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In the current study, we present a mathematical model for human immunodeficiency virus type-1 (HIV -1) transmission, incorporating Cytotoxic T-Lymphocyte immune impairment within a seasonal environment. The model divides the infected cell compartment into two sub-compartments: latently infected cells and productively infected cells. Additionally, we consider three possible routes of infection, allowing HIV to spread among susceptible cells via direct contact with the virus, latently infected cells, or productively infected cells. The system is analyzed, and the basic reproduction number is derived using an integral operator. We demonstrate that the HIV -free periodic trajectory is globally asymptotically stable if R 0 < 1 , while HIV persists when R 0 > 1. Several numerical simulations are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In the current study, we present a mathematical model for human immunodeficiency virus type-1 (HIV-1) transmission, incorporating Cytotoxic T-Lymphocyte immune impairment within a seasonal environment. The model divides the infected cell compartment into two sub-compartments: latently infected cells and productively infected cells. Additionally, we consider three possible routes of infection, allowing HIV to spread among susceptible cells via direct contact with the virus, latently infected cells, or productively infected cells. The system is analyzed, and the basic reproduction number is derived using an integral operator. We demonstrate that the HIV-free periodic trajectory is globally asymptotically stable if $ \mathcal{R}_0 < 1 $, while HIV persists when $ \mathcal{R}_0 > 1 $. Several numerical simulations are provided to validate the theoretical results.

A climate-based metapopulation malaria model is formulated by incorporating human travel between zones with varying climatic factors, effective and counterfeit drug treatments, and time-periodic parameters for the mosquito population to understand the effect of human travel on malaria transmission. We study the existence, uniqueness, and stability of positive periodic solutions in the model and carry out numerical simulations for three climatic zones of Ghana. The study shows that the climate effects introduce fluctuations in the solutions, while human travel between zones affects the disease prevalence in each zone and the local transmission dynamics of malaria. We observed different outcomes depending on various restrictions imposed on human travels. The study also suggests that it is essential to ban the sale, importation or manufacture of counterfeit drugs and punish the offenders to ensure the effective use of high-quality drugs in the population. [ABSTRACT FROM AUTHOR]

In order to study the impact of seasonality on Zika virus dynamics, we analyzed a non-autonomous mathematical model for the Zika virus (ZIKV) transmission where we considered time-dependent parameters. We proved that the system admitted a unique bounded positive solution and a global attractor set. The basic reproduction number, $ \mathcal{R}_0 $, was defined using the next generation matrix method for the case of fixed environment and as the spectral radius of a linear integral operator for the case of seasonal environment. We proved that if $ \mathcal{R}_0 $ was smaller than the unity, then a disease-free periodic solution was globally asymptotically stable, while if $ \mathcal{R}_0 $ was greater than the unity, then the disease persisted. We validated the theoretical findings using several numerical examples.

Hand, foot, and mouth disease (HFMD) is widely spread in China Mainland, seriously threatening the health of infants and young children. We develop a meta‐population model that includes both local and ecdemic populations to study the impacts of fomites and immigration on the transmission of HFMD in Shanghai. The model includes both direct transmission between susceptible and infected individuals (asymptomatic or symptomatic) and indirect transmission via fomites. The forward bifurcation shows that if the basic reproduction number is less than unity, the epidemic will die out; otherwise, the epidemic will spread. The fitting results show that the basic reproduction numbers of the local population and the ecdemic population are 1.4696 and 1.7288, respectively. Sensitivity analysis of the basic reproduction number showed that there was a high correlation between the basic reproduction number and the transmission rate of asymptomatic infected individuals, as well as the parameters related to the fomites (such as indirect transmission rate and virus shedding rate). Hence, asymptomatic infected individuals and fomites have a significant impact on the new HFMD infection. The numerical simulations on the prevention and control strategy show that reducing the transmission rate between susceptible and asymptomatic infected individuals or reducing the fomites can delay the outbreak of the epidemic and weaken the severity of HFMD. Our findings can provide guidance for the eradication of HFMD epidemic in the presence of indirect transmission and other factors. [ABSTRACT FROM AUTHOR]

