Existence for positive steady states of an eco-epidemiological model

An eco-epidemiological model with an epidemic in the predator and with a Holling type Ⅱ function is considered.A system with diffusion under the homogeneous Neumann boundary condition is studied.The existence for a positive solution of the corresponding steady state problem is mainly discussed.First,a prior estimates（positive upper and lower bounds） of the positive steady states of the reaction-diffusion system is given by the maximum principle and the Harnack inequation.Then,the non-existence of non-constant positive steady states by using the energy method is given.Finally,the existence of non-constant positive steady states is obtained by using the topological degree.

Global Stability of An Eco-epidemiological Model with Beddington-DeAngelis Functional Response and Delay

In this paper,an eco-epidemiological model with Beddington-DeAngelis functional response and a time delay representing the gestation period of the predator is studied.By means of Lyapunov functionals and Laselle’s invariance principle,sufficient conditions are obtained for the global stability of the interior equilibrium and the disease-free equilibrium of the system,respectively.

Asymptotic analysis of a nonlinear stochastic eco-epidemiological system with feedback control

This paper proposes a new stochastic eco-epidemiological model with nonlinear incidence rate and feedback controls.First,we prove that the stochastic system has a unique global positive solution.Second,by constructing a series of appropriate stochastic Lyapunov functions,the asymptotic behaviors around the equilibria of deterministic model are obtained,and we demonstrate that the stochastic system exists a stationary Markov process.Third,the conditions for persistence in the mean and extinction of the stochastic system are established.Finally,we carry out some numerical simulations with respect to different stochastic parameters to verify our analytical results.The obtained results indicate that the stochastic perturbations and feedback controls have crucial effects on the survivability of system.

Analysis of a Nonautonomous Eco-Epidemiological Model with Saturated Predation Rate

In this paper, a nonautonomous eco-epidemiological model with disease in the predator is formulated and analyzed, in which saturated predation rate is taken into consideration. Under quite weak assumptions, sufficient conditions for the permanence and extinction of the disease are obtained. Moreover, by constructing a Liapunov function, the global attractivity of the model is discussed. Finally, numerical simulations verified these results.

Analysis of an Eco-Epidemiological Model Considering Effect of Harvesting and Prey Refuge

In this work, we analyze an eco-epidemiological model with the disease in the prey, considering a constant proportion of harvesting of either species or a prey refuge. The positive invariant set, the conditions of existence, and locally asymptotically stability of the equilibria are studied using the stability theory of ordinary differential equation. The global stability of border equilibria by constructing Lyapunov functions and permanence of the system by comparison theoremare proved. The numerical simulation further proved the correctness of the theoretical analysis. The result indicates that overfishing would lead to population extinction and a reasonable fishing strategy should keep the coexistence of populations.

The effect of the fear factor on the dynamics of an eco-epidemiological system with standard incidence rate

In order to protect endangered prey,ecologists suggest introducing parasites into predators which have achieved the expected goal in practice.Then how to explain the inherent mechanism and validate the effectiveness of this approach theoretically?In response to this question,we propose an eco-epidemiological system with the standard incidence rate and the anti-predator behavior in this paper,where the predator population is infected by parasites.We show the existence and local stability of equilibria for the system,and verify the occurrence of Hopf bifurcation.Theoretical and numerical results suggest that the fear effect reduces the density of the predator population but has no effect on the density of prey population.In addition,the cost of fear may not only break the stability of the equilibrium of the system,but also induce the equilibrium to change from unstable to stable.Based on the theoretical analysis,we confirm that introducing parasites into the predator population is an effective method to protect endangered prey.

Competitive and exclusion and oscillation phenomena of an eco-epidemiological model: An invasive approach

Eco-epidemiological model plays a crucial role in maintaining ecological balance and biological invasions.In this paper,we propose a predator-prey model where the predator population is infected by the disease.The non-negativity and boundedness of the solution are addressed.The existence and stability of equilibrium are discussed by an invasive approach.Numerical simulations are used to demonstrate the competitive and exclusion,and oscillation behaviors of the model.Interestingly,the predation rate of the infected predator is shown to have far-reaching implications for the structure and stability of ecological communities.

