An eco-epidemic model with seasonal variability: a non-autonomous model

Seasonal variability strongly affects the animal population in wildlife. It becomes essential to model seasonality in eco-epidemic dynamics to know the effect of system parameters in a periodic environment. This article presents a set of non-autonomous differential equations with time-varying disease transmission rates among prey and predators, the mortality rate of a diseased predator, the predation rate of healthy prey, and an additional food supply. The positiveness, boundedness, and presence of solution are derived. We have proved that the infection-free state is stable if periodic basic reproduction number RC(t)<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_C(t)<1$$\end{document}. The stability of the coexistence state is shown at RC(t)>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_C(t)>1$$\end{document} using the Poincare map and comparison theory. The significance of the parameters related to disease transmission and prevalence is described using sensitivity analysis. Numerical simulation verified our analytical findings and proved that the predator control strategies in the periodic environment via controlling predation rate, disease transmission rate among predators, and death rate of diseased prey lead the system towards an infection-free environment.


Introduction
The population dynamic of different species is strongly affected by the seasonal variations [19]. The reproduction rate of the pathogen, the birth of species, emigration of birds, sunlight density, and dynamics of some diseases are the factors highly influenced by seasonal temperature deviation [17,18]. Anthrax, avian influenza, ebola, Escherichia coli are some zoonotic diseases with seasonality reported in wildlife and livestock [25]. The above facts reveal that periodic fluctuations significantly affect the system's dynamics. So, it becomes essential to examine the effect of seasonality on the behavior of population communities and model these diseases as periodically forced non-linear systems.
Eco-epidemic dynamics have been broadly introduced in population biology and achieved significant progress. Periodicity can be introduced to eco-epidemic systems via ordinary differential equations with time-varying parameters. The autonomous system with time-varying parameters becomes non-autonomous, showing a more complicated and natural phenomenon than the autonomous system [7,16]. More precisely, the permanent behavior of an eco-epidemic model can be studied through a non-autonomous model.
Thus, the non-autonomous eco-epidemiological system is required to be studied [17,19]. In the last few years, many scientists have made broad studies on the effect of seasonality on the different dynamics [17][18][19]. They have analyzed various seasonality sources like varying transmission rate, fluctuation in birthrates, vaccination rate, predation rate, death rate and, supply of additional food [17,19,26].
Disease control is a vital tool for the population balance of both prey and predator. Chemical control and biological control are two strategies that have been practiced for many years. Chemical control like pesticides (fungicides, bactericides, soil fumigants) are relatively fast-acting and reduce the damage done to crops. Nevertheless, these chemicals are not beneficial for the environment in the long term. They are toxic to target organisms and other organisms; the chemicals can pass through the food chain to the top predators or humans and harm them. Infected individual infects the surrounding; further, they get absorbed and remain as residue in the biomass of coexisting healthy species in the ecosystem [28]. Therefore, various researchers pay keen attention to later approaches for controlling infectious diseases.
The control strategies of infection rate, predation rate, and death rate (Culling or Hunting rate) play an essential part in the pray-predator dynamics. Some mathematicians discussed that the dynamics could be infection-free via appropriate management of predation rate and infection rate [17,21,22,36]. The disease can also be controlled via culling or hunting strategies in diseased prey or predator compartment [20].
The disease progression in ecosystems is a relevant area of research for its massive impact on the population. Research has shown that infected prey harvesting affects infection prevalence. Therefore, it is beneficial to consider disease progression while formulating a model for real-world ecosystems [1,13,24,29,34]. Some of the authors also developed model with both prey-predator are suffering from an infection [3,6,9,14,22]. It is discussed in [4,30] that healthy prey is more active than the infected one. Some articles [2,11] discussed models with the infection in the predator, where the diseased predator is incapable of predating on healthy prey [2]. Some researchers discussed the horizontal transmission of disease among predators [6,31]. The supply of additional food (non-prey food resources) to a predator is one of the biological control methods used in horticultural areas to intensify biological control levels. Some authors also consider this in their articles [10,27,28].
Based on these pieces of information, we are inclined to examine the outcome of predator regulation methods on diseased predator-prey dynamics with periodically varying additional food to predator, disease transmission rate, predation rate, and death rate. We have formulated a four-compartment non-autonomous model with infection in prey and predator to show the effect of these time-varying parameters.
The following Sect. 2 contains a set of four non-autonomous differential equations taking time-varying disease transmission rates, predation rate, and death rate representing diseased prey-predator dynamics. In Sect. 3, we study that all solutions are positive, and these solutions start and always remain in a bounded region for any initial conditions. Section 4 describes the classification and existence of steady states. We have done basic reproduction number and stability analysis in Sect. 5. Section 6 performs the sensitivity analysis to determine the impact of system parameters on reproduction number. Section 7 presents a numerical simulation to support our analytical findings. We have discussed and concluded the quantitative results in Sect. 8. This section also contains the biological interpretation of our results.