This paper concerns with the global threshold dynamics of a Tuberculosis (TB) model incorporating age‐space structure, treatment, and relapse. The original model is converted into a hybrid system comprising two Volterra integral equations and two partial differential equations by integrating along the characteristic line. The well‐posedness for the model is demonstrated by using the fixed point theory in conjunction with the induction method. In order to discuss whether a disease is persistent or extinct, we provide the explicit formulation of the basic reproduction number. By analyzing the distribution of the characteristic roots of the characteristic equations and constructing the proper Lyapunov functionals, the local and global stability for the steady states are addressed. Numerical simulations are conducted to confirm the conclusions of our analytical results and reveal that reduction of the TB transmission coefficient, reduction of infectiousness of treated individuals infected with TB, and increasing the treatment rate of infectious class are three feasible measures to control the transmission of TB. [ABSTRACT FROM AUTHOR]

Inspired by the transmission characteristics of the Coronavirus disease 2019 (COVID-19), an epidemic model with quarantine and standard incidence rate is first developed, then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals, which means that the epidemic is uniformly persistent if the control reproduction number ℛ c > 1 . This approach can be applied to the related biomathematical models, and some existing works can be improved by using that. In addition, the infection-free equilibrium V0 of the model is locally asymptotically stable (LAS) if ℛ c < 1 and linearly stable if ℛ c = 1 ; while V0 is unstable if ℛ c > 1 . [ABSTRACT FROM AUTHOR]

In order to study the impact of seasonality on Zika virus dynamics, we analyzed a non-autonomous mathematical model for the Zika virus (ZIKV) transmission where we considered time-dependent parameters. We proved that the system admitted a unique bounded positive solution and a global attractor set. The basic reproduction number, R0, was defined using the next generation matrix method for the case of fixed environment and as the spectral radius of a linear integral operator for the case of seasonal environment. We proved that if R0 was smaller than the unity, then a disease-free periodic solution was globally asymptotically stable, while if R0 was greater than the unity, then the disease persisted. We validated the theoretical findings using several numerical examples. [ABSTRACT FROM AUTHOR]

Seasonality is repetitive in the ecological, biological and human systems. Seasonal factors affect both pathogen survival in the environment and host behavior. In this study, we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate. We started by studying the autonomous system by investigating the global stability of steady states. Later, we proved the existence, uniqueness, positivity and boundedness of a periodic orbit in a non-autonomous system. We demonstrate that the global dynamics are determined using the basic reproduction number ℛ0 which is defined by the spectral radius of a linear integral operator. We showed that if ℛ0 < 1, then the disease-free periodic solution is globally asymptotically stable and if ℛ0 > 1, then the trajectories converge to a limit cycle reflecting the persistence of the disease. Finally, we present a numerical investigation that support our results. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and analyzed a five-dimensional system of ordinary differential equations modeling the competition of two competing bacteria in a chemostat under the influence of the leachate recirculation and in the presence of a pathogen associated only with the bacteria 1. We suppose that the nutriment is present into two forms, soluble and insoluble nutriment, and both two forms are continuously added to the chemostat. The proposed model takes the form of an “SI” epidemic model and uses general increasing growth functions and general increasing incidence rate. It admits multiple equilibria that we give the conditions under which we assure both the existence and the local stability of each equilibrium point. The possibility of periodic trajectory was excluded, and the uniform persistence of both types of bacteria was proved. Finally, several numerical examples confirming the theoretical findings are given. [ABSTRACT FROM AUTHOR]

To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When do the signs of the invasion growth rates determine coexistence? (ii) When signs are sufficient, which invasion growth rates need to be positive? We focus on deterministic models and equate coexistence with permanence, i.e., a global attractor bounded away from extinction. For models satisfying certain technical assumptions, we introduce invasion graphs where vertices correspond to proper subsets of species (communities) supporting an ergodic measure and directed edges correspond to potential transitions between communities due to invasions by missing species. These directed edges are determined by the signs of invasion growth rates. When the invasion graph is acyclic (i.e. there is no sequence of invasions starting and ending at the same community), we show that permanence is determined by the signs of the invasion growth rates. In this case, permanence is characterized by the invasibility of all [Formula: see text] communities, i.e., communities without species i where all other missing species have negative invasion growth rates. To illustrate the applicability of the results, we show that dissipative Lotka-Volterra models generically satisfy our technical assumptions and computing their invasion graphs reduces to solving systems of linear equations. We also apply our results to models of competing species with pulsed resources or sharing a predator that exhibits switching behavior. Open problems for both deterministic and stochastic models are discussed. Our results highlight the importance of using concepts about community assembly to study coexistence.