Global stability and Hopf bifurcation of an eco-epidemiological model with time delay

In this paper,an eco-epidemiological model with time delay representing the gestation period of the predator is investigated.In the model,it is assumed that the predator population suffers a transmissible disease and the infected predators may recover from the disease and become susceptible again.By analyzing corresponding characteristic equations,the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free and coexistence equilibria are established,respectively.By means of Lyapunov functionals and LaSalle’s invariance principle,sufficient condi tions are obtained for the global stability of the coexistence equilibrium,the disease-free equilibrium and the predator-extinct equilibrium of the system,respectively.

一类具有时滞的生态流行病模型的稳定性与Hopf分支

In this paper,an eco-epidemiological model with time delay is studied.The local stability of the four equilibria,the existence of stability switches about the predationfree equilibrium and the coexistence equilibrium are discussed.It is found that Hopf bifurcations occur when the delay passes through some critical values.Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem.Some numerical simulations are carried out to illustrate the theoretical results.

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.

Stability Analysis of Predator-Prey System with Consuming Resource and Disease in Predator Species

The present study is concerned with formulating a predator-prey eco-epidemiological mathematical model assuming that an infection exists in the predator species.The two classes of predator species(susceptible and infected)compete for the same sources available in the environment with the predation option.It is assumed that the disease does not spread vertically.The proposed model is analyzed for the stability of the coexistence of the predators and prey.The fixed points are carried out,and the coexisting fixed point is studied in detail by constructing the Lyapunov function.The movement of species in search of food or protection in their habitat has a significant influence,examined through diffusion.The ecological influences of self-diffusion on the population density of both species are studied.It is theoretically proved that all the under consideration species can coexist in the same environment.The coexistence fixed point is discussed for both diffusive and non-diffusive cases.Moreover,a numerical scheme is constructed for solving time-dependent partial differential equations.The stability of the scheme is given,and it is applied for solving presently modified eco-epidemiological mathematical model with and without diffusion.The comparison of the constructed scheme with two exiting schemes,Backward in Time and Central in Space(BTCS)and Crank Nicolson,is also given in the form of plots.Finally,we run a computer simulation to determine the effectiveness of the proposed numerical scheme.For readers’convenience,a computational code for the proposed discrete model scheme may be made available upon request.

Leishmania in an Ecotourism Area in Rio de Janeiro: Still a Concern for Public Health?

We evaluated some eco-epidemiological characteristics of the sand fly fauna in an ecotourism area in the Atlantic Forest located in Rio de Janeiro, southeastern Brazil. During a period of one year, sandflies were collected in three different locations, where the sampled residences were located, respectively, one inside the forest, the other two, respectively at the edge of the forest and the other in a more urbanized area. These three types of ecotopes were evaluated: home, peridomicile and kennel. Four hundred and fifty-six sandflies were collected and six species belonging to five genera were identified: Migonemyia migonei, Lutzomyia longipalpis, Nyssomyia intermedia, Evandromyia sallesi, Evandromyia edwardsi and Brumptomyia wedge. The two most abundant species collected were M. migonei and L. longipalpis, contributing 70% and 18% respectively, totaling 88% of the individuals collected. The results suggested that modifications of the natural environment due to anthropic action probably resulted in changes in the composition of the sand fly population. At point (3), where spraying occurred irregularly, even representing a degraded environment, only one species was captured, M. migonei. Differently at points (1) and (2), areas located respectively in the interior and on the edge of the Atlantic Forest, a greater number of sand fly species was observed. However, after a few years, anthropic actions ceased, followed by the implementation of reforestation projects and currently the landscape is very different, showing considerable forest recovery. For this reason, ecotourism activities are increasing in the area, creating potentially dangerous conditions caused by the exposure of greater numbers of people and dogs to insect vectors. Therefore, the implementation of environmental education projects is essential. However, we suggest that the use of warning signs to be placed at the entrances to the main traffic routes, alerting tourists to the risk of infection and indicating protective measures, would be very useful.

