The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

In this research, we discuss the influence of an infectious disease in the evolution of ecological species. A computational predator-prey model of fractional order is considered. Also, we assume that there is a non-fatal infectious disease developed in the prey population. Indeed, it is considered that the predators have a cooperative hunting. This situation occurs when a pair or group of animals coordinate their activities as part of their hunting behavior in order to improve their chances of making a kill and feeding. In this model, we then shift the role of standard derivatives to fractional-order derivatives to take advantage of the valuable benefits of this class of derivatives. Moreover, the stability of equilibrium points is studied. The influence of this infection measured by the transmission rate on the evolution of predator-prey interaction is determined. Many scenarios are obtained, which implies the richness of the suggested model and the importance of this study. The graphical representation of the mathematical results is provided through a precise numerical scheme. This technique enables us to approximate other related models including fractional-derivative operators with high accuracy and efficiency.


Introduction
Infectious diseases arise when contaminated foreign bodies enter the human body. These foreign bodies are germs, viruses, fungi, or parasites. These bodies are transmitted through infection from another human being, animals, contaminated food, or from exposure to any of the environmental factors that are contaminated with any of these bodies.
These infections have many symptoms on the body, including high body temperature and pain, in addition to other symptoms that differ according to the site of infection, type, and severity of the infection. It is possible to have an infection that causes mild symptoms, and therefore it does not need to be treated. On the other hand, there are serious cases that may cause death. In some cases, they may affect the population balance of many species in the environment. In a worse scenario, some species may even become extinct due to some deadly diseases developed in some highly sensible populations. Mathematical models for predicting the evolution of species have been used in an increasing way in the last few decades. It started from Lotka and Volterra models [27,37], where their usefulness in avoiding many worst scenarios for numerous species as extinction was proved. Nowadays scientists use this tool for revealing the outcome of a certain strategy taken by some governments in dealing with some species, which can be considered as an important tool for conserving all the species existing nowadays.
The ecological species are most sensitive to any infection that appears, which can influence the evolution of some species, as an epitome. We consider a predator-prey interaction. This infection can affect the strength of some predators and the efficiency of hunting, which puts some predators in danger of extinction. In the literature, many works have investigated the predator-prey interaction in the presence of infectious diseases. For instance, please refer to the papers [9,25,26,41]. On the other hand, there are many strategies that the predators can consider for achieving a successful hunt. One of the most effective strategies is the predator hunting cooperation, where many predators gather to hunt a single prey. This method is very useful in reducing the hunting fail rate. Many hunters behave in this way. For example, lions, hyenas, and wild dogs are known for the highefficiency rate in hunting. The mathematical modeling of this specific behavior of predator was modeled for the first time in [22], where a simple model was used for describing such a cooperation. Until now, there have been a few works that study such a behavior in the predator-prey interaction, we mention as an example the works of [8, 10, 29-33, 38, 39].
Hence, it is worth investigating the influence of an infectious disease in the predator-prey interaction in the presence of predator hunting cooperation, which is going to be the principal subject of this research. We consider a three-species model that considers an infection developed in a prey population (which means that the prey population is divided into two species: the susceptible prey and the infected prey), and because of the opportunism of the predator, we can consider that the predator hunts the two types of prey. Based on the obtained results in [40], it is obtained that the time-fractional derivative has extensive applications in describing many real-life situations, which is known by memory effect of a dynamical system; memory rate is called for the order of the derivative, and the memory function for the kernel of the factional derivative. The time-fractional derivative is used for modeling many real-world phenomena, we mention as an example [20,23,24]. So its role is indispensable in the mathematical modeling of some ecological-epidemiological behaviors. To get more models of prey and hunter-modeled under the influence of a disease, we encourage interested readers to refer to the relevant literature in [7, 9, 10, 13-21, 23-25, 29, 31, 34-36, 39, 41]. Some other general problems can be found in [1][2][3][4][5][6]28].
Following the above-mentioned trends, we consider the following eco-epidemiological model: where D α is the Caputo derivative with respect to time given by where S(t), I(t), P(t) are the densities of susceptible prey, infected prey, and predator populations, respectively. In this model, r is the reproduction number of the prey population, and it is assumed that this infection cannot be transmitted vertically, in other words, mother-child, and the predator cannot be infected by this infection after a direct contact (predation) with infected prey. We use e to denote the conversion rate of prey biomass (infected or susceptible) into predator biomass. Moreover, μ is the death rate of the prey population, and m denotes the natural mortality of the predator population. The transmission rate of the prey population (infection rate) is described by δ. The functional (λ + aP)PS (resp. (λ + aP)PI) is the hunting cooperation functional [22]. Our main motivation in this article is to study the influence of infection on the evolution of ecological species. Further, we study the influence of δ on the evolution of species. Moreover, the influence of the hunting cooperation functional on the spread of this infectious disease is studied. Accordingly, the general structure of this article is as follows. In the next section, we establish some useful tools for dealing with the fractional-order derivative operator. The equilibrium points of the system are calculated in Sect. 3. Section 4 is devoted to studying the stability of these equilibrium points. A numerical scheme is used in the fifth section for plotting the solution of model (1), where some graphical representations are used for verifying the expected mathematical results. Finally, the concluding section ends the paper.

