt}=\sqrt{S}\left[\sqrt{S}(r-m-c_{SS}S-c_{SI}I-\beta I)-\frac{aP}{1+bS}\right]\\ \hfill \approx \sqrt{S}\left\{\sqrt{S}[r-m-(c_{SI}+\beta )I]-aP\right\},\\ \hfill \frac{dI}{dt}=I[\beta S-\mu -\frac{aP}{1+bI}]\approx -\mu I<0,\\ \hfill \frac{dP}{dt}=P\left[s\left(1-\frac{P}{H}\right)+\frac{ea\sqrt{S}}{1+bS}+\frac{eaI}{1+bI}\right]\approx sP>0.\end{array}$$

(8)

Clearly, from the last estimate, unconditional instability follows.

For the equilibrium $E_2$ with nonvanishing P, we find estimates similar to the first two in (8),

$$\begin{array}{c}\hfill \frac{dS}{dt}=\sqrt{S}\left[\sqrt{S}(r-m-c_{SS}S-c_{SI}I-\beta I)-\frac{a(P-H+H)}{1+bS}\right]\\ \hfill \approx \sqrt{S}\left\{\sqrt{S}[r-m-(c_{SI}+\beta )I]-aH\right\}\approx -aH\sqrt{S}<0,\\ \hfill \frac{dI}{dt}=I[\beta S-\mu -\frac{aP}{1+bI}]\approx -(\mu +aH)I<0,\end{array}$$

as well as

$$\begin{array}{c}\hfill \frac{dP}{dt}=(P-H+H)\left[s\left(1-\frac{(P-H+H)}{H}\right)+\frac{ea\sqrt{S}}{1+bS}+\frac{eaI}{1+bI}\right]\\ \hfill \approx (P-H+H)\left[ea\sqrt{S}+eaI\right]\approx H\left[ea\sqrt{S}+eaI\right]>0\end{array}$$

(9)

because here $P\to H$ and $S,I\to 0^{+}$. Once again, $E_2$ is also always unstable.

Equilibria with nonvanishing prey

At $E_3$ the Jacobian becomes an upper triangular matrix, from which the eigenvalues are immediate:

$$-r+m,\beta S_3-\mu ,s+\frac{ea\sqrt{S_3}}{1+b{S}_3}>0,$$

so that this point is also unconditionally unstable.

The analysis of the point $E_4$ parallels the one performed in [23], since it is the same equilibrium point. In summary, the stability conditions arise from the negativity of the eigenvalue that is immediate, $J_{22}$ and the Routh–Hurwitz conditions on the remaining minor

$$\begin{array}{cc}\hfill \beta S_4& <\mu +a{P}_4,J_{11}\left(E_4\right)+J_{33}\left(E_4\right)<0,\hfill \\ \hfill & J_{11}\left(E_4\right)J_{33}\left(E_4\right)-J_{13}\left(E_4\right)J_{31}\left(E_4\right)>0,\hfill \end{array}$$

(10)

where

$$\begin{array}{c}\hfill J_{11}\left(E_4\right)=r-m-2c_{SS}{S}_4-\frac{a{P}_4(1-b{S}_4)}{2\sqrt{S_4}{(1+b{S}_4)}^2},\\ \hfill J_{33}\left(E_4\right)=s\left(1-2\frac{P_4}{H}\right)+\frac{ea\sqrt{S_4}}{1+b{S}_4}.\end{array}$$

Furthermore, $E_7$ is found to be always unstable. The Jacobian at this point factorizes to produce one explicit eigenvalue, which is

$$J_{33}\left(E_7\right)=s+\frac{ea\sqrt{S_7}}{1+b{S}_7}+\frac{ea{I}_7}{1+b{I}_7}>0.$$

Coexistence is investigated numerically. It can be attained; see Figure 3.

2.3. The Model for Contamination through the Environment

The system now introduced is based on (1) but also contains a pathogenic agent V that is present in the environment and possibly contributes to the spread of the infection. The model becomes

$$\begin{array}{ccc}\hfill \frac{dS}{dt}& =& rS-mS-c_{SS}{S}^2-c_{SI}SI-\frac{a\sqrt{S}P}{1+bS}-\beta SI-\lambda SV,\hfill \\ \hfill \frac{dI}{dt}& =& \beta SI+\lambda SV-\mu I-\frac{aIP}{1+bI},\hfill \\ \hfill \frac{dP}{dt}& =& sP\left(1-\frac{P}{H}\right)+\frac{ea\sqrt{S}P}{1+bS}+\frac{eaIP}{1+bI},\hfill \\ \hfill \frac{dV}{dt}& =& \gamma I-\lambda SV-\delta V.\hfill \end{array}$$

(11)

The main change here is represented by the introduction of the equation for the free pathogen. The fourth equation shows its dynamics. Pathogens are excreted by the infected individuals at rate $\gamma$, the first term, they are picked up at rate $\lambda$ by susceptible prey, the second term, and are washed out at rate $\delta$, the last term. During the process where prey intake pathogens, the susceptible individuals become infected. This mechanism is modeled via a simple mass action term, and we compare again the second term in this equation.