The systematic development of the model
The proposed system has two types of species; the prey and predator. We assumed an infectious disease in prey, and the disease of prey spreads in predators through predation. When there is no infection, prey and predator are denoted by X (t) and W (t), respectively. The present dynamics is developed based on the following assumptions: 1 When infection and predator are absent, the growth of prey follows the logistic differential equation dX dt = r X (1 − X K ), where r and K are given in Table 1. 2 The presence of infection decomposes the prey population X (t) into two categories: susceptible prey S(t) and infective prey I (t). We have considered that the disease is incurable, i.e., once a prey gets an infection, it will either die with disease-induced death rate δ 1 or be removed before having the possibility of reproducing. For this reason, we have not considered birth terms in infective class [24]. Thus, the logistic equation takes the form; dS dt = r S(1 − S K ). 3 In the absence of a predator, the infection spreads among healthy prey with a bilinear incidence rate γ S I , i.e., where γ is disease contact rate. 4 The infectious disease is transmitted in predators via predation of infected prey; the predator population W (t) can also be partitioned into two categories: Healthy predator P(t) and infected predator Y (t). The disease is also communicable among predators through direct transmission with Holling type-II functional response with the disease transmission rate σ . Here, we considered the disease transmission process identical to predation. We assumed that when a healthy predator comes in contact with an infected one, it could not be easily infected (encountered) due to its self-immunity. The time required to infect a healthy individual is considered handling time. For this reason, we used Holling type-II functional response as its used when a predator needs handling time to encounter the prey. 5 The healthy predator predates on both healthy and diseased prey with predation rates α and β respectively.
Here, the predation rate for infected prey is different from that of healthy prey because infected prey has restricted movement due to infection and can easily be encountered. Again, the conversion rates of healthy and infected prey to a healthy predator are 1 and 2 . The healthy predator dies with a natural death rate δ 2 . We assumed that infected predator predates only on infected prey [2,8] with predation rate ρ because the infected predator is weakened due to infection and cannot harvest the healthy prey. The conversion coefficient of the infected prey to the infected predator is 3 . All these interactions are with Holling type-I functional response. We further assumed that an infected predator never recovers and dies out with a disease death rate δ 3 . 6 The healthy predator also grows due to alternative food resources with a rate of b.
We have formulated the following non-autonomous model based upon the above assumption: The initial condition of model are All the parameters of the system are positive. The description of parameters and their range of values are shown in Table 1. We studied the parameter values of [4,23,24,29,30,33] and set parameter values of the model in such a way that all the value lies between the observed range except the value where we did not get the model solutions.
In the present model, disease transmission parameters γ (t) and σ (t), predation rate of healthy prey α(t), mortality rate of healthy and diseased predator δ 2 (t) and δ 3 (t) and rate of supplying additional food b(t) are considered as time-dependent parameters varying with one year period. We can write these parameters for all t ≥ 0 in the following way: Here, γ 0 and σ 0 are the average disease transmission rates between prey and predator respectively, α 0 is average predation rate, b 0 is the average rate of food supply to the predator and, (δ 2 ) 0 and (δ 3 ) 0 are the average death rate of healthy and diseased predator respectively. Further, ω 1 , ω 2 , ω 3 , ω 4 , ω 5 and ω 6 are relative amplitude of seasonal variation. The above-described parameter simulates the periodic fluctuations in the suggested system.
The subsequent Sect. 3 examined whether the solutions of the system (1) are positive and time-bounded with the initial conditions. The above means that the population of interacting species is always positive and grows logistically for a long time interval as resources in a particular patch are limited.

