key: cord-0056626-wu3zvk16 authors: Mathur, Kunwer Singh; Srivastava, Abhay; Dhar, Joydip title: Dynamics of a stage-structured SI model for food adulteration with media-induced response function date: 2021-02-20 journal: J Eng Math DOI: 10.1007/s10665-021-10089-4 sha: 3ef549e9967c54198f9f67d80b6b0125c9931faa doc_id: 56626 cord_uid: wu3zvk16 In this work, an eco-epidemic predator–prey model with media-induced response function for the interaction of humans with adulterated food is developed and studied. The human population is divided into two main compartments, namely, susceptible and infected. This system has three equilibria; trivial, disease-free and endemic. The trivial equilibrium is forever an unstable saddle position, while the disease-free state is locally asymptotically stable under a threshold of delay parameter [Formula: see text] as well as [Formula: see text] . The sufficient conditions for the local stability of the endemic equilibrium point are further explored when [Formula: see text] . The conditions for the occurrence of the stability switching are also determined by taking infection delay time as a critical parameter, which concludes that the delay can produce instability and small amplitude oscillations of population masses via Hopf bifurcations. Further, we study the stability and direction of the Hopf bifurcations using the center manifold argument. Furthermore, some numerical simulations are conducted to validate our analytical findings and discuss their biological inferences. Finally, the normalized forward sensitivity index is used to perform the sensitivity analysis of [Formula: see text] and [Formula: see text] . 1 of media, the susceptible persons restrict themselves from eating adulterated food. Hence the response function exponentially decreases. Thus the media-induced response function is more suitable and realistic to study the effect of food adulteration on the human population. The susceptible person may become infected by eating adulterated food. Thus conversion from susceptible to infected is not instantaneous, and actually, it requires a period, after that the susceptible may become infected. So, based on this concept, we consider an eco-epidemic model, where the delay period partitioned the human population into two stages: the early and later stages. Assume that the individuals in the early stage can transfer to the later stage. When susceptible human population eats the adulterated food, it will be counted in the early stage as it takes a constant time to develop the infection inside the susceptible population and then transfers into the later stage, i.e., infected individual class through a delay term. In the last decade, several researchers have increased their attention on the epidemic models with multiple phases of infections, where infectivity passes through consecutive phases of infection [21] [22] [23] [24] . Thus, we believe that this is the first time where an eco-epidemic prey-predator model with media-induced response function and multiples stages through time delay is considered to study the dynamics of diseases spread by food adulteration. The paper is organized as follows: Sect. 2 deals with the development of a mathematical model. The system dynamics and the existence of equilibria are discussed in Sect. 3 . The stability analysis is determined in Sect. 4 . The stability and direction of Hopf bifurcation are studied in Sect. 5. Further, the numerical simulations and discussion are given in Sect. 6 . Sensitivity analysis of the threshold R 0 is performed in Sect. 7. Finally, a conclusion is presented in the last section. In this section, we will develop a prey-predator model with media-induced interaction of adulterated food and human population, where the adulterated food will be used as a prey population while human as a predator population. Further, the predator will be divided into two compartments, namely susceptible/latently infected and infected. Here it is assumed that all the newborns are susceptible/latently infected and can be infected through the consumption of adulterated food, and this will be the only source of infection in the human population. Moreover, it is assumed that the infected predator population is only infected, but not infectious, and they cannot spread any disease. Let x(t), y(t), and z(t) be densities of prey, latently infected and infected human/predator population, respectively, at time t. The basic assumptions of our model are as follows: (A1) The prey population is growing in a logistic manner with an intrinsic growth rate r and carrying capacity K . (A2) The predation of adulterated foods by the susceptible human is influenced through the information spread