As they are intaken by the prey, pathogens disappear from the environment and migrate into the prey bodies, where they will replicate and infect the individual. This extra contamination term also appears in other equations of the model. In the first equation, it is detrimental for the susceptible prey; in the second equation, it represents an additional recruitment term for the infected. All other terms in the first three equations retain the meaning that they have in model (1).

2.4. Analysis of the Model for Contamination through the Environment

2.4.1. Boundedness

The considerations of Section 2.2.1 can be reproduced without changes in this case as well. For the viruses, neglecting the negative quadratic term involving susceptible prey, we have that, if $V>\gamma M{\delta }^{-1}+1$, the virus population continually decreases, and there is a time $t^{*}$ such that, for all $t>t^{*}$, $V\left(t\right)<\gamma M{\delta }^{-1}+1$, giving boundedness for this population as well.

2.4.2. The System’s Equilibria and Their Feasibility

Here, the equilibria are

$${\widehat{E}}_1=(0,0,0,0),{\widehat{E}}_2=(0,0,0,H),{\widehat{E}}_3=\left(\frac{r-m}{c_{SS}},0,0,0\right),$$

which are easily determined. The former two are always feasible, and the feasibility conditions of the latter are given by (2). Other possible points that are admissible are ${\widehat{E}}_4=({\widehat{S}}_4,0,{\widehat{P}}_4,0)$, ${\widehat{E}}_{15}=({\widehat{S}}_{15},{\widehat{I}}_{15},{\widehat{P}}_{15},0)$ and coexistence ${\widehat{E}}_{*}=({\widehat{S}}_{*},{\widehat{I}}_{*},{\widehat{P}}_{*},V_{*})$.

Now, ${\widehat{E}}_{15}$ is not examined, because as will be seen in the next section, it is unconditionally unstable, even if feasible. Coexistence is investigated numerically.

We study instead the disease-and-virus-free equilibrium ${\widehat{E}}_4$. The two equilibrium equations for $S^{\prime }=0$ and $P^{\prime }=0$ can, respectively, be rewritten as

$$P=\mathsf{\Phi }\left(S\right)=\frac{s}{H}\left[s+\frac{ea\sqrt{S}}{1+bS}\right],$$

(12)

while, from the second one, we find

$$\frac{a\sqrt{S}}{1+bS}=\frac{s}{e}P\left(1-\frac{P}{H}\right)$$

and upon substitution into the other one, we are ultimately led to the conic section

$$\mathsf{\Psi }(S,P)=(r-m)S-c_{SS}{S}^2-\frac{s}{e}P\left(1-\frac{P}{H}\right)=0.$$

(13)

To understand its nature, assuming nondegeneracy, we calculate the invariant

$$-c_{SS}\frac{s}{eH}<0,$$

and its negative sign shows that it is a hyperbola. Its intersections with the coordinate axes are the origin and the points

$$Q_H=(0,H),Q_0=({\widehat{S}}_3,0),{\widehat{S}}_3=\frac{r-m}{c_{SS}}.$$

Now, if $r<m$, from the first equation in (11), we have $S^{\prime }<0$ and ultimately $S\to 0$ so that ${\widehat{E}}_4$ cannot be attained. We thus consider (2) in what follows. Thus, ${\widehat{S}}_3>0$. Implicit differentiation with respect to S leads to

$$\frac{d\mathsf{\Psi }}{dS}(S,P)=r-m-2c_{SS}S-\frac{s}{e}\frac{dP}{dS}+\frac{2s}{eH}P\frac{dP}{dS}=0.$$

(14)

