Stability and bifurcation of a delayed prey-predator eco-epidemiological model with the impact of media

: In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What’s more, the correctness of the theoretical analysis is veriﬁed by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end.


Introduction
One of the main fields of mathematics in biology is to explore the interaction among species.The model can analyze population dynamics and predict the tendency.Considering the impact of diseases or the interaction between populations based on epidemiology, the model combining infectious diseases and population is called eco-epidemiological model.In 1989, Harder and Freedman [1] first simulated the spread of disease between species.They assumed that parasites spread to habitats through the infected prey, and susceptible predator were infected by eating the infected prey.Chattopadhyay and Bairagi [2] established a prey-predator model in which diseases are transmitted in prey populations, and first proposed "ecological epidemics".Since then, many scholars have investigated the predator-prey model with disease transmission in prey population in reference [2][3][4][5][6][7][8][9].Of course, some researchers have focused on the predator-prey model with disease in predator population in reference [10][11][12][13][14][15][16][17].For example, Li et al. [11] showed a spatial eco-epidemiological system with disease spread within the predator population in open advective environments, and obtained some dynamics of such model, such as, the impacts of advection rate on the net reproductive rate, sufficient conditions for the prevalence of disease and the local stability of disease-free attractor.Meng et al. [14] considered a food chain model with two infected predators, and discussed the stability of equilibrium and back bifurcation.
The prevention and control of infectious diseases is one of the top concerns of the health sector.The study found that there are lots of factors affecting the spread of the disease, including the interaction between susceptible and infected people (effective contact rate), media reports, vaccination, migration, etc.With the progress of economic globalization and technology, the dissemination of information is more convenient and faster, and the role of information in controlling the spread of infectious diseases is becoming more and more important.Therefore, many experts has paid attention to the infectious disease models.During the outbreak of infectious diseases, people received relevant information about the disease from media reports, and then took related measures to avoid infection, such as wearing masks, washing hands frequently, maintaining social distance and vaccinating [18][19][20][21].Hence, media reports can lead to change the personal behavior, and further affect the spread of infectious diseases.For example, in the early days of the outbreak of the coronavirus disease 2019  in December 2019, the media reported less on the epidemic and did not attract people's attention.Later, a large number of reports on the spread of the disease, which make people understand the main transmission routes of the COVID-19, and enhance the people's sense of self-protection and reduce the chance of infection [22].It can be seen that media reports play an important role in controlling the spread of the epidemic.
The media coverage is not only the most important factor affecting the spread of infectious diseases, but also is a very important issue.Due to a large number of infection cases, media reports may cause social panic, but they certainly reduce the probability of contact and transmission among susceptible populations, which is positive to control the spread of the disease [23].Thus, mathematical modeling plays an important role to understand the potential impact of media coverage on the spread of infectious diseases.After the outbreak of Severe Acute Respiratory Syndrome (SARS) in 2002, the impact of media reports on the spread of infectious diseases was introduced.Many scholars studied the psychological impact of the SARS outbreak on the public and the impact of the media on the control and prevention of SARS [24][25][26][27][28][29][30].In recent years, there are many studies on the media effect about the transmission of infectious diseases.For example, Liu et al. [26] described the impact of media reports on communication dynamics by introducing a media function e −a 1 E−a 2 I−a 3 H into the transmission coefficient, here E, I, H represent the number of reported exposed, infected and hospital individuals, respectively, a i (i = 1, 2, 3) is a constant.Cai and Li [31] considered a new incidence function that reflects the impact of media coverage on disease spread and control, and built a compartment model with an incidence rate.
Li and Cui [32] constructed an SIS epidemiological model with the incidence rate, here β 2 I m+I reflects the reduction in contact rate caused by media reports.Xiao et al. [33] used a piecewise smooth function β(t) = β 0 e −εM(t) to describe the media effect, here M(t) = max{0, p 1 I(t) + q 1 I q (t) + p 2 dI dt + q 2 dI q dt }, I(t) and I q (t), represent the infected and the isolated infected, respectively, p 1 , p 2 , q 1 and q 2 are non-negative parameters.
Next, a common assumption in the mediation functions that the influence of the media on the transmission is instantaneous, that is, the number of the infected people at a certain moment affects the transmission coefficient at that moment, resulting in a decrease in the incidence rate.However, there is a time delay in the impact of media coverage on transmission dynamics, which describes the timing of an individual's response to a reported infection.Song and Xiao [34] considered the time delay of media report in the infectious disease model and studied the global Hopf bifurcation.Mathur et al. [35] considered the impact of media coverage on the population balance and the size of the infected population.The media induced response function is established by combining the media influence and the predator.The model is as follows: where x(t), y(t) and z(t) are the densities of the prey, susceptible predator and infected predator, respectively.The term βe −mz(t) x(t) is the media induced response function, here β is the predation rate and the constant m is the media awareness corresponding to the predator population.When media awareness increases, the predation rate decreases exponentially.Lots of factors such as media coverage, lifestyle and population density may directly or indirectly affect the incidence.Therefore, the non-linear incidence β 1 − β 2 I m+I reflects the characteristics of media reports, where β 1 is the normal contact rate and β 2 is the minimum effective exposure rate reduced through media reports, and m is the half-saturation constant.Thus, β 2 I m+I reflects the reduced contact rate through media reports.Since media reports cannot completely block the spread of disease, we suppose that β 1 > β 2 .The infected people will reduce their contact with others in a certain area.The more infected people are reported, the less chance of the susceptible people contacting with the infected are.
On the basis of literature [35], we will introduce a nonlinear media influence term and study the impact of media on the spread of disease.The transition from the susceptible to the infected is not instantaneous, which takes some time to become the infected one.Thus, the lag stage in the model divides the population into two stages: early stage and late stage.It is assumed that early individuals can be transferred to the later.When the susceptible people eat the adulterated food, they are recorded as the early stage.Because the susceptible population turn into the infection needs a constant period, and then transfers to the later stage.
The remainder of the paper is organized as follows.In Section 2, we introduce an eco-epidemic model with media influence and present the positivity and boundedness of solutions.In Section 3, we give the basic reproduction number and discuss the asymptotical stability of the non-trivial equilibriums.We also study the existence of the endemic equilibriums and the stability of the positive equilibrium by using the stability switching criteria given by Kuang and Beretta [36].In Section 4, we obtain formulations to determine the direction of Hopf bifurcation and the stability of periodic solutions.In Section 5, some numerical simulations are carried out.In the last section, we give some discussions.