Equilibria of the model
In this section, the equilibrium points of system (1) along with their epidemiological relevancies are investigated. The equilibria of system (1) are the solution to the following system: Obviously, (S, I, P) = (0, 0, 0) is an equilibrium point of model (1) which models the extinction of the three species. Now we consider other particular cases.
Case 1 We seek equilibriums that have a zero density of predators (predator-free equilibriums), which are written as E 1 = (S 1 , I 1 , 0), which represents the extinction of the predator population. By replacing this result in the second equation of (2), we obtain S 1 = μ δ . By replacing these results in the first equation of (2), we get Hence, which exists if the following assumption holds: (H 1 ) (δS 1r > 0 and r > μ) or (δS 1r < 0 and r < μ).
Case 2 We look for equilibrium without infection (infection-free equilibrium), which is elaborated by the presence of the susceptible prey and predator population only. This equilibrium is expressed as E 2 = (S 2 , 0, P 2 ). Using the fact that I 2 = 0, system (2) becomes Multiplying the first equation of (3) by e and adding it to the second equation, we obtain e(rμ)S 2 -mP 2 = 0, hence, with r > μ. Replacing (4) in the second equation of system (3), we get Obviously, Eq. (5) has a unique positive solution The infection-free equilibrium exists if and only if the following condition holds: Case 3 In this case, we look for the positive equilibrium which models the existence of the three species. This equilibrium is indicated by E * = (S * , I * , P * ). Multiplying the first and second equations of system (2) by e, then adding them to the third equation, we obtain So, it gives where r > μ. Replacing (6) in the third equation of system (2), we obtain the following equation: By some straightforward calculations, we obtain Replacing (7) in (6), one gets Replacing P * in the second equation of system (2), one achieves Using the explicit expression of S * obtained in (9) along with (7) leads to For the epidemiological relevance of this equilibrium, we must have S * , I * , P * > 0. By taking a look at (8) and (9), we can guarantee that r > μ and m 2 + λμ > erλ are necessary and sufficient conditions for having S * , P * > 0. Now, we need an additional condition for guaranteeing the positivity of I * .
In fact, using (6) we have then I * > 0 if and only if Then we can summarize all the necessary existing conditions of the equilibrium point as follows: (H 3 ) r > μ, m 2 + λμ > erλ and δ > δ * .

The asymptotic behavior of equilibria
Our main interest in this section is to determine the local stability of the equilibria calculated in the previous section. For that purpose we calculate the Jacobian matrix of system (1) given by At the origin E 0 = (0, 0, 0), the Jacobian matrix (11) becomes Obviously, the Jacobian matrix (12) has the following eigenvalues: ϑ 1 = rμ < 0, ϑ 2 = -μ, ϑ 3 = -m. Note that ϑ 2 , ϑ 3 verify the condition | arg(ϑ i )| > απ 2 . Hence the eigenvalue ϑ 1 dominates the stability of the extinction equilibrium E 0 , hence we deduce the following results.

Lemma 1 If r < μ, then the equilibrium E 0 is locally stable and unstable if r > μ.
On the other hand, for the equilibrium point E 1 = (S 1 , I 1 , 0), the Jacobian matrix (11) is calculated as It is easy to see that ϑ 3 = eλ(S 1 + I 1 )m is an eigenvalue of the Jacobian matrix (13), hence the positivity of this quantity is expressed as Note that under the condition λ > λ 2 we have the instability of the equilibrium E 1 . Now, we presume that λ < λ 2 , which means that ϑ 3 < 0. This condition means that the other two eigenvalues of the Jacobian matrix (13) determine the stability (resp. instability) of the equilibrium E 1 . The remaining eigenvalues are the eigenvalues of the following matrix: Obviously, the trace and the determinant of matrix (14) are Using the fact that Det 1 < 1, we can deduce that one of the eigenvalues of matrix (14) verifies | arg(ϑ)| < απ 2 . Hence, the equilibrium E 2 = (S 2 , I 2 , 0) is always unstable. We resume the obtained results in the following lemma.
Note that in the case of the integer differential equation we deduce that this equilibrium is always unstable. But in the context of fractional differential calculus, this equilibrium point can be stable. In this regards, we presume that the eigenvalues of matrix (16) are written in the form ϑ = P + iQ, which verifies Hence we can deduce that the eigenvalue ϑ verifies the condition | arg(ϑ)| < απ 2 if the following condition holds: 4(λ + aP 2 )(λ + 2aP 2 ) > eaP 2 S 2 1 + tan 2 απ 2 .
Now we can resume the stability conditions of the equilibrium E 2 in the following lemma.
Now, let us study the stability of the interior equilibrium point. At this point, the Jacobin matrix is calculated as follows: By solving the following equation, the eigenvalues of the Jacobi matrix (17) can be determined as follows: Using the Routh-Hurwitz for fractional calculus, we arrive at the stability conditions of the positive equilibrium, which is resumed in the following theorem.
If we try to use the numerical method presented in formula (22) to solve problem (1), we obtain the following iterative schemes:

Graphical representations
In this section, we verify the obtained mathematical results using several graphical representations. The plots are offered in the following manner.

Conclusion
In this paper, we dealt with the influence of infectious disease on the evolution of an ecological species using a fractional-order model. The fractional-order derivative stands for the memory which is responsible for the ecological fluctuation generated by the environment. It is assumed that this infection is fatal neither for the prey (after the infection) nor for the predator (after consuming an infected prey). Also, it is considered that the predator has a social behavior deduced by the cooperation in hunting the two prey (healthy and infected ones). Indeed, we obtained that system (1) can undergo many scenarios, as the extinction of the three populations which is deduced by the existence of the equilibrium E 0 , the extinction of the predator population only without the infection. In this case, we have the persistence of the infection without the predator, which is deduced by the existence of the equilibrium E 1 . This scenario cannot exist because of the instability of the equilibrium E 1 , which is deducted from Lemma 3.2. Further, the extinction of the infection without predation is the most desired result, which represents the recovering of the ecological system from this infection. The stability of the infection-free equilibrium E 2 is the main result. We mention that in the case of the first-order derivative the equilibrium E 2 is always unstable because the eigenvalues of the Jacobian matrix verify Tr 2 > 0 and Det 2 > 0, but because of the expansion of the stability region generated by the time-fractional order derivative, we can have the stability of this equilibrium, which highlights the big influence of the memory on the ecological fluctuation. This result is supported numerically using graphical simulations plotted in Figs. 1 and 2. The existence of the three species is also possible where the stability of the equilibrium E * is possible. Figures 3, 4, 5, 6 confirm this result.