Thus,

$$\frac{d\mathsf{\Psi }}{dS}(0,0)=r-m-\frac{s}{e}{\left.\frac{dP}{dS}\right|}_{S=0}=0$$

gives

$${\left.\frac{dP}{dS}\right|}_{S=0,P=0}=\frac{r-m}{s}e>0.$$

This rules out that the three points $Q_H$, O and $Q_0$ lie on the same branch of the hyperbola and that $Q_H$ and $Q_S$ lie on the same branch because, in such cases, the slope at the origin would be negative. Thus, two points must lie on one branch with the other point on the other branch. Furthermore,

$$\frac{d\mathsf{\Psi }}{dS}(0,H)=r-m-\frac{s}{e}\frac{dP}{dS}+\frac{2s}{e}\frac{dP}{dS}=0,$$

so that at this point on the vertical axis, we have

$${\left.\frac{dP}{dS}\right|}_{S=0,P=H}=-\frac{r-m}{s}e<0.$$

Now,

$$\frac{d\mathsf{\Psi }}{dS}({\widehat{S}}_3,0)=r-m-2c_{SS}{\widehat{S}}_3-\frac{s}{e}\frac{dP}{dS}=0$$

from which

$${\left.\frac{dP}{dS}\right|}_{S={\widehat{S}}_3,P=0}=-\frac{r-m}{e}s<0.$$

The evaluation of

$$\mathsf{\Psi }(\tilde{S},H)=(r-m)\tilde{S}-c_{SS}{\tilde{S}}^2=0,\tilde{S}\in \left\{0,\frac{r-m}{c_{SS}}\right\}$$

gives $\mathsf{\Psi }(0,H)=\mathsf{\Psi }({\widehat{S}}_3,H)$; thus, for the abscissa ${\widehat{S}}_3$, there are two values of $\mathsf{\Psi }$, 0 and H. Now, there could be two alternatives:

(a)

either a branch of $\mathsf{\Psi }$ goes through O and $Q_H$ and another one joins $Q_S$ and $({\widehat{S}}_3,H)$ or,

(b)

alternatively, one branch goes through O and $Q_S$ and another one joins $Q_H$ and $({\widehat{S}}_3,H)$.

Seeking the point for which the tangent to $\mathsf{\Psi }$ is possibly horizontal, i.e., $P^{\prime }\left(S\right)$ vanishes, from (14), we find

$$S_{{\mathsf{\Psi }}_M}=\frac{r-m}{2c_{SS}}.$$

Thus, the point with this abscissa is well-defined for $\mathsf{\Psi }$; hence, alternative (b) holds true. The points corresponding to the extrema of $\mathsf{\Psi }$ are then obtained by substitution of $S_{{\mathsf{\Psi }}_M}$ into the equation of the hyperbola (13), thereby, obtaining

$$P\left(S_{{\mathsf{\Psi }}_M}\right)=P_{\pm }=\frac{H}{2}\left[1\pm \sqrt{1-\frac{e{\widehat{S}}_3}{sH}(r-m)}\right]$$

and recalling the condition (2), we find $0<P_{-}<P_{+}<H$ in line with the expectations of alternative (b).

Thus, in summary, all these results imply that O and $Q_S$ are on the same concave branch, positive in $[0,{\widehat{S}}_3]$, which attains a maximum height $P_{-}$ at $S_{{\mathsf{\Psi }}_M}$. The other convex one emanates from $Q_H$, begins descending to reach the minimum value $P_{+}<H$ at $S_{{\mathsf{\Psi }}_M}$ and then raises up to cross the point $({\widehat{S}}_3,H)$, remaining below the height H in the whole interval $0<S<{\widehat{S}}_3$ and then tending to infinity as $S\to +\infty$.

To investigate $\mathsf{\Pi }\left(S\right)$ instead, we calculate the derivative to find

$${\mathsf{\Pi }}^{\prime }\left(S\right)=\frac{eaH}{s{(1+bS)}^2}\left(\frac{1+bS}{2\sqrt{S}}-b\sqrt{S}\right)=\frac{eaH}{s{(1+bS)}^2}\frac{1-bS}{2\sqrt{S}},{\mathsf{\Pi }}^{\prime }\left(0\right)=+\infty ,$$

which annihilates only once for

$$S_{{\mathsf{\Pi }}_M}=\frac{1}{b}.$$

Note that, at this point, the hyperbola $\mathsf{\Psi }(S_{{\mathsf{\Pi }}_M},P)=0$ attains the values

$$P^{\pm }=\frac{H}{2}\left[1\pm \sqrt{1-\frac{4e}{b^{2}sH}[(r-m)b-c_{SS}]}\right],$$

so that, for ${\widehat{S}}_3>b^{-1}$, we find $0<P^{-}<P^{+}<H$, while for ${\widehat{S}}_3<b^{-1}$ we find $P^{-}<0$, $P^{+}>H$. Further,

$$\mathsf{\Pi }\left(0\right)=H,\underset{S\to +\infty }{\lim }\mathsf{\Pi }\left(S\right)=H^{+}.$$

Combining these results, the function $\mathsf{\Pi }$ raises up from the point $Q_H$ to a maximum at $S_{{\mathsf{\Pi }}_M}$, then decreasing steadily toward the horizontal asymptote $P=H$. Thus, it remains always above the asymptote.

