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In this paper, we investigated a new heroin–cocaine epidemic model which incorporates spatial heterogeneity and nonlinear incidence rate. The main project of this paper is to explore the threshold dynamics in terms of the basic reproduction number $ \mathcal {R}_0 $ R 0 , which was defined by applying the next-generation operator. The threshold type results shown that if $ \mathcal {R}_0 \lt 1 $ R 0 < 1 , then the drug-free steady state is globally asymptotically stable. If $ \mathcal {R}_0 \gt 1 $ R 0 > 1 , then heroin–cocaine spread is uniformly persistent. Furthermore, the globally asymptotic stability of the drug-free steady state has been established for the critical case of $ \mathcal {R}_0=1 $ R 0 = 1 by analysing the local asymptotic stability and global attractivity. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a population suppression model and a population replacement model with periodic impulsive releases of Nilaparvata lugens infected with wStri. The conditions for the stability of wild- $ N.\,lugens $ N. l u g e n s -eradication periodic solution of two systems are obtained by applying the Floquet theorem and comparison theorem. And the sufficient conditions for the persistence in the mean of wild $ N.\,lugens $ N. l u g e n s are also given. In addition, the sufficient conditions for the extinction and persistence of the wild $ N.\,lugens $ N. l u g e n s in the subsystem without wLug are also obtained. Finally, we give numerical analysis which shows that increasing the release amount or decreasing the release period are beneficial for controlling the wild $ N.\,lugens $ N. l u g e n s , and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats. [ABSTRACT FROM AUTHOR]

Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis – namely how a directed movement in response to an external chemical signal may occur based on purely reaction–diffusion equations reflecting inner working of the cells. The model is inspired by the well-studied role of the rho-GTPase Cdc42 regulator of cell polarity, in particular in yeast cells. We analyse several versions of the model to better understand its analytic properties and prove global regularity in one and two dimensions. Using computer simulations, we demonstrate that in the framework of this model, at least in certain parameter regimes, the speed of the directed movement appears to be proportional to the size of the gradient of signalling chemical. This coincides with the form of the chemical drift in the most studied mean field model of chemotaxis, the Keller–Segel equation. [ABSTRACT FROM AUTHOR]

Leishmaniasis, an infectious disease, manifests itself mostly in two forms, cutaneous leishmaniasis (CL) and, a more severe and potentially deadly form, visceral leishmaniasis (VL). The current control strategy for leishmaniasis relies on chemotherapy drugs such as sodium antimony gluconate (SAG) and meglumine antimoniate (MA). However, all these chemotherapy compounds have poor efficacy, and they are associated with toxicity and other adverse effects, as well as drug resistance. While research in vaccine development for leishmaniasis is continuously progressing, no vaccine is currently available. However, some experimental vaccines such as LEISH-F1+MPL-SE (V) have demonstrated some efficacy when used as drugs for CL patients. In this paper we use a mathematical model to address the following question: To what extent vaccine shots can enhance the efficacy of standard chemotherapy treatment of leishmaniasis? Starting with standard MA treatment of leishmaniasis and combining it with three injections of V , we find, by Day 84, that efficacy increased from 29% to 65–91% depending on the amount of the vaccine. With two or just one injection of V , efficacy is still very high, but there is a definite resurgence of the disease by end-time. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a leading cause of death worldwide. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation. Diabetic patients are twice as likely to have a heart attack or stroke. In this paper, we consider a simplified mathematical model for diabetic atherosclerosis involving LDL, HDL, glucose, insulin, free radicals (ROS), β cells, macrophages and foam cells, which satisfy a system of partial differential equations with a free boundary, the interface between the blood flow and the plaque. We establish the existence of small radially symmetric stationary solutions to the model and study their stability. Our analysis shows that the plague will persist due to hyperglycemia even when LDL and HDL are in normal range, hence confirms that diabetes increase the risk of atherosclerosis. [ABSTRACT FROM AUTHOR]

Worldwide, the recent SARS-CoV-2 virus disease outbreak has infected more than 691,000,000 people and killed more than 6,900,000. Surprisingly, Sub-Saharan Africa has suffered the least from the SARS-CoV-2 pandemic. Factors that are inherent to developing countries and that contrast with their counterparts in developed countries have been associated with these disease burden differences. In this paper, we developed data-driven COVID-19 mathematical models of two 'extreme': Cameroon, a developing country, and New York State (NYS) located in a developed country. We then identified critical parameters that could be used to explain the lower-than-expected COVID-19 disease burden in Cameroon versus NYS and to help mitigate future major disease outbreaks. Through the introduction of a 'disease burden' function, we found that COVID-19 could have been much more severe in Cameroon than in NYS if the vaccination rate had remained very low in Cameroon and the pandemic had not ended. [ABSTRACT FROM AUTHOR]