by media, which is represented in terms of the media-induced response function βe −mz(t) x(t), where β is the predation rate and m is the density of media awareness corresponding to the predator population. This term represents the predation, which influences through the information spread by media in proportion to the infected person. Thus, when the media increases awareness, the predation rate exponentially decreases. Hence, media-induced response function is exponentially decreasing due to media awareness in the presence of infective person (as suggested in [25] ). (A3) In this paper, we discussed about only those susceptible human population, whose growth is completely dependent on adulterated food. Hence the growth of this susceptible population is proportional to the predation rate β f (x(t), z(t))y(t)), with proportionality constant k as conversion rate. The same population dies out naturally with mortality rate d 1 . (A4) The term βe −mz(t) x(t)y(t) does not represent the incidence or disease transmission. It shows the predation, where kβe −mz(t) x(t)y(t) represents the growth of susceptible predator in form of energy conversion. Also, a fraction of this population dies out naturally with the rate d 1 and the remaining part of susceptible human population becomes infected after a time period τ as the conversion from one stage to another is not instantaneous (similar to that considered by several researchers in [26] [27] [28] , where the predator has two stages: immature and mature). Hence the term kβe −d 1 τ f (x(t − τ ), z(t − τ ))y(t − τ ) represents the total number Keeping the above assumptions in mind, our proposed mathematical model is ruled by the following system of differential equations: (2.1) All the system parameters are positive and their description is given in Table 1 . The initial population densities for system (2.1) take the form: . From the second equation of system (2.1), we get Thus, we can impose the following continuity condition: Assume the continuous solution of (2.1) is X (t) = (x(t), y(t), z(t)) T , is defined as X : R + → R 3 + and is also Lipschitzian in a compact set R + with initial conditions (2.2) and (2.3). Therefore, [29, Theorem 2.3] Proof Suppose (x(t), y(t), z(t)) is any solution of system (2.1) with initial conditions (2.2) . For 0 ≤ t ≤ τ , the last equation of (2.1) gives Clearly z(t) ≥ z(0)e −d 2 t z * (t) > 0 for all t ≥ 0. Now first differential equation of (2.1), ∀t ≥ t 0 for some t 0 > 0, we get which gives For 0 ≤ t ≤ τ , the second equation of system (2.1) can be written as Let v(t) be the solution of Hence We thus have v(τ ) = 0, and therefore y(t) ≥ 0 for t ∈ [0, τ]. By induction, we can show that y(t) ≥ 0 for all t ≥ 0. This gives (i) Trivial equilibrium E 1 = (0, 0, 0), which always exists. (ii) Disease-free or human-free equilibrium E 2 = (K , 0, 0), which also always exists. (iii) Endemic equilibrium E 3 = (x * , y * , z * ) exists whenever min{R 0 , and The existence of E 1 and E 2 is trivial, hence omitted. Here, we discuss only the existence of endemic equilibrium in detail. If endemic equilibrium point E 3 (x * , y * , z * ) exists, it must satisfy the equations: Clearly, if there exists a positive equilibrium, it is a positive solution of To find sufficient conditions for the uniqueness of a positive solution of (3.4), we follow the work of Lie et al. cited in [30] , which provides that Thus, the two curves G(z) and H (z) have at least one positive intersection. In order to determine the number of other positive intersections, we consider the tangency of the above two curves G(z) and H (z). If the two curves intersect, it must have G(z) = H (z) and G (z) = H (z), i.e., and The difference of equations (3.5) and (3.6) provides Substituting the above value in (3.5), we obtain that Squaring both sides of (3.9), we get (3.10) 1 From (3.7) and (3.10), we get It can be simply determined that (3.12) has no root for 0 < m < m 0 , and hence the system (2.1) has a unique endemic equilibrium. For m = m 0 , (3.12) has one unique root and the system (2.1) has one endemic equilibrium point of multiplicity at least two. Again, if m > m 0 , then (3.12) has two roots and hence the system (2.1) has three endemic equilibria. Thus, the existence of unique endemic equilibrium is stated in the following: The above lemma suggests that a unique endemic equilibrium point exists whenever the delay parameter crosses a threshold (i.e., τ > τ * := max{ τ 1 , τ 2 }), where In this part, we perform the local stability analysis and existence of the Hopf bifurcation for the model (2.1), which is governed by a crucial threshold R 0 . Clearly the characteristic equation of trivial equilibrium E 1 (0, 0, 0) takes the