The equilibrium point is the possible intersection of these two curves. The lower branch of $\mathsf{\Psi }$ lies always below the height H, and thus it cannot intersect $\mathsf{\Pi }$. The upper one instead lies below H in $[0,{\widehat{S}}_3]$ and then raises up to infinity. Thus, an intersection ${\widehat{E}}_4$ with $\mathsf{\Pi }$ occurs, and furthermore we have the bounds

$${\widehat{S}}_4>{\widehat{S}}_3,{\widehat{P}}_4>H.$$

2.4.3. Local Stability Analysis

The Jacobian of (11) is

$$\widehat{\mathbf{J}}=\left[\begin{array}{cccc}{\widehat{J}}_{11}& -c_{SI}S-\beta S& -\frac{a\sqrt{S}}{1+bS}& -\lambda S\\ \beta I+\lambda V& \beta S-\mu -\frac{aP}{{(1+bI)}^2}& -\frac{aI}{1+bI}& \lambda S\\ \frac{eaP(1-bS)}{2\sqrt{S}{(1+bS)}^2}& \frac{eaP}{{(1+bI)}^2}& {\widehat{J}}_{33}& 0\\ -\lambda V& \gamma & 0& -\lambda S-\delta \end{array}\right]$$

(15)

where

$$\begin{array}{cc}\hfill {\widehat{J}}_{11}& =r-m-2c_{SS}S-c_{SI}I-\frac{aP(1-bS)}{2\sqrt{S}{(1+bS)}^2}-\beta I-\lambda V,\hfill \\ \hfill {\widehat{J}}_{33}& =s(1-2\frac{P}{H})+\frac{ea\sqrt{S}}{1+bS}+\frac{eaI}{1+bI}.\hfill \end{array}$$

Equilibria with nonvanishing prey

It is immediately seen that the point ${\widehat{E}}_{15}$ is unstable. Indeed, from (15) evaluated at this equilibrium, the eigenvalue ${\widehat{J}}_{33}$ is immediate, and we found that it is positive.

$${\widehat{J}}_{33}\left({\widehat{E}}_{15}\right)=s+\frac{ea\sqrt{{\widehat{S}}_{15}}}{1+b{\widehat{S}}_{15}}+\frac{ea{\widehat{I}}_{15}}{1+b{\widehat{I}}_{15}}>0.$$

We investigate now the disease-and-virus-free point ${\widehat{E}}_4$.

As for the stability of ${\widehat{E}}_4$, from (15) evaluated at this equilibrium, the Jacobian factorizes into the product of two minors of order two—namely,

$${\widehat{\mathbf{J}}}_{\mathbf{4}}^{\left(\mathbf{1}\right)}=\left[\begin{array}{cc}{\widehat{J}}_{11}& -\frac{a\sqrt{{\widehat{S}}_4}}{1+b{\widehat{S}}_4}\\ \frac{ea{\widehat{P}}_4(1-b{\widehat{S}}_4)}{2\sqrt{{\widehat{S}}_4}{(1+b{\widehat{S}}_4)}^2}& {\widehat{J}}_{33}\end{array}\right]{\widehat{\mathbf{J}}}_{\mathbf{4}}^{\left(\mathbf{2}\right)}=\left[\begin{array}{cc}\beta {\widehat{S}}_4-\mu -a{\widehat{P}}_4& \lambda {\widehat{S}}_4\\ \gamma & -\lambda {\widehat{S}}_4-\delta \end{array}\right].$$

(16)

For the first minor, the Routh–Hurwitz conditions can be written as

$${\widehat{J}}_{11}+{\widehat{J}}_{33}<0,{\widehat{J}}_{11}{\widehat{J}}_{33}-{\widehat{J}}_{13}{\widehat{J}}_{31}>0.$$