A Mathematical Study for the Stability of Two Predator and One Prey with Infection in First Predator using Fuzzy Impulsive Control

In this study,we develop a set of ordinary differential equations that represents the dynamics of an ecosystem with two predators and one prey,but only the first predator population is affected by an infectious disease.The Lotka-Volterra predator-prey system’s model stability have been examined using the Takagi-Sugeno(T-S)impulsive control model and the Fuzzy impulsive control model.Following the formulation of the model,the global stabilities and the Fuzzy solution are carried out through numerical simulations and graphical representations with appropriate discussion for better understanding the dynamics of our proposed model.

An optimal control model with cost effectiveness analysis of Maize streak virus disease in maize plant

In this paper we formulated and analyzed an optimal deterministic eco-epidemiological model for the dynamics of maize streak virus(MSV)and examine the best strategy to fight maize population from maize streak disease(MSD).The optimal control model is developed with three control interventions,namely prevention(u_(1)),quarantine(u_(2))and chemical control(u_(3)).To achieve an optimal control strategy,we used the Pontryagin’s maximum principle obtain the Hamiltonian,the adjoint variables,the characterization of the controls and the optimality system.Numerical simulations are performed using Forward-backward sweep iterative method.The findings show that each integrated strategy is able to mitigate the disease in the specified time.However due to limited resources,it is important to find a cost-effective strategy.Using Incremental Cost-Effectiveness Ratio(ICER)a cost-effectiveness analysis is investigated and determined that the combination of prevention and quarantine is the best cost-effective strategy from the other integrated strategies.Therefore,policymakers and stakeholders should apply the integrated intervention to stop the spread of MSV in the maize population.

Salton Sea: An ecosystem in crisis

Salton Sea (a destination resort) is to be saved from being converted into a skeletonfilled wasteland. The critical amount of water flowing into the sea to maintain its level and salinity has been diverted since January 2018. This will lead to shrinking volumes and increasing salinities. Ecological consequences and public health impacts of altered conditions will be phenomenal. We design and analyze a minimal eco-epidemiological model to figure out future journey of this sea;a way station for fish-eating migratory birds. The mathematical model has been assembled in terms of prey-predator interaction. The salient feature of the proposed model is its seasonally varying contact rate which represents rate of conversion of susceptible fishes into infectives. We have analytically investigated the global stability, disease persistence and periodic solutions of the proposed model system. Susceptible prey-induced periodic solution is globally asymptotically stable when R01< 1, otherwise unstable and hence disease persists for R01> 1.Global stability and Hopf bifurcation (HB) analysis help us extract parameter values to explore the dynamical behavior of the model system. Two-dimensional parameter scans and bifurcation diagrams reveal that the model displays propensity towards chaotic dynamics, which is associated with extinction-sized population densities. In the presence of stochastic external forces, this implies extinction of most of the fish species. This, in turn, suggests that resident birds will have to migrate to other destinations. The fish-eating migratory birds will be forced to switch over to invertebrates. Ecological consequences and public health impacts of this transition would be severe. Conservation groups are solicited to draw attention of the government to avert this impounding danger. It is important to plan for an ecosystem-wide transition such that impacts on birds and on human inhabitants living adjacent to the shrinking and salinizing sea are minimized.

Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.

Infectious diseases and climate:Case of Morocco

It is predicted that the life cycle,incidence and spread of several infectious diseases will be increasingly and adversely affected by climate change.Morocco,designated as an area of significant impact by numerous reports of Intergovernmental Panel on Climate Change,is notably susceptible to such drastic climate-related health consequences.The present work thus examines the increasing risk of vector-borne diseases in hazard-prone localities,while also highlights the current lack of dedicated scientific research in this critical area.It further identifies the severe challenges both of health adaptation to climate change and of consequent policy responses,before providing a more detailed overview of Morocco’s adaptive capacity to such crises.