This paper aims to apply an optimal control theory for the autonomous model of the leptospirosis epidemic to examine the effect of four time-dependent control measures on the model dynamics with cost-effectiveness. Pontryagin's Maximum Principle was used to derive the optimality system associated with the optimal control problem. Numerical simulations of the optimality system were performed for different control strategies and the results were presented graphically with and without controls. The optimality system was simulated using the Forward–Backward Sweep method in the Matlab programme. The numerical results revealed that the combination of all optimal control measures is the most effective strategy for minimizing the spread and impact of disease in the community. Furthermore, a cost-effectiveness analysis was performed to determine the most cost-effective strategy using the incremental cost-effectiveness ratio approach and we observed that the rodenticide control-only strategy is most effective to combat the spread of disease when available resources are limited. [ABSTRACT FROM AUTHOR]

This paper studies a 2-dimensional discrete-time competition model of Ricker type with reproductive delay. The model is examined under the assumption that species 1 and 2 have the same properties except that a fraction η of species 1 individuals delays the initiation of reproduction. This assumption ensures that species 1 is dominated by species 2 in the sense that species 2 is increasing whenever species 1 is increasing. It is shown that, even under this assumption, delayed reproduction can be adaptive, i.e. species 1 can invade the monoculture system of species 2 while species 2 cannot invade the monoculture system of species 1, if the population is fluctuating. The result is obtained by analytically examining the species invasibility at boundary 2-cycles, whose coordinates can be estimated by assuming $ \eta \approx 0 $ η ≈ 0. [ABSTRACT FROM AUTHOR]

Populations are generally censused daily, weekly, monthly or annually. In this paper, we introduce a discrete-time nutrients-phytoplankton-oysters (NPO) model that describes the interactions of nutrients, phytoplankton and oysters in a bay ecosystem. We compute the threshold parameter $ \mathcal {R}_N $ R N for persistence of phytoplankton with or without oysters. When $ \mathcal {R}_{N} \lt 1 $ R N < 1 , then both phytoplankton and oysters populations go extinct. However, when $ \mathcal {R}_N \gt 1 $ R N > 1 , we show that the model may exhibit two scenarios: (1) a locally asymptotically stable equilibrium with positive values of nutrients and phytoplankton with oysters missing, and (2) a locally asymptotically stable interior equilibrium with positive values of nutrients, phytoplankton and oysters. We use sensitivity analysis to study the impact of human and environmental factors on the model. We use examples to illustrate that some human activities and environmental factors can force the interior equilibrium to undergo a Neimark–Sacker bifurcation which generates phytoplankton blooms with oscillations in oysters population and nutrients level. [ABSTRACT FROM AUTHOR]

An intraguild predator–prey model including prey refuge and hunting cooperation is investigated in this paper. First, for the corresponding ordinary differential equation model, the existence and stability of all equilibria are given, and the existence of Hopf bifurcation, direction and stability of bifurcating periodic solutions are investigated. Then, for partial differential equation model, the diffusion-driven Turing instability is obtained. What is more, the existence and non-existence of the non-constant positive steady state of the reaction–diffusion model are established by the Leray–Schauder degree theory and some priori estimates. Next, some numerical simulations are performed to support analytical results. The results showed that prey refuge can change the stability of model and even have a stabilizing effect on this model, meanwhile the hunting cooperation can make such model without diffusion unstable, but make such model with diffusion stable. Lastly, a brief conclusion is concluded in the last section. [ABSTRACT FROM AUTHOR]

In this paper, we consider a flux-limited Keller–Segel model derived in [16, 18] in a one-dimensional bounded domain and give a refined asymptotic result of the spiky steady state by using the Sturm oscillation theorem in a more meticulous way based on the existence result of spiky steady state in [4], showing a more accurate characterization of the cell aggregation phenomenon. [ABSTRACT FROM AUTHOR]