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution. [ABSTRACT FROM AUTHOR]

In the proposed study, a nonlinear model is developed to explore the interactive dynamics between cattle and invertebrates when they coexist in a grassland system and compete with one another for the same resource — the grass biomass. The constructed model is theoretically investigated using the qualitative theory of differential equations to show the system's rich dynamical properties, which are crucial for maintaining the ecosystem's balance in grasslands. The qualitative findings show that, depending on the parameter combinations, the system not only displays stability of many equilibrium states but also experiences transcritical and Hopf bifurcations. The model results support the idea that inter-specific competition between cattle and invertebrates does not always produce regular dynamic patterns but may also produce periodic and destabilizing patterns. The model's outputs may assist in striking a balance between pasture and natural grass biomass in grassland with the invertebrates. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Zika transmission model which considers human-to-human sexual transmission, the extrinsic incubation period of mosquitoes, and the vector-bias effect. Firstly, the explicit expression of the basic reproduction number R0 is given by using the next-generation operator method, and the global dynamics of the model are established by taking R0 as the threshold condition, that is, if R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, if R0 > 1, the model has a unique endemic equilibrium that is locally asymptotically stable and the disease persists. And when we ignore the vector-bias effect, the global asymptotic stability of the endemic equilibrium is proved by constructing a Lyapunov function. Then, we select the reported epidemic data from Brazil for fitting, which verifies the obtained theoretical results. Meanwhile, we study the impact of human-to-human sexual transmission rate and mosquito-to-human transmission rate on the spread and prevalence of Zika. In addition, we calculate the sensitivity indices of R0 to the model parameters and provide effective measures to control Zika transmission. The simulation results indicate that extending the extrinsic incubation period of mosquitoes is beneficial for disease control while ignoring the vector-bias effect will underestimate the risk of Zika transmission. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and studied a simple five-dimensional mathematical model that describes the second and third stages of the anaerobic degradation process under the influence of leachate recirculation. The state variables are the concentration of insoluble substrate, soluble substrate, produced hydrogen, acetogenic bacteria and hydrogenotrophic-methanogenic bacteria. The growth rates of used bacteria will be of general nonlinear form. The stability of the steady states will be studied by reducing the model to a 3D system. According to the operating parameters of the bioreactor described by the added insoluble substrate, soluble substrate and hydrogen input concentrations and the dilution rate, we proved that the model can admit multiple equilibrium points and we gave the necessary and sufficient assumptions for their existence, their uniqueness and their stability. In particular, the uniform persistence of the system was satisfied under some natural assumptions on the growth rates. Then, a question was answered related to the management of renewable resources where the goal of was to propose an optimal strategy of leachate recirculation to reduce the organic matter (either soluble or insoluble) and keep a limitation of the costs of the recirculation operation during the process. The findings of this work were validated by an intensive numerical investigation.

Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number $ \mathcal{R}_0 $ is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case $ \mathcal{R}_0 > 1 $ when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given $ \mathcal{R}_0 > 1 $, if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring.

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by $ \mathcal{R}_0 $ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if $ \mathcal{R}_0 < 1 $ and we also show the persistence of the disease if $ \mathcal{R}_0 > 1 $ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.