Model description
In this part, we will give a prey-predator model including adulterated food interacts with a human population and nonlinear media effects, where the adulterated food is the prey population and the human is the predator.In addition, predators are divided as susceptible predators and infected predators.It is assumed that humans can only be infected by eating the adulterated food, but are not infectious.Let X(t), S (t) and I(t) be the density of prey population, susceptible population and infected population at time t, respectively.
(A1) The prey population grows in a logistic form with an intrinsic growth rate r and the environmental carrying capacity K.The growth of susceptible predators is completely dependent on the prey.Thus, the growth of the susceptible population is proportional to predation with the proportionality constant k denoting the conversion rate and the natural mortality rate d 1 .(A2) The susceptible predation is affected by the information transmitted by the media.We use the nonlinear influence term β 1 − β 2 , here β 1 is the general predation rate, β 2 is the predation rate affected by the media, and m is the half-saturation constant reflecting the effect of media reports on disease transmission.The term β 2 I(t) m+I(t) indicates the impact of media reports on communication.When the awareness of media reports increases, the predation rate decreases.(A3) The term β 1 − β 2 I(t) m+I(t) X(t)S (t) represents predation, and then k β 1 − β 2 I(t) m+I(t) X(t)S (t) represents the growth of susceptible predators in the form of energy coverion.Moreover, since the predator is not instantaneous from the predation stage to the infection stage, it becomes infected after the delay τ.Thus, the term ke −d 1 τ f X(t − τ), S (t − τ), I(t − τ) represents the number of infected persons joining the susceptible predator at time t − τ.The term e −d 1 τ denotes the maximum survival probability, here 0 < e −d 1 τ < 1.
Based on the above assumptions, the flow chart of the model is as follows (see Figure 1): Thus, the corresponding model is given as (2.1) All the parameters are positive and their descriptions are given in Table 1.Suppose that The initial conditions of the model (2.1) are given as where the initial function φ belongs to the Banach space C. By the fundamental theory of functions differential equations [37], there is a unique solution (X(t), S (t), I(t)) of model (2.1) with initial conditions (2.2).