This paper presents an age-structured mathematical model for malaria transmission dynamics with asymptomatic carrier and temperature variability. The temperature variability function is fitted to the temperature data, and the malaria model is then fitted to the malaria cases and validated to check its suitability. Time-dependent controls were considered, including Long Lasting Insecticide Nets, treatment of symptomatic, screening and treatment of asymptomatic carriers and spray of insecticides. Pontryagin's Maximum Principle is used to derive the necessary conditions for optimal control of the disease. The numerical simulations of the optimal control problem reveal that the strategy involving the combination of all four controls is the most effective in reducing the number of infected individuals. Furthermore, the cost-effectiveness analysis shows that treatment of symptomatic, screening and treatment of asymptomatic carriers and insecticide spraying is the most cost-effective strategy to implement to control malaria transmission when available resources are limited. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process is formulated and analysed, which is used to obtain a better understanding of the population dynamics. At first, we validate that the stochastic system has a unique global solution with any initial value. Then we analyse the stochastic dynamics of the model in detail, including pth moment boundedness, asymptotic pathwise estimation in turn. After that, we obtain sufficient conditions for the existence of a stationary distribution of the system by adopting stochastic Lyapunov function methods. In addition, under some mild conditions, we derive the specific form of covariance matrix in the probability density near the quasi-positive equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we developed a mathematical model to simulate virus transport through a viscous background flow driven by the natural pumping mechanism. Two types of respiratory pathogens viruses (SARS-Cov-2 and Influenza-A) are considered in this model. The Eulerian–Lagrangian approach is adopted to examine the virus spread in axial and transverse directions. The Basset–Boussinesq–Oseen equation is considered to study the effects of gravity, virtual mass, Basset force, and drag forces on the viruses transport velocity. The results indicate that forces acting on the spherical and non-spherical particles during the motion play a significant role in the transmission process of the viruses. It is observed that high viscosity is responsible for slowing the virus transport dynamics. Small sizes of viruses are found to be highly dangerous and propagate rapidly through the blood vessels. Furthermore, the present mathematical model can help to better understand the viruses spread dynamics in a blood flow. [ABSTRACT FROM AUTHOR]

HIV/AIDS-COVID-19 co-infection is a major public health concern especially in developing countries of the world. This paper presents HIV/AIDS-COVID-19 co-infection to investigate the impact of interventions on its transmission using ordinary differential equation. In the analysis of the model, the solutions are shown to be non-negative and bounded, using next-generation matrix approach the basic reproduction numbers are computed, sufficient conditions for stabilities of equilibrium points are established. The sensitivity analysis showed that transmission rates are the most sensitive parameters that have direct impact on the basic reproduction numbers and protection and treatment rates are more sensitive and have indirect impact to the basic reproduction numbers. Numerical simulations shown that some parameter effects on the transmission of single infections as well as co-infection, and applying the protection rates and treatment rates have effective roles to minimize and also to eradicate the HIV/AIDS-COVID-19 co-infection spreading in the community. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with nonuniform predator dispersal, called predation-induced dispersal (PID), which represents predator motility depending on the maximal predation rate and the predator death rate in a spatially heterogeneous region. We study the local stability of the semitrivial steady state when predators are absent for models with PID and linear dispersal. We then investigate the local/global bifurcation from the semitrivial steady state of these models. Finally, we compare the results of the model with PID to the results of the model with linear dispersal. We conclude that the nonuniform dispersal of predators obeying PID increases fitness for predator invasion when rare; thus, predators with PID can invade a region with an increased probability even in cases wherein predators dispersed linearly cannot invade a certain region. Based on the results, we provide an ecological interpretation with the simulations. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

In a recent paper, Che et al. [5] used a continuous-time Ordinary Differential Equation (ODE) model with risk structure to study cholera infections in Cameroon. However, the population and the reported cholera cases in Cameroon are censored at discrete-time annual intervals. In this paper, unlike in [5], we introduce a discrete-time risk-structured cholera model with no spatial structure. We use our discrete-time demographic equation to 'fit' the annual population of Cameroon. Furthermore, we use our fitted discrete-time model to capture the annually reported cholera cases from 1987 to 2004 and to study the impact of vaccination, treatment and improved sanitation on the number of cholera infections from 2004 to 2019. Our discrete-time cholera model confirms the results of the ODE model in [5]. However, our discrete-time model predicts a decrease in the number of cholera cases in a shorter period of cholera intervention (2004–2019) as compared to the ODE model's period of intervention (2004–2022). [ABSTRACT FROM AUTHOR]