form (λ − r )(λ + d 1 )(λ + d 2 ) = 0, and hence E 1 is an unstable saddle. Further the characteristic equation for E 2 (K , 0, 0) is given as It is clear that λ = −r and λ = −d 2 are always two negative eigenvalues. All other eigenvalues are given by the − d 1 must intersect at a negative value of λ, and hence the predator-free equilibrium E 2 is locally asymptotically stable provided that R 0 < 1, i.e., for all τ ∈ (0, τ 1 ). Now the characteristic equation of endemic equilibrium E 3 (x * , y * , z * ) takes the form The above equation can be rewritten as Lemma 3.3 ensures the existence of a unique endemic equilibrium for τ > τ * . Therefore, existence of purely imaginary roots of the characteristic equation (4.1) is established for τ > τ * by using the geometric criterion for delay-dependent coefficients [31] , which is stated as follows: , if all the properties given in [31] are satisfied. Proof For τ ∈ I , it is clear that We have Let χ = ω 2 , then (4.4) is rewritten as follows: Thus, there are three cases for the existence of solution of (4.5). (a) Equation (4.5) has a pair of complex roots and a real root when D > 0. The positivity condition of the real root is given by (b) Again for D = 0, (4.5) has all real roots, with two being equal. Moreover, if A > 0, it has only one positive real root, , and there exist three positive roots for A 6 < 3 − g 2 < − A 3 , given by (c) If D < 0, there are three distinct real roots given as has to be calculated in radians. Furthermore, if A > 0, there exists only one positive root. Otherwise, if A < 0, there may exist either one or three positive real roots. If there is only one positive real root, it is equal to max{χ 1 , χ 2 , χ 3 }. Hence, the sign of the discriminant D of (4.5) determines the number of positive roots. Hence, the existence of at least one positive real root is given in the following region: Thus (4.4) has at least two real roots ω ± (τ ) = ± √ χ(τ ) for all τ ∈ I . Since the function F(ω) is a sixth-degree polynomial, it has at most six real zeros for all τ ∈ I . (v) From implicit function theorem each root of F(ω, τ ) = 0 is continuous and differentiable in I , because F(ω, τ ) is differentiable with respect to ω and continuous in ω and τ . Hence, all the conditions given in [31] are satisfied, and thus the Lemma 4.1 ensures the existence of purely imaginary roots of characteristic equation (4.2) for all τ ∈ I . Now assume that λ = iω (ω > 0) is a purely imaginary characteristic root of (4.2). By substituting λ = iω into (4.2) and separating real and imaginary parts, we obtained the following transcendental equations: It follows from (4.7) that As cited in [31] , a well and unique θ(τ ) ∈ [0, 2π ], ∀τ ∈ I , is defined by (4.9) One can check that iω * with ω * = ω(τ * ) > 0 is a purely imaginary root of (4.2) if and only if τ * is a roots of S n defined by following map: Following Beretta and Kuang [31] , we can state a theorem as given below: Here, we can easily find that S n (0) < 0 and S n (τ ) > S n+1 (τ ) ∀τ ∈ I , n ∈ N 0 . Thus, if S 0 has no zero in I , the function S n also has no zero in I and if the function S n has positive zeros, denoted by τ j n for some τ ∈ I, n ∈ N 0 , then without loss of generality, we may assume that dS n (τ j n )/dτ = 0 with S n (τ j n ) = 0 and applying the similar logic as in [31] , it is obtained that the stability switches occur at the zeros of S 0 (τ ), denoted by τ j 0 . Thus using [29] , we can conclude the dynamics of stability switches in the following theorem: The local behavior of the system (2.1) at endemic equilibrium E 3 is described as follows: 1. If the function S 0 (τ ) has no positive zero in I , then the endemic equilibrium E 3 (x * , y * , z * ) is locally asymptotically stable for all τ > τ * or R 0 > 1, 0 < m < m 0 . 2. If the function S n (τ ) has some positive zeros at τ * 1 , τ * 2 , τ * 3 . . . in I for some n ∈ N 0 , then E 3 is locally asymptotically stable for τ ∈ (τ * , τ * 1 ) ∪ (τ * 2 , τ * 3 ) ∪ . . . and unstable with existence of a Hopf bifurcation for τ ∈ (τ * 1 , τ * 2 ) ∪ (τ * 3 , τ * 4 ) ∪ . . ., i.e., stability switching occurs from stability-instability-stability and so on. In the previous section, we have got some sufficient conditions under which a class of periodic solutions bifurcated from the steady state of the system (2.1), when the delay is more than the critical level τ * . Now we shall study the direction of these Hopf bifurcations and stability of bifurcated periodic solutions using the method discussed in [32] . Using Appendix A, we can compute the following values: which determine the behavior of periodic solution in the center manifold at τ * , i.e., again if μ 2 > 0 (μ 