Parameters Descriptions Unit r
The intrinsic growth rate of prey time −1 K The carrying capacity of prey ind β 1 The general predation efficiency of the predator ind −1 time −1 β 2 The maximum effective predation efficiency after media influence ind −1 time −1 m The half-saturation constant reflecting the impact of media coverage on the spread of the disease − k The conversion rate The death rate of susceptible predator time −1 d 2 The death rate of infected time −1 τ Time period time

Preliminaries
In order to show that model (2.1) is epidemiologically meaningful, we will present some preliminaries, including positive invariance and boundedness of solutions.We have the following results.Proof.First, we give some symbols as follows:

Boundedness of solutions
Theorem 2.2.All positive solutions of model (2.1) with initial condition (2.2) are ultimately bounded.
Proof.Let (X(t), S (t), I(t)) be any solution of model (2.1) with initial condition (2.2).We suppose that Calculating the derivative of V along solutions of (2.1), we obtain where Then there exists an M > 0, depending only on the parameters of model (2.1), such that V(t) < M for all t large enough.Then X(t), S (t), I(t) have an ultimately bound.This proof is completed.

The basic reproductive number
Supposing that the right-hand sides of model (2.1) are zero, we have that It is straightforward to see that model (2.1) has a trivial equilibrium E 0 = (0, 0, 0) and a disease-free equilibrium E 1 = (K, 0, 0).Next, we use the next generation matrix method [39] to obtain the basic reproduction number R 0 of model (2.1).Here, we have the new infection matrix F (x) and the transition matrix V(x).Let x = (I, S ) T , then model (2.1) can be rewritten as: where The Jacobian matrices F (x) and V(x) at the disease-free equilibrium E 1 are, respectively, Therefore, the basic reproductive number R 0 is

Stability of the non-trivial equilibria
Let E( X, S , I) be any arbitrary equilibrium.The Jacobian matrix evaluated at E leads us to the following characteristic equation: where I is an identity matrix and The characteristic roots of Eq (3.4) are r, −d 1 and −d 2 .Since the characteristic root r is positive, E 0 is unstable.
Proof.The characteristic equation of model (2.1) at the disease-free equilibrium E 1 is Thus, the two eigenvalues of Eq (3.5) are that λ 1 = −r, λ 2 = −d 2 and the other are determined by , must intersect at a negative value of λ, and hence the disease-free equilibrium E 1 is locally asymptotically stable provided that R 0 < 1, i.e., for all τ ∈ (0, τ).Therefore, when R 0 < 1, the disease-free equilibrium E 1 is locally asymptotically stable, but the diseased-free equilibrium E 1 is unstable when R 0 > 1.

Existence of the endemic equilibrium
Theorem 3.2.If model (2.1) satisfies the following condition: then model (2.1) has the endemic equilibrium.
Proof.We assume that E * = (X * , S * , I * ) is a solution of Eq (3.1), that is, From the second and third equation of (3.7), we have that Substituting S * and I * into the first equation of (3.7), we have where According to the theory of the distribution of cubic equations roots [40], we have the following results. Let Thus, there are three cases for the existence of roots of (3.8).
(a) Equation (3.8) has a pair of complex roots and a real root when D > 0. The positivity condition of the real root is given by (b) When D = 0, Eq (3.8) has all real roots with two being equal.Moreover, if A > 0, then Eq (3.8) has only one positive real root X 1 = 2 3 − q 2 − A 3 ; if A < 0, then Eq (3.8) has only one positive real root X 1 = 2 3 − q 2 − A 3 when 3 − q 2 > A 3 , and there exist two positive roots when A 6 < 3 − q 2 < A 3 , defined as (c) If D < 0, then there are three distinct real roots given as 3 ), has to be calculated in radians.Furthermore, if A > 0, then there exists only one positive root.
Furthermore, the endemic equilibrium is E * (X * , S * , I * ), where Remark 1. Theorem 3.2 indicates that a unique endemic equilibrium exists, whenever the delay parameter cross a threshold τ > τ, here