In this paper we assess the effectiveness of different non-pharmaceutical interventions (NPIs) against COVID-19 utilizing a compartmental model. The local asymptotic stability of equilibria (disease-free and endemic) in terms of the basic reproduction number have been determined. We find that the system undergoes a backward bifurcation in the case of imperfect quarantine. The parameters of the model have been estimated from the total confirmed cases of COVID-19 in India. Sensitivity analysis of the basic reproduction number has been performed. The findings also suggest that effectiveness of face masks plays a significant role in reducing the COVID-19 prevalence in India. Optimal control problem with several control strategies has been investigated. We find that the intervention strategies including implementation of lockdown, social distancing, and awareness only, has the highest cost-effectiveness in controlling the infection. This combined strategy also has the least value of average cost-effectiveness ratio (ACER) and associated cost. [ABSTRACT FROM AUTHOR]

In this paper, we use a mean-reverting Ornstein-Uhlenbeck process to simulate the stochastic perturbations in the environment, and then a modified Leslie–Gower Holling-type II predator-prey stochastic model in a polluted environment with interspecific competition and pulse toxicant input is proposed. Through constructing V-function and applying It o ^ ′ s formula, the sharp sufficient conditions including strongly persistent in the mean, persistent in the mean and extinction are established. In addition, the theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

Diabetes mellitus is a noncommunicable disease, which is a serious threat to human health around the world. In this paper, we propose a simple glucose–insulin model with Michaelis–Menten function as insulin degradation rate to mimic the pathogenic mechanism of diabetes. By theoretical analysis, a unique positive equilibrium of model exists and it is globally asymptotically stable. The four strategies are designed for diabetes patients based on the sensitivity of parameters, including insulin injection and medicine treatments. Numerical simulations are given to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper mainly explores the complex impacts of spatial heterogeneity, vector-bias effect, multiple strains, temperature-dependent extrinsic incubation period (EIP) and seasonality on malaria transmission. We propose a multi-strain malaria transmission model with diffusion and periodic delays and define the reproduction numbers R i and R ^ i (i = 1, 2). Quantitative analysis indicates that the disease-free ω-periodic solution is globally attractive when R i < 1 , while if R i > 1 > R j ( i ≠ j , i , j = 1 , 2), then strain i persists and strain j dies out. More interestingly, when R 1 and R 2 are greater than 1, the competitive exclusion of the two strains also occurs. Additionally, in a heterogeneous environment, the coexistence conditions of the two strains are R ^ 1 > 1 and R ^ 2 > 1. Numerical simulations verify the analytical results and reveal that ignoring vector-bias effect or seasonality when studying malaria transmission will underestimate the risk of disease transmission. [ABSTRACT FROM AUTHOR]

The novel Coronavirus (COVID-19) infection has become a global public health issue, and it has been a cause for morbidity and mortality of more people throughout the world. In this paper, we investigated the impacts of vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment strategies simultaneously using a deterministic mathematical modelling approach. No one has considered these intervention strategies simultaneously in his/her modelling approach. We examined all the qualitative properties of the model such as the positivity and boundedness of the model solutions, the disease-free and endemic equilibrium points, the effective reproduction number using next-generation matrix method, local stabilities of equilibrium points using the Routh–Hurwitz method. Using the Centre Manifold criteria, we have shown the existence of backward bifurcation whenever the COVID-19 effective reproduction number is less than unity. Moreover, we have analysed both sensitivity and numerical simulation using parameter values taken from published literature. The numerical results show that the transmission rate is the most sensitive parameter we have to control. Also vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment have great effects to minimize the COVID-19 transmission in the community. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R 0 and show that if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, whereas if R 0 > 1 , then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves. [ABSTRACT FROM AUTHOR]