The goal of this paper is to introduce a non-autonomous environmental transmission model for most respiratory and enteric infectious diseases to study the impact of periodic environmental changes on related infectious diseases. The transmission and decay rates of pathogens in the environment are set as periodic functions to summarize the influence of environmental fluctuations on diseases. The solutions of the model are qualitatively analyzed, and the equilibrium points and the reference criterion, $ R_0 $, for judging the infectivity of infectious diseases are deduced. The global stability of the disease-free equilibrium and the uniform persistence of the disease are proved by using the persistence theory. Common infectious diseases such as COVID-19, influenza, dysentery, pertussis and tuberculosis are selected to fit periodic and non-periodic models. Fitting experiments show that the periodic environmental model can respond to epidemic fluctuations more accurately than the non-periodic model. The periodic environment model is reasonable and applicable for seasonal infectious diseases. The response effects of the periodic and non-periodic models are basically the same for perennial infectious diseases. The periodic model can inform epidemiological trends in relevant emerging infectious diseases. Taking COVID-19 as an example, the sensitivity analysis results show that the virus-related parameters in the periodic model have the most significant influence on the system. It reminds us that, even late in the pandemic, we must focus on the viral load on the environment.

This paper is devoted to the study of spreading properties and traveling wave solutions for a vector-host disease system, which models the invasion of vectors and hosts to a new habitat. Combining the uniform persistence idea from dynamical systems with the properties of the corresponding entire solutions, we investigate the propagation phenomena in two different cases: (1) fast susceptible vector; (2) slow susceptible vector when the disease spreads. It turns out that in the former case, the susceptible vector may spread faster than the infected vector and host under appropriate conditions, which leads to multi-front spreading with different speeds; while in the latter case, the infected vector and host always catch up with the susceptible vector, and they spread at the same speed. We further obtain the existence and nonexistence of traveling wave solutions connecting zero to the endemic equilibrium. We also conduct numerical simulations to illustrate our analytic results. [ABSTRACT FROM AUTHOR]

We propose a general model to investigate the effect of the distinct dispersal coefficients infected and susceptible hosts in the pathogen dynamics. The mathematical challenge lies in the fact that the investigated model is partially degenerate and the solution map is not compact. The spatial heterogeneity of the model parameters and the distinct diffusion coefficients induce infection in the low-risk regions. In fact, as infection dispersal increases, the reproduction of the pathogen particles decreases. The dynamics of the investigated model is governed by the value of the basic reproduction number R0. If R0 ≤ 1, then the pathogen particles extinct, and for R0 > 1 the pathogen particles persist, and there is at least one positive steady state. The asymptotic profile of the positive steady state is shown in the case when one or both diffusion coefficients for the host tends to zero or infinity. [ABSTRACT FROM AUTHOR]

In this study, we consider a system of nonlinear differential equations modeling the human immunodeficiency virus type-1 (HIV-1) in a variable environment. Infected cells were subdivided into two compartments describing both latently and productively infected cells. Thus, three routes of infection were considered including the HIV-to-cell contact, latently infected cell-to-cell contact, and actively infected cell-to-cell contact. The nonnegativity and boundedness of the trajectories of the dynamics were proved. The basic reproduction number was determined through an integral operator. The global stability of steady states is then analyzed using the Lyapunov theory together with LaSalle's invariance principle for the case of a fixed environment. Similarly, for the case of a variable environment, we showed that the virus-free periodic solution is globally asymptotically stable once R 0 ≤ 1 , while the virus will persist once R 0 > 1 . Finally, some numerical examples are provided illustrating the theoretical investigations. [ABSTRACT FROM AUTHOR]

The spatial-temporal diffusion dynamics of infectious disease with vaccination therapy are studied through a mathematical model. We have investigated the well-posedness, disease-free equilibrium, disease equilibrium, the existence and the uniqueness of solutions, and the calculation of basic reproduction numbers by Jacobian matrix. After that, the positivity, as well as boundedness of solutions, are also established. The global stability of diseasefree and steady-state disease results is established by utilizing compatible Lyapunov functions and LaSalle's invariance principle. Illustration of the numerical examples to show the dynamics of different population groups over time. The effects of different parameters on the compartments are shown in detail. [ABSTRACT FROM AUTHOR]