Positivity and boundedness
We consider that the initial conditions of model (1)

Theorem 3.1 If the initial conditions for model
In earlier case, all solutions are obviously positive. In later case, all solutions are also positive. To prove this, we take the contradiction that at least one of S (T ), Now, we write the equations one of the system (1) in the following form: Thus S (T ) > 0 ⇒ S (t) > 0 for any value of t > 0. Which contradict our assumption that S (T ) = 0, hence S (t) > 0, ∀t > 0. Now, consider I (T ) = 0, P (T ) = 0 and Y (T ) = 0 simultaneously. Again, the rest of equations of system (1) can be written as: this leads to following solutions,  Proof We represent the total population of the system (1) as The total derivative of this expression with respect to time t is, Since, 1 , 2 , 3 ≤ 1, therefore we get, The above expression indicates that as t → ∞, e −ωt → 0 which implies that, Hence, all populations of the model (1) starts and remain in bounded domain = {(S, I, P, Y ) ∈ R 4 + : The subsequent Sect. 4 reveals the steady-states and their existing conditions for biological relevance.

The classification and existence of periodic equilibrium
The equilibriums of non-autonomous system are time-varying equilibria determined by pullback attraction [15,18]. The model (1) has following periodic steady states: The periodic infected predator-free steady state E 5 (S, I, P, 0) where, exists when following conditions hold: The periodic coexistence steady state E 6 S ,Ï ,P,Ÿ .  (1) exists under the following conditions: Proof At the periodic endemic equilibrium E 6 S ,Ï ,P,Ÿ the model (1) can be express in the following form: On solving (2), (3) and (5), we obtain following values: Now, on substituting all these values in (4), we get the following nonlinear algebraic polynomial in P, Where, 3 .
We clearly see that F 3 is always positive. F 0 is a negative if condition (2) holds. Again, F 2 is negative when F 0 < 0 and the condition (1) holds. Finally, F 1 is negative, when F 0 and F 2 is negative and the condition (1) holds. Thus, there is only one sign change in the equation (7) if condition (1) and (2) holds. Hence, Eq. (7) has only one positive real solutionP. After having value ofP, we can easily find the value ofS,Ï andŸ from expressions (6). Thus, the system (1) has a unique periodic coexistence equilibrium if conditions (1) and (2) holds.

Basic reproduction number and stability
This section investigates the periodic basic reproduction number R C (t) and stability of the periodic disease-free state E 2 (S, 0,P, 0) and coexistence state E 6 (S,Ï ,P,Ÿ ).
The basic reproduction R 0 number is the measurement of severity of any disease. This number provides the total infective cases; one primary infective has, over its infectious period in an uninfected population. We determined the periodic basic reproduction number R C (t) through the next generation method [12].
The following two compartments of (1) are responsible for infection: Now, we define the matrices for new infections, i.e, F(t) and transfer terms, i.e., V (t) at disease-free equilibrium E 2 S , 0,P, 0 as follows: and, So that, The reproduction numberR 0 is the largest positive root of the following characteristic polynomial G(x) of matrix (12), Where, Here, R 0γ and R 0σ are the infections caused by prey and predators, respectively in the environment. Equation (13) always has at least one positive root as (R 0γ + R 0σ ) > 0. Now, whenR 0 = 1, the largest root of Eq. (13) will be 1. Thus, we see from Eq. (13) that, The above equation shows that R 0γ + R 0σ = 1 + R 0γ R 0σ which is true for R 0γ = 1 and R 0σ = 0. Thus, R 0γ + R 0σ = 1 if and only ifR 0 = 1. Hence,R 0 and R 0σ + R 0γ show identical behavior. Moreover, Hence, the periodic basic reproduction number R 0 of the system (1) can be considered as: Or, .
Again, we defined the matrix M(t) related to the healthy compartment of model (1) at disease-free equilibrium E 2 (S, 0,P, 0) with the help of lemma [35]: The monodromy matrix φ M (T ) of the T-periodic linear system, dZ dt = M(t)Z , is given as, Where, . Which always lesser than one for the existence of a disease-free equilibrium. Hence, disease-free equilibrium E 2 (t) is linearly asymptotically stable in space . Now, we consider the linear system (17) and used the approach [35] to analyzed the threshold dynamics of the eco-epidemiological system (1) periodically:

The spectrum of φ M (T ) is given as, Sp(φ M (T )) = {e
Where Y (t, s) and I represent the 2 × 2 square matrix and identity matrix respectively. The monodromy matrix of linear system (17) ∀t ≥ 0 is, Now, we see from the expression that λ 1 and λ 2 are always lesser than one for the existence of disease-free equilibrium. Hence, both the eigenvalues of the monodromy matrix (18) are lesser than one if R C (t) < 1. So that, ρ(φ −V (T )) < 1 for the disease-free state.
Let us consider (s) as the initial distribution of infected individuals, which is T-periodic in s. Then is the distribution of the new infections cumulated at time t produced by all infected individual (s) introduced with previous time t. let C T be the ordered Banach space of all T-periodic function of R to R 2 with the norm · ∞ on the positive cone, Then, we can define the linear operator L of C T → C T by, Now, the global asymptotic stability of disease-free equilibrium E 2 (t) will be examined through the following fundamental theorem of Wang and Zhao [35].