Latently infected CD 4 + T cells represent one of the major obstacles to HIV eradication even after receiving prolonged highly active anti-retroviral therapy (HAART). Long-term use of HAART causes the emergence of drug-resistant virus which is then involved in HIV transmission. In this paper, we develop mathematical HIV models with staged disease progression by incorporating entry inhibitor and latently infected cells. We find that entry inhibitor has the same effect as protease inhibitor on the model dynamics and therefore would benefit HIV patients who developed resistance to many of current anti-HIV medications. Numerical simulations illustrate the theoretical results and show that the virus and latently infected cells reach an infected steady state in the absence of treatment and are eliminated under treatment whereas the model including homeostatic proliferation of latently infected cells maintains the virus at low level during suppressive treatment. Therefore, complete cure of HIV needs complete eradication of latent reservoirs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic delay mutualistic model of leaf-cutter ants with stage structure and their fungus garden, in which we explore how the discrete delay and white noise affect the dynamic of the population system. The existence and uniqueness of global positive solution are proved, and the asymptotic behaviours of the stochastic model around the positive equilibrium point of the deterministic model are also investigated. Furthermore, the sufficient conditions for the persistence of the population are established. Finally, some numerical simulations are performed to show the effect of random environmental fluctuation on the model. [ABSTRACT FROM AUTHOR]

HIV is a virus that weakens a person's immune system. HIV has three stages, and AIDS is the most severe stage of HIV (Stage 3). People with HIV should take medicine (called ART) recommended by WHO as soon as possible to reduce the amount of virus in the body. In this paper, we formulate a mathematical model for HIV/AIDS with a new approach by focusing on two groups of infectious individuals, HIV and AIDS. We also introduce a controlled class (treated patients and being monitored), and people in this class can spread the disease. We further investigate the essential dynamics of the model through an equilibrium analysis. Optimal control theory is applied to explore effective treatment strategies by combining two control measures: standard antiretroviral therapy and AIDS treatments. Numerical simulation results show the effects of the two time-dependent controls, and they can be used as guidelines for public health interventions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fear effect predator–prey model with mutual interference or group defense. For the model with mutual interference, we show the interior equilibrium is globally stable, and the mutual interference can stabilize the predator–prey system. For the model with group defense, we discuss the singular dynamics around the origin and the occurrence of Hopf bifurcation, and find that there is a separatrix curve near the origin such that the orbits above which tend to the origin and the orbits below which tend to limit cycle or the interior equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamical behaviour of a stochastic coronavirus (COVID-19) susceptible-infected-removed epidemic model with the inclusion of the influence of information intervention and Lévy noise. The existence and uniqueness of the model positive solution are proved. Then, we establish a stochastic threshold as a sufficient condition for the extinction and persistence in mean of the disease. Based on the available COVID-19 data, the parameters of the model were estimated and we fit the model with real statistics. Finally, numerical simulations are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a stage-structured wild and sterile mosquito interaction impulsive model. The aim is to study the feasibility of controlling the population of wild mosquitoes by releasing sterile mosquitoes periodically. The existence of trivial periodic solutions is obtained, and the corresponding local stability and global stability conditions are proved by Floquet theory and Lyapunov stability theorem, respectively. And we prove the existence conditions of non-trivial periodic solutions and their local stability. We can find that the system has the bistable phenomenon in which the trivial periodic solution and the non-trivial periodic solution can coexist under certain threshold conditions. All the results show that the appropriate release period and release amount of sterile mosquitoes can control the wild mosquito population within a certain range and even make them extinct. Finally, numerical simulation verifies our theoretical results. [ABSTRACT FROM AUTHOR]

Habitat loss and harvesting of non-timber forest products (NTFPs) significantly affect the population dynamics. In this paper, we propose a general mathematical modelling approach incorporating the impact of habitat size reduction and non-lethal harvesting of NTFP on population dynamics. The model framework integrates experimental data of Pentadesma butyracea in Benin. This framework allows us to determine the rational non-lethal harvesting level and habitat size to ensure the stability of the plant ecosystem, and to study the impacts of distinct levels of humidity. We suggest non-lethal harvesting policies that maximize the economic benefit for local populations. [ABSTRACT FROM AUTHOR]

This paper is concerned with a modified Leslie–Gower and Holling-type II two-predator one-prey model with Lévy jumps. First, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and prove that there is a unique positive solution to the system. Moreover, sufficient conditions for persistence in the mean and extinction of each species are established. Finally, we give some numerical simulations to support the main results. [ABSTRACT FROM AUTHOR]