We investigate the large time behaviour of solutions to a non-autonomous Fisher–KPP equation with non-local diffusion, involving a thin-tailed kernel. In this paper, we are concerned with both compactly supported and exponentially decaying initial data. As far as general time heterogeneities are concerned, we provide upper and lower estimates for the location of the propagating front. As a special case, we derive a definite spreading speed when the time varying coefficients satisfy some averaging properties. This setting covers the cases of periodic, almost periodic and uniquely ergodic variations in time, in particular. Our analysis is based on the derivation of suitable regularity estimates (of uniform continuity type) for some particular solutions of a logistic equation with non-local diffusion. Such regularity estimates are coupled with the construction of appropriated propagating paths to derive spreading speed estimates, using ideas from the uniform persistence theory in dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and studied a simple five-dimensional mathematical model that describes the second and third stages of the anaerobic degradation process under the influence of leachate recirculation. The state variables are the concentration of insoluble substrate, soluble substrate, produced hydrogen, acetogenic bacteria and hydrogenotrophic-methanogenic bacteria. The growth rates of used bacteria will be of general nonlinear form. The stability of the steady states will be studied by reducing the model to a 3D system. According to the operating parameters of the bioreactor described by the added insoluble substrate, soluble substrate and hydrogen input concentrations and the dilution rate, we proved that the model can admit multiple equilibrium points and we gave the necessary and sufficient assumptions for their existence, their uniqueness and their stability. In particular, the uniform persistence of the system was satisfied under some natural assumptions on the growth rates. Then, a question was answered related to the management of renewable resources where the goal of was to propose an optimal strategy of leachate recirculation to reduce the organic matter (either soluble or insoluble) and keep a limitation of the costs of the recirculation operation during the process. The findings of this work were validated by an intensive numerical investigation. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic analysis of spatially heterogeneous viral transmission, incorporating cell‐to‐cell transmission, virus nonlocal dispersal, and intracellular delay. Due to the noncompactness of the semiflow, we used the Kuratowski measure of noncompactness to demonstrate the existence of a global compact attractor. This noncompactness issue generates difficulties in calculating the basic reproduction number R0$$ {R}_0 $$, which is the principal eigenvalue of the next‐generation operator. The threshold role of this number is determined, where we derived two different cases: (i) the global stability of the virus‐free steady state, which is globally stable for R0<1$$ {R}_0<1 $$ by the Lyapunov direct method, and (ii) the global stability of the virus steady state for R0>1$$ {R}_0>1 $$. Indeed, the second case is demonstrated through several steps that include uniform persistence, the existence of a virus steady state, and the global stability of the virus steady state. The results are supported by different graphical representations with proper biological justifications. [ABSTRACT FROM AUTHOR]

Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number R 0 is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case R 0 > 1 when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given R 0 > 1 , if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring. [ABSTRACT FROM AUTHOR]

The increase of animal transportation and livestock lead to repeated outbreaks of brucellosis, and its transmission process is extremely complex. According to the clinical symptoms and infectious differences of the diseased sheep, it is further divided into acute infection and chronic infection, and a reaction-diffusion SLICB (susceptible-latent-acute infected-chronic infected-brucella) model with seasonality, spatial heterogeneity and nonlocal delay is proposed to understand the transmission law of brucellosis and analyze its transmission risk. Based on the basic reproduction number 0 of the model, the final development trend of brucellosis transmission among sheep is analyzed by persistence theory. The 0 of the model is numerically calculated by the generalized power method, and simulation analysis shows that: (i) extending the latent period and increasing the random walk rate of infected sheep can effectively prevent brucellosis from developing into an endemic disease; (ii) the greater the density of acute infections, the higher the risk of brucellosis transmission, and the density of infected sheep and the time to reach the stable state will have a large deviation if only consider acute or chronic sheep; (iii) reaching the peak time will be delayed if the peak time of sheep birth is delayed. These results can provide some suggestions for the control of brucellosis. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena for a basic spatial vaccination model posed on unbounded domain. The main features of the current three-dimensional model system are the lack of comparison principle and the unboundedness of bilinear incidence (mass action incidence), which brings about some difficulties in analyzing the long term behavior of the corresponding solutions. The asymptotic speed of spread is first established by weak dissipativity of system and uniform persistence idea on dynamical system. The full information on the traveling wave solutions is then investigated, which further indicates the coincidence with the asymptotic speed of spread. [ABSTRACT FROM AUTHOR]