Stability switches at the endemic equilibrium
In the following, we will use the stability switching criteria given by Kuang and Beretta [36], which provides practical guidelines for combining graphical information with analytical work to study effectively the local stability of those models involving delay-dependent parameters.Specifically, we will show that the stability of a given steady state is determined only graphically by a number of functions including delays which can be explicitly represented.
First, we give the stability switching criterion for the third-order characteristic equation.The characteristic equation of the model (2.1) can be rewritten as: where In order to use the criterion due to Kuang and Beretta [36], we need to verify the following conditions for all τ ∈ ( τ, +∞).
Let τ ∈ ( τ, +∞), it is easy to see that This implies that condition (1) is satisfied.And condition (2) is obviously true because From (3.11) and (3.12), we know that Thus, condition (3) is true.Let F(ω, τ) be defined as in condition (4).From and where It is obvious that condition (4) is satisfied.By implicit function theorem, condition ( 5) is also satisfied.Now let λ = iω(ω > 0) be a root of Eq (3.10).Substituting it into Eq (3.10) and separating the real and imaginary parts yields where we deliberately omit the dependence of the A i and B i on τ.Equations (3.15) and (3.16) can be written cos(ωτ) = Im P(iω, τ) Q(iω, τ) , and sin(ωτ A necessary condition for this model to have solutions is that the sum of the squares of the right hand side terms equals one.Adding the squares of both sides of (3.13) and (3.14), a necessary condition for the existence of solution is that That is, Let z = ω 2 .The polynomial function F can be written as Thus, Eq (3.18) is defined by h(z, τ) = 0. We set ∆(τ) = a 2 1 (τ) − 3a 2 (τ).When ∆(τ) > 0, we have that In the following, we will use the following lemma [41].
Then, θ(τ) ∈ [0, 2π] can be defined for τ ∈ I by Since F(ω, τ) = 0 for τ ∈ I, it follows that θ is well and uniquely defined for all τ ∈ I. Using (3.13) and (3.19), iω * with ω * = ω(τ * ) is a purely imaginary characteristic root of Eq (3.10) if and only if τ * is a zero point of the function S n (τ) defined by The following theorem is true due to Kuang and Beretta [36].We can easily observe that S n (0) < 0.Moreover, S n (τ) > S n+1 (τ) with n ∈ N for all τ ∈ I. Therefore, if S 0 has no root in I, then the functions S n have no root in I.If the function S n (τ) has positive zeros, denoted τ i n for some τ ∈ I, n ∈ N, then without loss of generality, we may assume that Applying the similar method as in [36], it is obtained that the stability switches occur at the zeros of S 0 (τ), denoted by τ i 0 .We can conclude the existence of a Hopf bifurcation which is stated in the next theorem.
(i) If function S 0 (τ) has no positive zero point in I, then the endemic equilibrium E * is locally asymptotically stable for all τ ≥ τ.