Letting

$$\begin{array}{c}\hfill Z=r-m-2c_{SS}{\widehat{S}}_4<0,W=a{\widehat{P}}_4\frac{1-b{\widehat{S}}_4}{2\sqrt{{\widehat{S}}_4}{(1+b{\widehat{S}}_4)}^2},\\ \hfill Y=s\left(1-\frac{2}{H}{\widehat{P}}_4\right)<0,X=\frac{a{\widehat{S}}_4}{1+b\sqrt{{\widehat{S}}_4}}>0,\end{array}$$

they explicitly become

$$\begin{array}{c}\hfill r-m+eX+W<2c_{SS}{\widehat{S}}_4+\frac{s}{H}\left({\widehat{P}}_4-H\right),\end{array}$$

(17)

$$\begin{array}{c}\hfill ZY+W(eX-Y)>-eXZ+eXW.\end{array}$$

(18)

Note that W is negative for ${\widehat{S}}_3>b^{-1}$, as ${\widehat{S}}_4>{\widehat{S}}_3$ while, conversely, its sign is uncertain. The Routh–Hurwitz conditions on the second minor are:

$$\beta {\widehat{S}}_4<\mu +a{\widehat{P}}_4+\lambda {\widehat{S}}_4+\delta ,(\beta {\widehat{S}}_4-\mu -a{\widehat{P}}_4)(\lambda {\widehat{S}}_4+\delta )<\gamma \lambda {\widehat{S}}_4.$$

(19)

At ${\widehat{E}}_3$, two eigenvalues are immediate—namely, ${\widehat{J}}_{11}$ and ${\widehat{J}}_{33}$—from which, we find

$${\widehat{J}}_{11}=r-m-2c_{SS}{\widehat{S}}_3,{\widehat{J}}_{33}=s+\frac{ea\sqrt{{\widehat{S}}_3}}{1+b{\widehat{S}}_3}>0,$$

and instability is ensured by the last result.

Again, as for model (1), the equilibria with $S=0$ contain singularities in the Jacobian and are examined separately.

Equilibria with vanishing prey

Consider $E_1$. From the first equation of (11), we consider the dominant terms for $S,I,P,V\to 0$:

$$\begin{array}{c}\hfill \frac{dS}{dt}=\sqrt{S}\left[\sqrt{S}(r-m-c_{SS}S-c_{SI}I-\beta I-\lambda V)-\frac{aP}{1+bS}\right]\\ \hfill \approx \sqrt{S}\left\{\sqrt{S}[r-m-(c_{SI}+\beta )I]-\lambda V-aP\right\},\\ \hfill \frac{dI}{dt}=I[\beta S-\mu -\frac{aP}{1+bI}]+\lambda SV\approx -\mu I<0,\\ \hfill \frac{dP}{dt}=P\left[s\left(1-\frac{P}{H}\right)+\frac{ea\sqrt{S}}{1+bS}+\frac{eaI}{1+bI}\right]\approx sP>0,\\ \hfill \frac{dV}{dt}=\gamma I-\lambda SV-\delta V\approx \gamma I-\delta V.\end{array}$$

(20)

From the third estimate, this equilibrium is always unstable.

The equilibrium $E_2$ with nonvanishing P gives similar estimates to (20),

$$\begin{array}{c}\hfill \frac{dS}{dt}=\sqrt{S}\left[\sqrt{S}(r-m-c_{SS}S-c_{SI}I-\beta I)-\frac{a(P-H+H)}{1+bS}\right]\\ \hfill \approx \sqrt{S}\left\{\sqrt{S}[r-m-(c_{SI}+\beta )I]-aH\right\}\approx -aH\sqrt{S}<0,\\ \hfill \frac{dI}{dt}=I[\beta S-\mu -\frac{aP}{1+bI}]\approx -(\mu +aH)I<0,\\ \hfill \frac{dV}{dt}=\gamma I-\lambda SV-\delta V\approx \gamma I-\delta V\end{array}$$

except for the third one, which is exactly (9), as in the first model (1), because no viruses appear in it. Thus, $E_2$ is unconditionally unstable too.

Coexistence is shown to be achieved numerically.

3. Results

3.1. Results of the Model for Direct Disease Transmission among Individuals

Table 3. Equilibria $E_k=(S_k,I_k,P_k)$ of (1) and their stability properties.

Equilibrium	Stability Conditions
$E_1=(0,0,0)$	unstable
$E_2=(0,0,H)$	unstable
$E_3=((r-m)c_{SS}^{-1},0,0)$	unstable
$E_4=(S_4,0,P_4)$	(10)
$E_7=\left(\frac{\mu }{\beta },\frac{\beta (r-m)-c_{SS}\mu }{\beta (\beta +c_{SI})},0\right)$	unstable
$E_{*}=(S_{*},I_{*},P_{*})$	numerical

summarizes the stability findings for (1).