We investigate a mosquito population suppression model, which includes the release of Wolbachia-infected males causing incomplete cytoplasmic incompatibility (CI). The model consists of two sub-equations by considering the density-dependent birth rate of wild mosquitoes. By assuming the release waiting period T is larger than the sexual lifespan T ¯ of Wolbachia-infected males, we derive four thresholds: the CI intensity threshold s h ∗ , the release amount thresholds g ∗ and c ∗ , and the waiting period threshold T ∗ . From a biological view, we assume s h > s h ∗ throughout the paper. When g ∗ < c < c ∗ , we prove the origin E 0 is locally asymptotically stable iff T < T ∗ , and the model admits a unique T-periodic solution iff T ≥ T ∗ , which is globally asymptotically stable. When c ≥ c ∗ , we show the origin E 0 is globally asymptotically stable iff T ≤ T ∗ , and the model has a unique T-periodic solution iff T > T ∗ , which is globally asymptotically stable. Our theoretical results are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the influence of the waning immunity on the dynamics of Tilapia Lake Virus (TiLV) transmission in wild and farmed tilapia within freshwater. We formulate a model for which susceptible individuals can contract the disease in two ways: (i) direct mode caused by contact with infected individuals; (ii) indirect mode due to the presence of pathogenic agents in the water. We obtain an age-structured model which combines both age since infection and age since recovery. We derive an explicit formula for the reproductive number R 0 and show that the disease-free equilibrium is locally asymptotically stable when, R 0 < 1. We discuss on the form of the waning immunity parameter and show numerically that a Hopf bifurcation may occur for suitable immunity parameter values, which means that there is a periodic solution around the endemic equilibrium when, R 0 > 1. [ABSTRACT FROM AUTHOR]

In this paper, we derive a delayed epidemic model to describe the characterization of cytotoxic T lymphocyte (CTL)-mediated immune response against virus infection. The stability of equilibria and the existence of Hopf bifurcation are analysed. Theoretical results reveal that if the basic reproductive number is greater than 1, the positive equilibrium may lose its stability and the bifurcated periodic solution occurs when time delay is taken as the bifurcation parameter. Furthermore, we investigate an optimal control problem according to the delayed model based on the available therapy for hepatitis B infection. With the aim of minimizing the infected cells and viral load with consideration for the treatment costs, the optimal solution is discussed analytically. For the case when periodic solution occurs, numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences. [ABSTRACT FROM AUTHOR]

In this paper, the actual background of the susceptible population being directly patients after inhaling a certain amount of PM 2.5 is taken into account. The concentration response function of PM 2.5 is introduced, and the SISP respiratory disease model is proposed. Qualitative theoretical analysis proves that the existence, local stability and global stability of the equilibria are all related to the daily emission P 0 of PM 2.5 and PM 2.5 pathogenic threshold K. Based on the sensitivity factor analysis and time-varying sensitivity analysis of parameters on the number of patients, it is found that the conversion rate β and the inhalation rate η has the largest positive correlation. The cure rate γ of infected persons has the greatest negative correlation on the number of patients. The control strategy formulated by the analysis results of optimal control theory is as follows: The first step is to improve the clearance rate of PM 2.5 by reducing the PM 2.5 emissions and increasing the intensity of dust removal. Moreover, such removal work must be maintained for a long time. The second step is to improve the cure rate of patients by being treated in time. After that, people should be reminded to wear masks and go out less so as to reduce the conversion rate of susceptible people becoming patients. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic and stable dynamics of an interactive wild and sterile mosquito population model with density-dependent survival probability. We find a release amount upper bound G ∗ , depending on the release waiting period T, such that the model has exactly two periodic solutions, with one stable and another unstable, provided that the release amount does not exceed G ∗ . A numerical example is also given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

COVID-19 is a disease caused by infection with the virus 2019-nCoV, a single-stranded RNA virus. During the infection and transmission processes, the virus evolves and mutates rapidly, though the disease has been quickly controlled in Wuhan by 'Fangcang' hospitals. To model the virulence evolution, in this paper, we formulate a new age structured epidemic model. Under the tradeoff hypothesis, two special scenarios are used to study the virulence evolution by theoretical analysis and numerical simulations. Results show that, before 'Fangcang' hospitals, two scenarios are both consistent with the data. After 'Fangcang' hospitals, Scenario I rather than Scenario II is consistent with the data. It is concluded that the transmission pattern of COVID-19 in Wuhan obey Scenario I rather than Scenario II. Theoretical analysis show that, in Scenario I, shortening the value of L (diagnosis period) can result in an enormous selective pressure on the evolution of 2019-nCoV. [ABSTRACT FROM AUTHOR]