2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ > τ * (τ < τ * ); β 2 determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if β 2 < 0 (β 2 > 0) and, finally, the period increases (decreases) if T 2 > 0 (T 2 < 0). Our analytical results indicate that the time delay must be accountable for the observed regular cycles of disease occurrence. To examine the effect of time delay on the system (2.1), let us consider the following model: Here, some numerical results of system (6.1) will be discussed at different parametric values of τ, m, β, k, d 2 . Let m = 2, β = 0.24, k = 0.3, d 2 = 0.2, then it is clear that all the solutions of (6.1) are positive and bounded. A disease-free or human-free equilibrium point E 2 (10, 0, 0) exists and it is locally asymptotically stable for all τ < τ 1 = 1.49532 or R 0 < 1. If we choose τ = 1.48 < τ 1 , then R 0 = 0.990496 < 1, and hence the equilibrium point E 2 (10, 0, 0) satisfies usual analytical criteria for stable equilibrium points (see Fig. 1 ). The equilibrium point E 2 (10, 0, 0) is not feasible for the society, as the human population may never be extinct in reality. Hence our main objective in this paper is to determine the situation where the human population will survive with minimal effect of adulterated foods on his health. Let us take τ * = max{τ 1 , τ 2 }, where We can see that all the key parameters including predation rate β, media effect m, carrying capacity K , growth rate r , death rates d 1 , d 2 , and conversion rate k are involved in the above threshold condition. This allows us to address the effect of controlling parameter τ as well as adulterated food on human health. Fig. 2 ). Further, the periodic fluctuations in disease occurrence arise for a different range of delay parameter τ , which ensure that the stability of endemic equilibrium is switching from stability-instability-stability-instability-stability (see Fig. 3 ). Similarly, it is observed that the predator coefficient β, and the coefficient of media effect m can influence the dynamics of the system (2.1). Figures 4 and 5 , respectively, show the effect of predation rate and media coefficient on destabilization of endemic equilibrium E 3 (Fig. 6) . Thus the above numerical simulations suggest that the usually delay parameter destabilized the steady state, whereas substantial delays have a stabilizing one. On the other hand, high predation and media awareness can destabilize the steady state. As the spread of infections depends on different factors, therefore, real-time information 1 For real-life application, more attention should be paid towards highly sensitive parameters because these type of parameters will produce significant qualitative changes by a slight variation in their respective values. Thus, in this section, we discuss the sensitivity analysis of R 0 and R * 0 using normalized forward sensitivity index, which is defined as follows: Definition (see [33] ) The normalized forward sensitivity index of a variable, x, which depends upon a parameter, y, is defined as By taking the numerical values of all the parameters, the normalized forward sensitive indices for R 0 and R * 0 are calculated in Table 2 . From the above, it reveals that the parameters K , β, k, and τ have a positive impact on R 0 , which means that when these parameters increase (or decrease) by keeping the others are constant, then the value of R 0 will also increase (or decrease). Similarly, the parameter d 1 has a negative impact on R 0 , i.e., when the value of d 1 increases (or decreases) while others are being constant, then the value of R 0 decreases (or increases). On the other hand, parameters r , K , m, and k have negative impact on R * 0 while d 1 , d 2 , and τ have positive impact. For example, 10% increase (or decrease) in τ will produce 9.278% increase (or decrease) in R 0 , and 10% increase (or decrease) in d 1 will produce 0.722% decrease (or increase) in R 0 . Moreover, it can be easily seen that the parameters, which have either a positive or a negative impact on R 0 and R * 0 , are most sensitive. Therefore, these parameters should be dealt very carefully as they play an adaptive role in disease outbreak. Over the last few decades, it has been observed that food adulteration has a severe effect on human health. The adulteration is applying not only in the food but also in all things. Hence humans have nothing else to eat except adulterated items. Therefore, we have proposed an eco-epidemic prey-predator model for contaminated food and human with media awareness in the form of media-induced response function, where the growth of human is completely dependent on adulterated foods and due to consumption of these adulterants, human is assumed to be infected, and