Thus, from this theoretical analysis, the only possible outcomes at steady state are the disease-free environment at $E_4$ or population coexistence with endemic disease at $E_{*}$. These points can indeed be achieved, as shown in Figure 2 and Figure 3, using suitable sets of parameters. Specifically, those for $E_4$ are

$$\begin{array}{c}\hfill r=4.0,m=0.3,c_{SS}=0.1,c_{SI}=0.1,a=0.2,b=0.3,\\ \hfill \beta =0.01,\mu =0.5,s=0.4,e=0.4,H=20,\end{array}$$

(21)

with initial conditions

$$\begin{array}{c}\hfill S=40,I=2,P=10.\end{array}$$

(22)

The parameters for which $E_{*}$ is attained are

$$\begin{array}{c}\hfill r=0.8,m=0.03,c_{SS}=0.001,c_{SI}=0.001,a=0.002,\\ \hfill b=30,\beta =0.001,\mu =0.23,s=0.4,e=0.4,H=20,\end{array}$$

(23)

with the same initial conditions (22).

In addition, the numerical exploration of the system behavior shows that persistent oscillations are found, Figure 4. They occur for the following parameter values:

$$\begin{array}{c}\hfill r=2.5497,m=0.2719,c_{SS}=0.5454,c_{SI}=0.2068,a=1.6460,\\ \hfill b=10.9755,\beta =0.7620,\mu =0.7131,s=0.1819,e=0.9064,H=1.2606,\end{array}$$

(24)

and initial conditions

$$\begin{array}{c}\hfill S\left(0\right)=0.136685,I\left(0\right)=0.170899,P\left(0\right)=0.013987.\end{array}$$

(25)

3.2. Results of the Model for Contamination through the Environment

Table 4. Equilibria ${\widehat{E}}_k=({\widehat{S}}_k,{\widehat{I}}_k,{\widehat{P}}_k,{\widehat{V}}_k)$ of (11) and their ecological interpretation.

Equilibrium	Interpretation
${\widehat{E}}_1=(0,0,0,0)$	system collapse
${\widehat{E}}_2=(0,0,H,0)$	prey-virus-and-disease-free environment
${\widehat{E}}_3=((r-m)c_{SS}^{-1},0,0,0)$	disease-virus-and-predator-free environment
${\widehat{E}}_4=({\widehat{S}}_4,0,{\widehat{P}}_4,0)$	disease-and-virus-free environment
${\widehat{E}}_{15}=\left({\widehat{S}}_{15},{\widehat{I}}_{15},0,{\widehat{V}}_{15}\right)$	predator-free environment
${\widehat{E}}^{*}=({\widehat{S}}^{*},{\widehat{I}}^{*},{\widehat{P}}^{*},{\widehat{V}}^{*})$	populations coexistence

gives an interpretation of the feasible outcomes of the system (11), while

Table 5. Equilibria ${\widehat{E}}_k=({\widehat{S}}_k,{\widehat{I}}_k,{\widehat{P}}_k,{\widehat{V}}_k)$ of (11) and their feasibility.

Equilibrium	Feasibility Conditions
${\widehat{E}}_1=(0,0,0,0)$	-
${\widehat{E}}_2=(0,0,H,0)$	-
${\widehat{E}}_3=((r-m)c_{SS}^{-1},0,0,0)$	$r\ge m$
${\widehat{E}}_4=({\widehat{S}}_4,0,{\widehat{P}}_4,0)$	$r\ge m$
${\widehat{E}}_{15}=\left({\widehat{S}}_{15},{\widehat{I}}_{15},0,{\widehat{V}}_{15}\right)$	?
${\widehat{E}}^{*}=({\widehat{S}}^{*},{\widehat{I}}^{*},{\widehat{P}}^{*},{\widehat{V}}^{*})$	numerical

summarizes all the feasibility results, and

Table 6. Equilibria ${\widehat{E}}_k=(S_k,I_k,P_k,V_k)$ of (11) and their stability properties.

Equilibrium	Stability Conditions
${\widehat{E}}_1=(0,0,0,0)$	unstable
${\widehat{E}}_2=(0,0,H,0)$	unstable
${\widehat{E}}_3=((r-m)c_{SS}^{-1},0,0,0)$	unstable
${\widehat{E}}_4=({\widehat{S}}_4,0,{\widehat{P}}_4,0)$	(17)–(19)
${\widehat{E}}_{15}=\left({\widehat{S}}_{15},{\widehat{I}}_{15},0,{\widehat{V}}_{15}\right)$	unstable
${\widehat{E}}^{*}=({\widehat{S}}^{*},{\widehat{I}}^{*},{\widehat{P}}^{*},{\widehat{V}}^{*})$	numerical

contains, instead, the stability requirements.