Lemma 5.2 (See [35], Theorem 2.2)
The following proposition is true where ρ(φ F−V (T )) indicates the spectral ray of the monodromy matrix of the system,

Theorem 5.3 The basic reproduction number R C (t) < 1 assures the asymptotic stability of T -periodic infection-free state E 2 of (1) while this state become locally unstable if R C
is defined by (15).
Proof The following expression shows the monodromy matrix of the (19), Here,

It is clear from (15) that to make R C (t) < 1, both γ (T )S − βP − δ 1 and σ (T )P (a+P)
− δ 3 (T ) must be lesser than zero. Thus, both eigenvalues of matrix (20) are lesser than one when R C (t) < 1. This concludes that ρ(φ (F−V ) (T )) < 1, when R C (t) < 1. Which enable us to conclude the local asymptotic stability of T-periodic infection-free state E 2 . Again, when R C (t) > 1, either of λ 1 and λ 2 must be positive. Thus one eigenvalue of monodromy matrix (20) is greater than one when This results the instability of the steady state E 2 .

Theorem 5.4
For any solution of (1), R C (t) < 1, guarantees the global asymptotic stability of periodic steady state E 2 (S, 0,P, 0) in the bounded region while R C (t) > 1 guarantees the instability of E 2 .
Proof Theorem 5.3 already described the stability of infection-free state E 2 when R C (t) < 1. So, it is only remains to that show that E 2 (t) is globally attractive for R C (t) < 1. Let us suppose that R C (t) < 1. We know that the set = {(S, I, P, Y ) ∈ R 4 + : (1) is positively invariant, then for all > 0, ∃ T 1 > 0 such as + , ∀t > T 1 . Now, we have following equations from the system (1) for t > T 1 : Subsequently, we assume the complementary system: We write this system in the formW (t) = M (t)W (t). WhereW (t) = (Ĩ (t),Ỹ (t)) and, Where, 1 It is clear that, We know that the spectral radius is continuous and lemma 5.

Theorem 5.5 If R C (t) > 1, the periodic coexistence state E 6 (S,Ï ,P,Ÿ ) of system (1) exist and there exist θ * > 0 in such a way that any solution S (t),P(t),Ï (t),Ÿ (t) with initial condition
Proof Clearly, the coexistence state E 6 (S,Ï ,P,Ÿ ) of system (1) defined in theorem (4.1) is periodic and positive for R C (t) > 1. For clarity, the arrangement of the compartments is done such that the last two compartments corresponds to diseased compartments (i.e., (S,P,Ï ,Ÿ )). Now, it is remains to show that ∃ θ * > 0 such as any solution (S,P,Ï ,Ÿ ) of (1) with initial condition (S 0 , P 0 , Suppose that: Let Poincare map P : X → X related to the system (1), such that It is clear from the definition of Poincare map that: , the first equation of the model (1) takes the following form, Now, with help of the value of S, I and Y in (6), we will change the Eq. (26) in the following form, Where, . We consider following perturbation equation for Eq. (27), Equation (28) lead to the following solution, with arbitrary initial conditionS(0, π). This system also have a single periodic solution, Where,S * (0, π) is given by,S * (0, π) = −Ae −At Bπ(1 − e −At ) .
Let us consider a fixed point M 0 ∈ ∂ X 0 Now, according to continuity of solution with initial condition, for any π > 0, ∃π * > 0 such as ∀ (S 0 , P 0 , I 0 , Y 0 ) ∈ X 0 , verifying (S 0 , P 0 , I 0 , Y 0 ) − M 0 ≤ π * . Thus, we have: Let us suppose that (29) does not hold good. Then ∃ at least one (S 0 , we will follow the process given in [18] and conclude that the periodic solutionS * (t, π) of (28) is global attractive on R + andS * (t, π) > S * − ε. Thus we have S(t) > S * − ε for sufficiently large t. The differential equations for diseased compartment of system (1) can be written in the following form for amply large t.