thus the predator population was divided into two compartments: susceptible and infected. The conversion from the susceptible to infected cannot be instantaneous. It requires a period, after that the susceptible become infected. In the analysis, it has obtained that the system has three equilibria, namely, trivial, boundary, and interior. The trivial equilibrium is an unstable saddle point, while the boundary equilibrium point is locally asymptotically stable for R 0 < 1, i.e., for τ < τ 1 , which shows that due to adulterants if the conversion of the susceptible population into infected is in short time, then both the populations will become extinct. In the next, by taking the delay as a bifurcation parameter, we have obtained certain sufficient conditions on the existence of the switching stability of the positive steady state. It shows that the delay can induce a small amplitude oscillations of population densities, i.e., Hopf bifurcations, by applying the geometric approach for stability switches on the characteristic equation with delay-dependent parameters. It has been recognized that the time-varying delay-dependent parameters play an important role in the dynamics of disease propagation and the endemic steady state has a periodic fluctuation in disease occurrence arise for different ranges of delay parameter τ , which ensure that the stability of endemic equilibrium is switching, whereas large delays have a stabilizing effect. Furthermore, by employing the center manifold argument, the Hopf bifurcations' direction and stability have ascertained. Finally, the sensitivity analysis of threshold R 0 and R * 0 is performed, which shows that media helps in lowering the predation rate of adulterated foods as the predation rate is highly sensitive. Thus, when predation is reduced, the conversion rate of k will also be reduced. Moreover, we have proposed a new media-induced response function in the form of media awareness to represent the interaction between adulterated food and human, which is exponentially decreasing with the increase of media awareness, as in the presence of media, the susceptible persons restrict themselves from eating the adulterated food. This response function is beneficial to know the various effects of food adulteration either by the predation or media awareness. The thresholds of predation rate β and media effect coefficient m suggest to consume adulterated food at a low quantity, otherwise due to higher quantity the system may destabilize. Since adulteration in fruits and vegetables is a paramount problem for our society, the present work can be helpful in terms of showing the impact of media awareness on adulteration along with the behavioral change in population. 1 By the change of variables , τ = τ * + μ and dropping the bars for simplifications of notations, system (2.1) is transformed into a functional differential respectively, are given by By the Riesz representation theorem, there exists a 3 × 3 matrix η(θ, μ) : [−1, 0] → R 3 whose elements are of bounded variation such that In fact, we can choose where δ is a Dirac delta function. and (A.5) Then the system (A.1) is equivalent to the following operator equation: and a bilinear form where η(θ) = η(θ, 0). Then A(0) and A * are adjoint operators. From the discussion in previous section, we know that ±iω * τ * are eigenvalues of A(0) and therefore they are also eigenvalues of A * , corresponding to iω * τ * and −iω * τ * , respectively. Suppose that q(θ ) = (1, α, γ ) T e iθω * τ * is the eigenvector of A(0) corresponding to iω * τ * , then A(0)q(θ ) = iω * τ * q(θ ). It follows from the definition of A(0) and η(θ, μ) that Then, we can easily obtain q(0) = (1, α, γ ) T , where Again, let q * (θ ) = D(1, α * , γ * )e −iθω * τ * be the eigenvector of A * corresponding to −iω * τ * , then similarly we can obtain that By (A.8), we get Then we choosē D = 1 1 + αᾱ * + γγ * + τ * (γ * −ᾱ * )(y * + αx * − mγ x * y * )kβe −d 1 τ * e −mz * e −iω * τ * , 1 such that q * (s), q(θ ) = 1 and q * (s),q(θ ) = 0. In the following, we use the ideas in Hassard et al. [32] to compute the coordinates describing center manifold C 0 at μ = 0. Define on the center manifold C 0 , and we have where z andz are local coordinates for C 0 in C in the direction of q * andq * . Note that W is real if x t is real. We deal only with the real solution. For solution x t ∈ C 0 of (A.6), since μ = 0, we havė where g(z,z) =q * (0), and F 0 (z,z) = g 20 z 2 2 + g 11 zz + g 02z From (A.9) and (A.10), we have Thus, we can easily obtain that 20 (−1) 20 (−1) 20 (−1) From the definition of F(μ, x t ), we have −αβe −mz * (1 − kᾱ * ) + mkβγ y * e −d 1 τ * e −mz * e −2iθω * τ * (ᾱ * −γ * ) +mkαβγ x * e −d 1 τ * e −mz * e −2iθω * τ * (ᾱ * −γ * ) −kαβe −d 1 τ * e −mz * e −2iθω * τ * (ᾱ * −γ * ) +zzz 1 +z 2 z 2 + z 2z z 3 , where Δ 11 = − r K x 2 1t (0) + βe −mz * (my * x 1t (0)x 3t (0) + mx * x 2t (0)x 3t (0) − x 1t (0)x 2t (0)), +kβe −d 1 τ * e −mz * (my * x 1t (−1)x 3t (−1) + mx * x 2t (−1)x 3t (−1) − x 1t (−1)x 2t (−1)), Δ 33 = −kβe −d 1 τ * e −mz * (my * x 1t (−1)x 3t (−1) + mx * x 2t (−1)x 3t (−1) − x 1t (−1)x 2t (−1)), z 1 = − 2r K + β(γ +γ )my * e −mz * (1 − kᾱ * ) + β(αγ +ᾱγ )mx * e −mz * (1 − kᾱ * ) −(α +ᾱ)βe −mz * (1 − kᾱ * ) + (γ +γ )mkβy * e −d 1 τ * e −mz * (ᾱ * −γ * ) +(αγ +ᾱγ )mkβx * e −d 1 τ * e −mz * (ᾱ * −γ * ) − (α +ᾱ)kβe −d 1 τ * e −mz * (ᾱ * −γ * ), +mkβγ y * e −d 1 τ * e −mz * e 2iθω * τ * (ᾱ * −γ * ) + mkβᾱγ x * e −d 1 τ * e −mz * e 2iθω * τ * (ᾱ * −γ * ) −kᾱβe −d 1 τ * e −mz * e 2iθω * τ * (ᾱ * −γ * ), 20 (0)) − βe −mz * (1 − kᾱ * )(W (2) 11 (0) + αW (1) 11 (0) 20 (0)) + mkβy * e −d 1 τ * e −mz * (ᾱ * −γ * )(e −iθω * τ * W +γ e −iθω * τ * W (1) 11 (−1) + αe −iθω * τ * W On comparing the coefficients with (A.11), we obtain that g 20 = 2τ * D z 2 , g 11 = 2τ * D z 1 , g 02 = 2τ * D z 2 g 21 = 2τ * D z 3 . In order to determine g 21 , we need to compute W 20 (θ ) and W 11 (θ ). From (A.6) and (A.9), we havė Noting that q(θ ) = q(0)e iω * τ * θ , hencė W 20 (θ ) = ig 20 ω * τ * q(0)e iω * τ * θ + iḡ 02 3ω * τ * q (0)e −2iω * τ * θ + E 1 e 2iω * τ * θ , (A. 16) Food adulteration: sources, health risks, and detection methods Food adulteration and its problems (intentional, accidental and natural food adulteration) The effects of food safety issues released by we media on consumers awareness and purchasing behavior: a case study in China Capturing human behaviour Awareness of sexually transmitted disease among women and service providers in rural Bangladesh Media and education play a tremendous role in mounting aids awareness among married couples in Bangladesh Epidemic dynamics on an adaptive network Contact switching as a control strategy for epidemic outbreaks The impact of information transmission on epidemic outbreaks The impact of media coverage on the dynamics of infectious disease The impact of media coverage on the transmission dynamics of human influenza Media impact switching surface during an infectious disease outbreak Elements of physical biology Variations and fluctuations of the number of individuals in animal species living together Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator Hopf bifurcation of a predator-prey system with stage structure and harvesting A prey-dependent consumption two-prey one predator eco-epidemic model concerning biological and chemical controls at different pulses Stability and permanence of an eco-epidemiological sein model with impulsive biological control The natural control of animal populations Functional responses with predator interference: viable alternatives to the Holling type II model Global stability of a stage-structured epidemic model with a nonlinear incidence The stage-structured epidemic: linking disease and demography with a multi-state matrix approach model Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate Persistence in a discrete-time, stage-structured epidemic model Global Hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease Bifurcation and stability analysis in predator-prey model with a stage-structure for predator Periodic solutions of a delayed predator-prey model with stage structure for predator Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure Introduction to functional differential equations Modelling the impact of media in controlling the diseases with a piecewise transmission rate Geometric stability switch criteria in delay differential systems with delay dependent parameters Theory and applications of Hopf bifurcation Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity The first author would like to acknowledge and extend heartfelt gratitude to the Science and Engineering Research Board, New Delhi, India, for the financial support. 1 , E (2) 1 , E (3) 1 ) ∈ R 3 is a constant vector. Similarly, from (A.14) and (A.15), we obtain W 11 (θ ) = − ig 11 ω * τ * q(0)e iω * τ * θ + iḡ 11 ω * τ * q (0)e −iω * τ * θ + E 2 , (A. 17) where E 2 = (E (1) 2 , E (2) 2 , E2 ) ∈ R 3 is also a constant vector. In the following we shall find out E 1 and E 2 . From the definition of A and (A.14), we can obtainwhere η(θ) = η(0, θ). From (A.12) and (A.13), we havewhere which leads to ⎛ ⎝ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 31 σ 33It follows thatSimilarly, substituting (A.17) and (A.18) into (A.19), we can getσ 21 = −kβe −mz * y * (1 + e −d 1 τ * ), σ 23 = kβe −mz * y * e −d 1 τ * .Thus, we can determine W 20 and W 11 from (A.16) and (A.17).