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by R0 and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if R0 < 1 and we also show the persistence of the disease if R0 > 1 where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings. [ABSTRACT FROM AUTHOR]

An epidemic model is proposed to comprehend the disease dynamics between humans and animals and back to humans with a culling intervention strategy. The proposed model is separated into two cases with two different culling rates: (1) at a per-capita constant rate and (2) constant population being culled. The global asymptotic stability of equilibria is determined in terms of the basic reproduction numbers. Further, we find that the culling rate (2) considered in the model could change the dynamics by having multiple positive equilibria. Sensitivity analysis recommends developing a strategy that promotes animals' natural and disease-related death rates. By ranking the efficacies of various intervention strategies, we obtain that vaccination in the human population, isolation and public awareness are the largely effective control interventions. Our general theory raises concerns about both human and animal populations becoming reservoirs of the disease and affecting each other dynamically. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce a non-autonomous environmental transmission model for most respiratory and enteric infectious diseases to study the impact of periodic environmental changes on related infectious diseases. The transmission and decay rates of pathogens in the environment are set as periodic functions to summarize the influence of environmental fluctuations on diseases. The solutions of the model are qualitatively analyzed, and the equilibrium points and the reference criterion, R 0 , for judging the infectivity of infectious diseases are deduced. The global stability of the disease-free equilibrium and the uniform persistence of the disease are proved by using the persistence theory. Common infectious diseases such as COVID-19, influenza, dysentery, pertussis and tuberculosis are selected to fit periodic and non-periodic models. Fitting experiments show that the periodic environmental model can respond to epidemic fluctuations more accurately than the non-periodic model. The periodic environment model is reasonable and applicable for seasonal infectious diseases. The response effects of the periodic and non-periodic models are basically the same for perennial infectious diseases. The periodic model can inform epidemiological trends in relevant emerging infectious diseases. Taking COVID-19 as an example, the sensitivity analysis results show that the virus-related parameters in the periodic model have the most significant influence on the system. It reminds us that, even late in the pandemic, we must focus on the viral load on the environment. [ABSTRACT FROM AUTHOR]

This paper aims to the investigation of the global threshold dynamics of an infection age-space structured HIV infection model. The model is formulated in a bounded domain involving two infection routes (virus-to-cell and cell-to-cell) and Neumann boundary conditions. We first transform the original model to a hybrid system containing two partial differential equations and a Volterra integral equation. By appealing to the theory of fixed point problem together with Picard sequences, the well-posedness of the model is shown by verifying that the solution exists globally and the solution is ultimately bounded. Under the Neumann boundary condition, we establish the explicit expression of the basic reproduction number. By analyzing the distribution of characteristic roots of the associated characteristic equation in terms of the basic reproduction number, we achieve the local asymptotic stability of the steady states. The global asymptotic stability of the steady states is established by the technique of Lyapunov functionals, respectively. Numerical simulations are performed to validate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the dynamical behavior of an age-structured epidemic model including a nonlinear incidence and immune loss. For this the well-posedness of the considered model is obtained and the existence of the equilibria is obtained. Then the stability and persistence problems are investigated for this system. More precisely, the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $ R_0<1 $, while if $ R_0>1 $, the solutions of the system are uniformly persist and the positive equilibrium has locally asymptotical stability. Additionally a simple three-way partition for the global attractor of the solution semi-flow for the considered system is also obtained. In the end some numerical simulations are provided to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We derive a mathematical model that describes the competition of two populations in a chemostat in the presence of a virus. We suppose that only one population is affected by the virus. We also suppose that the substrate is continuously added to the bioreactor. We obtain a model taking the form of an "SI" epidemic model using general increasing growth rates of bacteria on the substrate and a general increasing incidence rate for the viral infection. The stability of the steady states was carried out. The system can have multiple steady states with which we can determine the necessary and sufficient conditions for both existence and local stability. We exclude the possibility of periodic orbits and we prove the uniform persistence of both species. Finally, we give some numerical simulations that validate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a deterministic model to study the transmission dynamics of anthrax disease, which includes live animals, carcasses, spores in the environment and vectors. We derive three biologically plausible and insightful quantities (reproduction numbers) that determine the stability of the equilibria. We carry out rigorous mathematical analysis on the model dynamics, the global stability of the disease-free and vector-free equilibrium, the disease-free equilibrium and the vector-free disease equilibrium is proved. The global stability of the endemic equilibrium as the basic reproduction number is greater than one is derived in the special case in which the disease-related death rate is zero. The possibility of backward bifurcation is briefly discussed. Numerical analyses are carried out to understand the transmission dynamics of anthrax and investigate effective control strategies for the outbreaks of the disease. Our studies suggest that the larval vector control measure should be taken as early as possible to control the vector population size, a vaccination policy and an animal carcass removal policy are useful methods to control the prevalence of the diseases in infected animal populations, the adult vector control measure is also necessary to prevent the transmission of anthrax. [ABSTRACT FROM AUTHOR]