In this paper, a general predator–prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order- 1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out. [ABSTRACT FROM AUTHOR]

In this paper, the invasive speed selection of the monostable travelling wave for a three-component lattice Lotka–Volterra competition system is studied via the upper and lower solution method, as well as the comparison principle. By constructing several special upper and lower solutions, we establish sufficient conditions such that the linear or nonlinear selection is realized. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function E f (S) is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number R 0 is calculated. When R 0 ≤ 1 the global asymptotical stability of the disease-free equilibrium is obtained, while when R 0 > 1 the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when R 0 > 1 the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems. [ABSTRACT FROM AUTHOR]

This paper aims to analyse stability and Hopf bifurcation of the HIV-1 model with immune delay under the functional response of the Holling II type. The global stability analysis has been considered by Lyapunov–LaSalle theorem. And stability and the sufficient condition for the existence of Hopf Bifurcation of the infected equilibrium of the HIV-1 model with immune response are also studied. Some numerical simulations verify the above results. Finally, we propose a novel three dimension system to the future study. [ABSTRACT FROM AUTHOR]

In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end. [ABSTRACT FROM AUTHOR]

This paper suggests a new way to think about a famous question: what explains cooperation in nature and in particular in humans? I argue that, for an evolutionary biologist as well as a quantitative social scientist, the triangle of two 'teammates' in the presence of a predator (passing and shooting in two-on-one situations) is one of the fundamental conceptual building-blocks for understanding these phenomena because in such a situation the fact that life is packaged in many distinct enclosures (and not in one big monolithic blob) can unfold its comparative advantage. I show how, in the presence of a predator, cooperative equilibria emerge among entirely selfish teammates if we infinitesimally bias the lead player in the selfish direction or assign a computational burden on the predator due to the presence of a teammate. I argue that 'predators' are common in the biological jungle but also in everyday human settings. Intuitively, this paper builds on the simple idea - a familiar one to a biologist observing the natural world but perhaps less so to social scientists - that everybody has enemies. [ABSTRACT FROM AUTHOR]

Epidemiological models with the identical basic reproduction number R 0 may behave differently on both short time scale and long time scale. In this paper, we compare the predicted final sizes for several deterministic epidemic models and estimate the probabilities of a minor/major outbreak for continuous-time Markov chain (CTMC) models, all epidemic models have the identical R 0 . It is proved that the final size predicted by the epidemic model with homogeneous mixing is larger than with heterogeneous mixing. For CTMC models with heterogeneous mixing, the probabilities of a minor outbreak initiated by superspreaders and non-superspreaders are calculated and compared. For both deterministic modelling and stochastic modelling, numerical simulations are performed to support the mathematical analysis. [ABSTRACT FROM AUTHOR]

The outbreak of COVID-19 was first experienced in Wuhan City, China, during December 2019 before it rapidly spread over globally. This paper has proposed a mathematical model for studying its transmission dynamics in the presence of face mask wearing and hospitalization services of human population in Tanzania. Disease-free and endemic equilibria were determined and subsequently their local and global stabilities were carried out. The trace-determinant approach was used in the local stability of disease-free equilibrium point while Lyapunov function technique was used to determine the global stability of both disease-free and endemic equilibrium points. Basic reproduction number, R 0 , was determined in which its numerical results revealed that, in the presence of face masks wearing and medication services or hospitalization as preventive measure for its transmission, R 0 = 0.698 while in their absence R 0 = 3.8. This supports its analytical solution that the disease-free equilibrium point E 0 is asymptotically stable whenever R 0 < 1 , while endemic equilibrium point E ∗ is globally asymptotically stable for R 0 > 1. Therefore, this paper proves the necessity of face masks wearing and hospitalization services to COVID-19 patients to contain the disease spread to the population. [ABSTRACT FROM AUTHOR]

In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. 2019 (to appear).], we embed the non-autonomous difference equations in an autonomous discrete dynamical systems in a higher dimension space, which is the product space of the phase space and the space of the functions defining the non-autonomous system. Our current approach applies to two scenarios. In the first scenario, we assume that the trait equations are decoupled from the equations of the populations. This requires specialized biological and ecological assumptions which we clearly state. In the second scenario, we do not assume decoupling, but rather we assume that the dynamics of the trait is known, such as approaching a positive stable equilibrium point which may apply to a much broader evolutionary dynamics. [ABSTRACT FROM AUTHOR]