Sensitivity Analysis
Here, we performs the sensitivity analysis of the reproduction number R C using normalized forward sensitivity index method [5]. This is used to find out the impact of system parameter on reproduction number. For this analysis, we have taken parameter values shown in Table 2. When there is no seasonality i.e., Table 2 shows that for this set of parameter values, parameters r , K ,β, 1 , α 0 ,b 0 and a are indirectly proportional to R C and have negative impact on it while σ 0 , (δ 2 ) 0 and γ 0 are directly proportional to R C and have positive impact on it and, rest of all the parameters do not affect R C .
The incoming section 7 consists of the numerical simulation of the system (1), which is extremely important to verify our theoretical findings.

Numerical simulation
It is important to execute any system numerically to verify its theoretical results. We can find out how the parameters of any system affect the system with the help of numerical simulation. We have used hypothetical values based on a real scenario such as the predation rate of healthy prey S is less than that of infected prey I since diseased prey is weak and predator can easily catch them compared to one healthy prey [32]. Hence, ρ and β must be greater than α. Similarly, the conversion coefficient of healthy prey is assumed to be greater than the infected one. Moreover, we took the conversion coefficient of healthy predator is also greater than infected one.
The numerical simulation of proposed system reveals that for the default parameter set r = 6, K = 100, ρ = 0.4, α 0 = 0.01, β = 0.9, δ 1 = 0.1, (δ 2 ) 0 = 0.8, (δ 3 ) 0 = 0.65, 1 = 0.9, 2 = 0.85, 3 = 0.3, a = 25, b 0 = 0.1, γ 0 = 1.05,σ 0 = 1.55 the system (1) shows periodic oscillation around interior equilibrium which is shown in Fig. 1. If we increase the average predation rate of healthy prey α 0 to 0.4, keeping the rest of the parameter value the same as in Fig. 1, the diseased prey and predator die out, and we will get a disease-free environment as shown in Fig. 2. To know the effect of α on infected populations, we keep other parameters the same as in Fig. 1 and variate average predation rates of healthy prey α 0 . Figure 3 clearly shows this situation that increasing predation of healthy prey, periodic oscillation occurs and later the infected prey and predation extinct from the environment at α 0 = 0.33. Again, we see the effect of the death rate of diseased prey δ 3 on the infected population by variation in the average death rate of diseased prey (δ 3 ) 0 and keeping the rest of the parameters are the same as in Fig. 1. We can see in Fig. 4 that increment in δ 3 leads to periodic oscillation first, then the diseased population to die out at (δ 3 ) 0 = 1.4.
Finally, the effect of disease transmission coefficient σ on infected population can be seen if we variate average disease transmission rate σ 0 and keep the rest of the parameter set same as in Fig. 1. Figure 5 shows that a decrement in the disease transmission rate leads the infected community to die out at σ 0 = 0.7. In Fig.  6, we have shown the graphical representation of sensitivity analysis.
Incoming Sect. 8; we will discuss and conclude our theoretical and numerical findings and their conclusions. We also explain the ecological perception of our results.

Conclusion
We have investigated a diseased prey-predator non-autonomous model with infection in prey and predator. We have considered that disease transmission parameters σ and γ , the death rate of healthy and diseased prey δ 2 and δ 3 , rate of alternative food b, predation rate of healthy prey α are time-dependent parameters that vary seasonally with one year period. The non-autonomous system is analyzed to understand these parameter's effect on present prey-predator dynamics in a periodic environment. This article contains an onerous analysis applying the comparison theory and Poincare map theory. We have analyzed the system (1) and found that a periodic disease-free and coexistence state is locally and globally stable in the domain R 4 + . Moreover, the periodic basic reproduction number R C (t) lesser than one shows the local stability of the disease-free equilibrium, and R C (t) greater than one shows the instability of the same. Further, the periodic coexistence state is stable when R C (t) > 1. In the real world, the present analysis reveals that the predator's preference to predate on infected prey may lead the system to an endemic situation. Further, increasing predation of healthy prey leads the system toward a disease-free situation with many predators.
Consequently, predation of healthy prey helps healthy predator population grow, and predator chooses most contagious prey so that contagious prey is removed from the population, and only healthy prey remains, thereby preventing spreading disease [21,28]. Decrement of disease contact rate between predators also results in an infection-free environment as it helps to grow healthy predator who prefers to predate on infected prey resulting a diseased-free environment. We get similar conditions by increasing diseased predator's death rate, which again helps to grow healthy predator population resulting in the disease-free condition.     Fig. 1 Funding This study and all authors have received no funding.
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Conflict of interest
The authors declare that they have no conflict of interest.