In this article, we formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H(x) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.

In this article, based on the result of the uniform persistence for $ 4 $-dimensional autonomous system with interactions in a rocky intertidal community by Hsu et al.[7], the extinction of the model is further explored. According to the existence of the positive equilibrium, we rigorously classify three categories corresponding to extinct states of the three species Mussel, Algae and Barnacle, respectively. The extinction results of the $ 4 $-dimensional model in this paper exactly verify further the uniform persistence obtained in [7], and the classifications are total for the uniform persistence. What is rather more significant is that the distinguished criteria characterize the three extinct steady state. The global extinctions of the model are helpful to understand the mechanism of the the three species coexistence and the cyclic succession fluctuation observed in [1]. [ABSTRACT FROM AUTHOR]

Distributed delay is included in a simple predator–prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent "discount" factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the "discounting" term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions. [ABSTRACT FROM AUTHOR]

In the current study, we formulated and analyzed a nonlocal diffusive cholera model incorporating the spatial heterogeneous structure, and seasonally forced intrinsic incubation period for demonstrating the infection dynamics between humans and vibrios in the environment. Here the intrinsic incubation period is the developmental duration for the vibrios within human populations, which is assumed to be $ \omega $-periodic in time. We establish the well-poseness of the model and study the threshold-type result in terms of the basic reproduction number by introducing the next generation operator. Specifically, we confirm the result that in a homogeneous case, the unique positive equilibrium is globally attractive, achieved by Lyapunov functional. [ABSTRACT FROM AUTHOR]

Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by R 0 and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if R 0 < 1 , and we show also the persistence of the disease once R 0 > 1 . Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if R 0 > 1 . [ABSTRACT FROM AUTHOR]

This paper studies the efficiency of protection in containing an epidemic in the case of immigration using a triple age structured model. For determining this efficiency, we split our study into two different cases, (i) with immigration flow into all model classes, (ii) without infected immigrants flow. For (i), we prove that the investigated model has no threshold dynamics and the infection always persists and the unique endemic equilibrium is globally stable. For (ii), we prove that by stopping the infected immigration flow, we obtain the threshold dynamics again. Therefore, we confirm that the infected immigration flow is responsible for eliminating the threshold dynamics. The global stability results in each case are shown with the help of the Lyapunov method. Some graphical representations are used to confirm the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction-diffusion model of chronic infectious diseases with a special nonlinear incidence rate describing the mechanism of some chronic infections diseases. We first prove that the model has a unique nonnegative global classical solution, and admits a connected global attractor. Further, the threshold dynamics such as the uniform persistence of the model solution and the global stability of the steady-state solution are studied. The theory is verified by numerical simulation. Our results show that chronic infectious diseases have a strong ability to survive, and it is difficult to eradicate chronic infectious diseases completely. [ABSTRACT FROM AUTHOR]