From the feasibility Table 5, it would seem that both ${\widehat{E}}_3$ and ${\widehat{E}}_4$ could coexist, giving rise to the possible insurgence of a bistable situation. In such a case, the outcome of the system toward one or the other equilibrium would be determined for the same model parameter choice by only the initial conditions. However, Table 6 shows that this is not possible as ${\widehat{E}}_3$ is unconditionally unstable.

The parameters showing that the feasibility and stability conditions for ${\widehat{E}}_4$ hold are those in (21) with the further requirements

$$\begin{array}{c}\hfill \lambda =0.002,\gamma =0.1,\delta =0.08\end{array}$$

(26)

and the initial conditions (22) to which we add

$$\begin{array}{c}\hfill V=30.\end{array}$$

(27)

The plot of Figure 5 shows the equilibrium ${\widehat{E}}_4$.

The parameters for showing the stable occurrence of coexistence ${\widehat{E}}^{*}$ are those in (23) but for

$$\begin{array}{c}\hfill \lambda =0.003,\gamma =0.1,\delta =0.08\end{array}$$

(28)

and the initial conditions (22)–(27).

Equilibrium ${\widehat{E}}^{*}$ appears in Figure 6.

Furthermore, through simulations, we discover the existence of a Hopf bifurcation because, for the following parameter values,

$$\begin{array}{c}\hfill r=4.0575,m=0.25,c_{SS}=0.099339,c_{SI}=0.26449,\\ \hfill a=0.021243,b=30.188,\beta =0.014957,\lambda =0.43197,\mu =0.31088,\\ \hfill s=0.53835,e=0.72692,\gamma =0.41323,\delta =0.45329,H=20.095,\end{array}$$

(29)

and initial conditions

$$S\left(0\right)=0.536685,I\left(0\right)=0.970899,P\left(0\right)=0.013987,V\left(0\right)=0.445127,$$

(30)

sustained oscillations for the populations other than the predators exist; see Figure 7.

In Figure 8, we show a bifurcation diagram in terms of the parameter m, the prey natural mortality. The Hopf bifurcation arises at the threshold $m^{†}=2.0$. The other parameters and initial conditions are, respectively, given in (29) and (30).

4. Discussion

The two models introduced in the paper have some specific features. They are of ecoepidemic type. This means that there are multiple interacting populations. In addition, a communicable disease must be present, affecting one or more of the populations. In our case, they are only two species, a predator and a prey. The epidemic spreads only in the latter. However, it cannot cross the species barrier and be transmitted to the predators. This can be the case in nature. For instance, foxes may be EHBS virus carriers without being affected by the syndrome. Models with these characteristics are now common in the literature. Among the early papers in this domain, see, e.g., [30].

In more recent years, the phenomenon of prey herding has also been considered. This implies that predation affects mainly the individuals situated on the boundary of the herd. This mathematically translates into the use of a fractional power function of the population instead of considering the whole population. All these features are present in the systems (1) and (11). The distinguishing feature of the ones introduced here with respect to former similar systems in the literature is the assumption that large numbers of prey are able to reduce the predator pressure.

The models (1) and (11) differ in formulation, as the former assumes only disease transmission via direct contact between individuals, while the latter additionally considers a pathogenic agent present in the environment. In both models, only two stable outcomes are possible. First is the disease-free (and virus-free, for the second model) point, (respectively, $E_4$ and ${\widehat{E}}_4$) where predators and only healthy prey thrive. Second is coexistence (respectively, $E_{*}$ and ${\widehat{E}}^{*}$), in which, again, the two animal populations persist with the disease becoming endemic in the ecosystem.

Another possible outcome is the onset of limit cycles, through a Hopf bifurcation that originates from the coexistence equilibria $E_{*}$ and ${\widehat{E}}^{*}$. We studied this occurrence via the bifurcation diagram of Figure 8. When the natural prey mortality decreases, the ${\widehat{E}}^{*}$ equilibrium loses its stability, and persistent oscillations arise. As their amplitude grows larger, the prey natural mortality becomes smaller. This means that the epidemic outbreaks become more acute if the disease-related mortality has a predominant effect on the prey.