The direction of Hopf bifurcation and stability of periodic solutions
In the previous section, we obtained the conditions that the periodic solution bifurcates from the endemic equilibrium E * at the critical value τ * 1 .In this section, we follow the idea of Hassard et al. [42] derive an normal formula determining the properties of Hopf bifurcation at the critical values of τ * 1 .Thus, the direction, stability, and period of the periodic solution bifurcated from the endemic equilibrium E * are determined.We always assume that model (2.1) has purely imaginary roots ±iω * at τ = τ * 1 corresponding to the characteristic equation of the endemic equilibrium where where By the Riesz representation theorem, there exists a function η(θ, µ) of bounded variation for θ ∈ [−1, 0], such that In fact, we choose where δ is the Dirac delta function.For φ ∈ C 1 ([−1, 0], R 3 ), we define the operator A(µ) as Further, we can define the operator R(µ) as Then, we can rewrite model (4.1) as the following operator equation where ), the adjoint operator A 0 * of A 0 can be define as where η T is the transpose of η, and the domains of A 0 and , a bilinear from is given by where η(θ) = η(θ, 0).Then A 0 and A 0 * are a pair of adjoint operators.By the results obtained in the previous section, we know that ±iω * are the eigenvalues of A 0 .Thus they are also eigenvalues of A 0 * .What's more, we suppose that q(θ) is the eigenvector of A 0 corresponding to iω * , then A 0 q(θ) = iω * q(θ). (4.8) By Eq (4.3), Eq (4.8) can be rewritten as From Eq (4.9) we can get where V = (1, v 1 , v 2 ) T ∈ C 3 is a constant vector.By virtue of Eq (4.9), we obtain On the other hand, −iω * is the eigenvalue of A * 0 .Thus we have A * 0 q * (ξ) = −iω * q * (ξ).For the non-zero vector q * (ξ), ξ ∈ [0, τ * 1 ], we have that , and T be a constant vector.Similarly, we get that In order to assure q * (ξ), q(θ) = 1, we need to determine the value of D. From Eq (4.7), we have 2 )kg 3 .Next, we use the same notations as the previous part to study the stability of bifurcating periodic solution.We will calculate the coordinates to express the center manifold C 0 at µ = 0. Let x t be the solution of Eq (4.5) when µ = 0. Define On the center manifold C 0 , we have where z and z are the local coordinates for the center manifold C 0 in the direction of q * and q * .Note that W is real if x t is real.Thus we only consider the real solution.
By Eqs (4.20) and (4.23), we have and where , Substituting Eqs (4.25) and (4.29) into Eq (4.27), we obtain It follows that where Similarly, it follows that where Thus, we can determine W 20 (θ) and W 11 (θ) from Eqs (4.25) and (4.26).Furthermore, g 21 can be expressed by the parameters and delay.Thus, we can compute the followings values: which determine the properties of bifurcating periodic solution in the center manifold at the critical value τ * 1 .From the conclusion of Hassard et al. [42], we give the following main findings.
Theorem 4.1.The values of the parameters µ 2 , β 2 and T 2 of (4.30) will determine the properties of Hopf bifurcation of system.(i) The sign of µ 2 determines the direction of Hopf bifurcation: The Hopf bifurcation is supercritical if µ 2 > 0 and the Hopf bifurcation is subcritical if µ 2 < 0.
(ii) The sign of β 2 determines the stability of bifurcation periodic solution: The periodic solutions are stable if β 2 < 0 and unstable if β 2 > 0.
(iii) The sign of T 2 determines the period of bifurcation periodic solution: The period increases if T 2 > 0 and the period decreases if T 2 < 0.
With the increase of m, the susceptible population increases and the infected population decreases.According to Figure 9(a), when m increases, the number of the susceptible population increases and the number of people who eat adulterated food decreases.Figure 9(b) shows that the number of the infected population gradually decreases with the increase of m, which means that the number of the susceptible population decreases and the intake of adulterated food decreases.Therefore, increasing the media influence coefficient reduces the intake of adulterated food.
Further, when m = 1.2, τ = 1.74, the endemic equilibrium E * loses stability (see Figure 10).Thus, the low media awareness damages the stability of the model.As the improve of disease depends on many factors, we spread disease news through the media, which has a positive impact on controlling the spread of disease.

Discussion and conclusions
In this paper, we give an eco-epidemic model with adulterated food and media effects, where human growth is entirely dependent on adulterated food.By using adulterated food, humans are divided into susceptible and infected populations.The model (2.1) has three equilibria, namely the trivial equilibrium, the disease-free equilibrium and the endemic equilibrium.The trivial equilibrium is an unstable saddle point.When R 0 < 1, the disease-free equilibrium is locally asymptotically stable.This suggests that due to adulteration, the susceptible population becomes infected within a short period of time and both populations become extinct.The time delay τ is taken as the bifurcation parameter to obtain the local stable switching conditions of the endemic equilibrium.By determining the stability switching conditions, the time delay can cause a small oscillation of the population density, that is, Hopf bifurcation takes place.It means that time delay plays an important role in the spread of disease, which cases the endemic equilibrium take the form of periodic fluctuations.In addition, by using the center manifold theory, we obtain the direction of Hopf bifurcation and stability of periodic solutions.
This media influence helps people to realize the various hazards of adulterated food.When the media increases, the susceptible population increases and the infected population decreases, which indicates that the media influence makes people reduce their intake of adulterated food and reduce infection.The threshold values of media influence coefficient m indicate that higher intake of adulterated food can lead to instability of the model.
In the future, we will not only take media reports as an independent variable, that is, the term M(t) represents the average number of news related to the epidemic, but also introduce the time delay caused by the response time of individuals to current media reports.

Figure 1 .
Figure 1.Flow chart of the epidemic model with the impact of media.

Theorem 2 . 1 .
The solutions of model (2.1) with the initial condition (2.2) are positive.

Figure 4 .
Figure 4. Graph of function S 1 (τ) with the same values as in Figure 3.

Figure 10 .
Figure 10.Approaching towards the periodic fluctuations in disease occurrence with media awareness coefficient m = 1.2:(a) prey, (b) susceptible predator, and (c) infected predator.