A discrete stage-structured tick population dynamical system with diapause is studied, and spraying acaricides as the control strategy is considered in detail. We stratify vector populations in terms of their maturity status as immature and mature subgroups. The immature subgroup is divided into two categories: normal immature and diapause immature. We compute the net reproduction number $ R_0 $ and perform a qualitative analysis. When $ R_0 < 1 $, the global asymptotic stability of tick-free fixed point is well proved by the inherent projection matrix; there exists a unique coexistence fixed point and the conditions for its asymptotic stability are obtained if and only if $ R_0 > 1; $ the model has transcritical bifurcation if $ R_0 = 1. $ Moreover, we calculate the net reproduction numbers of the model with constant spraying acaricides and periodic spraying acaricides, respectively, and compare the effects of the two methods on controlling tick populations.

In this paper, the asymptotical behaviour of an age-structured Human Immunodeficiency Virus infection model with general non-linear infection function and logistic proliferation term is studied. Based on the existence of the equilibria and theory of operator semigroups, linearized stability/instability of the disease-free and endemic equilibria is investigated through the distribution of eigenvalues of the linear operator. Then persistence of the solution semi-flow of the considered system is studied by showing the existence of a global attractor and the obtained result shows that the solution semi-flow is persistent as long as the basic reproduction number |$R_{0}>1$|⁠. Moreover, the Hopf bifurcations problem around the endemic equilibrium is also considered for the situation with a specific infection function. Since the system has two different delays, four cases are discussed to investigate the influence of the time delays on the dynamics of system around the endemic equilibrium including stability and Hopf bifurcations. At last, some numerical examples with concrete parameters are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We introduce here a predator-prey model where the prey are affected by a disease. The prey are assumed to gather in herds, while the predators are loose and act on an individualistic basis. Therefore their hunting affects mainly the prey individuals occupying the outermost positions in the herd, which is modeled via a square root functional response. The conditions of boundedness and uniform persistence are established. Stability and bifurcation analysis of all feasible equilibrium are carried out. Conditions on the model parameters for the possible existence of limit cycles are derived, global stability analysis is also shown in proper choice of suitable Lyapunov function. Numerical simulation of the various bifurcations validate the theoretical results. It is found that the system ultimate behavior depends mainly on two crucial parameters, the force of infection and predator average handling time. A discussion of the biological significance of the investigation concludes the paper. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the competition of two species for a single essential growth-limiting nutriment with viral infection that affects only the first species. Although the classical models without viral infection suggest competitive exclusion, this model exhibits the stable coexistence of both species. We reduce the fourth-dimension proposed model to a three-dimension one. Thus, the coexistence of the two competing species is demonstrated using the theory of uniform persistence applied to the three-variable reduced system. We prove that there is no coexistence of both species without the presence of the virus and the satisfaction of some assumptions on the growth rates of species. Finally, we give some numerical simulations to confirm the obtained theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we first propose a mathematical model to study the impact of awareness programs on dengue transmission. The basic reproduction number ℛ 0 is derived. The existence and stability of equilibria are investigated. The uniform persistence is established when ℛ 0 is larger than one. Our results suggest that awareness programs have significant impacts on dengue transmission dynamics although they cannot affect ℛ 0 . When ℛ 0 is less than one, awareness programs can shorten the prevailing time effectively. When ℛ 0 is larger than one, awareness programs may destabilize the unique interior equilibrium and a stable periodic solution appears due to Hopf bifurcation. In particular, we find that the occurrence of Hopf bifurcation depends not only on the intensity of awareness programs but also on the level of ℛ 0 . Besides, large fluctuations in the number of infected individuals caused by the stable periodic solution may bring pressure on limited medical resources. Therefore, different from intuitive ideas, blindly increasing the intensity of awareness programs is not necessarily conducive to control the transmission of dengue. The decision-making department should decide to adopt different publicity strategies according to the current level of ℛ 0 . Finally, we consider the optimal control problem of the model and find the optimal control strategy corresponding to awareness programs by Pontryagin's Maximum Principle. The results manifest that the optimal control strategy can effectively mitigate the transmission of dengue. [ABSTRACT FROM AUTHOR]