Recall that the range of the bifurcation figure contains all possible ranges of the prey natural mortality m. However, m must be smaller than $\mu$, the natural-plus-disease-related mortality of the infected. Thus, the closer m is to zero, the larger the impact of the disease-related mortality. This can perhaps be explained by observing that a large impact of the disease-induced mortality drastically reduces the number of infected. In turn, this favors a rapid increase in susceptible prey. Consequently, both the predator and the infected populations are amplified, and then the cycle begins again.

Comparing the results for $E_4$ and ${\widehat{E}}_4$, obtained with the same set of basic parameters (21) additionally with (26), a slight reduction of the prey population and a corresponding increase in the predators is observed; see their final levels attained in Figure 2 and Figure 5. The time when the infected vanish is slightly larger in the case of the free virus because the latter take some time to disappear from the environment as seen in Figure 5 around 30.

Furthermore, the comparison of the population values in Figure 3 and Figure 6 obtained with the parameter values (23) together with (28) shows that the presence of pathogens in the environment has some influence in this case. The infected are slightly higher in the model with no virus, reaching values slightly above 250. When the virus is present, they attain values around 250. However, the number of susceptible individuals is much higher in the former case, above 200, while they settle to 150 in the presence of free pathogen. Thus, the addition of a viral agent in the environment does not appear to have a great consequence on the ecosystem behavior in these conditions, although it certainly contributes to reducing the number of healthy prey. Predators do not seem to be affected, as they always settle to the carrying capacity.

However, to better observe the influence of the free pathogen, we ran further simulations, increasing the rate at which the transmission among susceptible and infected prey occurs—namely, varying $\lambda$ and $\gamma$. Figure 9 and Figure 10 show the bifurcations arising in these cases. These graphs are drawn using the baseline values (23) as well as (28).

As expected, higher values of pathogen pickup from the environment significantly reduce both the free viral load and slightly the level of the susceptible prey, in the range $\lambda \in [0,0.002]$. Past the critical threshold ${\lambda }^{†}\approx 0.008$, population oscillations are triggered, leading to high disease prevalence due to increasing numbers of infected and susceptible that alternate between very high and very low values. Predators are essentially not affected by these dynamics, as they typically settle near their carrying capacity.

A corresponding effect is noted by changes in the parameter $\gamma$. In this case, however, the decrease in the susceptible prey is more prominent until the oscillation onset, while the same behavior occurs also for the infected. This is surprising because a higher discharge of the pathogen in the environment by the infected individuals reduces the negative effect on the whole prey population. In Figure 10, during this phase, the virus accumulates in the environment with a steady increase. In this situation, the onset of limit cycles occurs above the critical value ${\gamma }^{†}\approx 0.16$.

In summary, the environmental contamination has the effect of destabilizing the endemic equilibrium and may lead to large amplitude oscillations. These are well-known to be potentially harmful to ecosystems. Stochastic perturbations that are always present in natural environments may drive species to extinction when the populations attain levels close to the troughs of the oscillations.

The boundedness results on these models prevents population explosions. As the disease-free and the endemic equilibria cannot simultaneously be present, they must be globally asympotically stable, unless oscillations arise through the numerically discovered Hopf bifurcation.

5. Conclusions

We introduced two ecoepidemic models whose specific feature incorporates the possible herd retaliation of predator attacks. The prey are assumed to gather in herds, and this parallels other recent investigations in the literature. The two systems both contain the above characteristic features. They differentiate from each other in manner in which the disease is transmitted, either by direct contact only or, additionally, via environmental factors.

The two models presented here are of deterministic type. It is well-known that stochastic perturbations do arise in reality, which may affect the system outcome. The effects on the ecosystem may only be slight equilibria perturbations or may even be catastrophic. Indeed, if they arise during the oscillations when some of the population is at the trough, it may push to values close to zero. This, therefore, entails its disappearance. Stochastic models in this research field are present in the literature such as [31,32]. Others can also be constructed along these lines to better investigate such behavior and might constitute directions of further investigations.

The results of this investigation could provide some insight to the theoretical ecologists. The models are applicable to a wide variety of diseases spreading in the environment and affecting wild populations. The evolution of such a situation clearly depends on the prey and predators involved as well as on the infection mechanism. The models are of theoretical nature and are not tailored to a particular practical situation. However, in specific instances, for an assessed situation involving a disease and known species, the gathering of field data may help in determining the model parameters. Hence, the ultimate ecoepidemiological outcome can be simulated, thereby, providing useful indications for possibly containing the epidemic spread.