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Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds R 0 (the basic reproduction number) and R c (a threshold less than one). Our investigations show that when R 0 > 1 , the system has an endemic equilibrium, which is globally asymptotically stable. When R c < R 0 < 1 , the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number R 0 is less than R c . The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions. [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]

Due to the random search of species and from the economic point of view, combined harvesting is more suitable than selective harvesting. Thus, we have developed and analyzed a prey–predator model with the combined effect of nonlinear harvesting in this research paper. Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system. We have examined the local stability analysis of all the equilibrium points. Besides these various types, rich and complex dynamical behaviors such as backward, saddle-node, Hopf and Bogdanov–Takens (BT) bifurcations, homo-clinic loop and limit cycles appear in this model. Furthermore, interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points. Also, we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species. We have observed that the extinction of predator species may not happen due to backward bifurcation, although a stable predator-free equilibrium (PFE) exists. Finally, numerical simulations are discussed using MATLAB to verify all the theoretical results. [ABSTRACT FROM AUTHOR]

Whenever a disease spreads in the population, people have a tendency to alter their behavior due to the availability of knowledge concerning disease prevalence. Therefore, the incidence term of the model must be suitably changed to reflect the impact of information. Furthermore, a lack of medical resources affects the dynamics of disease. In this paper, a mathematical model of malaria of type ShIhRh − SvIv with media information and saturated treatment is considered. The analysis of the model is performed and it is established that when the basic reproduction number, ℛ0, is less than unity, the disease may or may not die out due to saturated treatment. Furthermore, it is pointed out that if medical resources are accessible to everyone, disease elimination in this situation is achievable. The global asymptotic stability of the unique endemic equilibrium point (EEP) is established using the geometric approach under parametric restriction. The sensitivity analysis is also carried out using the normalized forward sensitivity index (NFSI). It is difficult to derive the analytical solution for the governing model due to it being a system of highly nonlinear ordinary differential equations. To overcome this challenge, a specialized numerical scheme known as the non-standard finite difference (NSFD) approach has been applied. The suggested numerical method is subjected to an elaborate theoretical analysis and it is determined that the NSFD scheme maintains the positivity and conservation principles of the solutions. It is also established that the disease-free equilibrium (DFE) point has the same local stability criteria as that of continuous model. Our proposed NSFD scheme also captures the backward bifurcation phenomena. The outcomes of the NSFD scheme are compared to two well-known standard numerical techniques, namely the fourth-order Runge–Kutta (RK4) method and the forward Euler method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a partial differential equation (PDE) model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans. The model features the variable infectiousness during the infectious period, which we call the infection age of the infectious host. We define two thresholds ℛ1j and ℛ2j,j = 1, 2, and show that the strain j can not invade the system if ℛ1j +ℛ 2j < 1. Further, the disease-free equilibrium of the system is globally asymptotically stable if maxj{ℛ1j +ℛ 2j} < 1. When ℛ1j > 1, strain j dominance equilibrium ℰj exists, and is locally asymptotically stable when ℛ1j > 1, ℛ1i < ςℛ 1j,i,j = 1, 2,i≠j, ς ∈ (0, 1). Then, by applying Lyapunov–LaSalle techniques, we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain j. This implies strain j eliminates the other strain as long as ℛ1i/ℛ 1j < b i/bj < 1,i≠j, where bj denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain j. Finally, we study the existence of the coexistence equilibria under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19. The basic reproduction number is calculated using the next generation matrix method. First, the existence of equilibrium points is investigated, and different kinds of equilibrium points indicate that the disease may disappear, or exist that result in different quantity of susceptible individuals, pre-symptomatic infected individuals and symptomatic infected individuals. The stability of the equilibria is discussed by a geometric approach, and it is found that controlling reproduction number to be lower than 1 is not sufficient for eradication of COVID-19. Second, transcritical bifurcation is explored, and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions, which indicates that two endemic equilibrium points appear. Third, to investigate the influence of parameters on the basic reproduction, sensitivity analysis is done to choose relatively sensitive parameters, and the parameters for treatment and media coverage are selected. An optimal control model is established to balance the treatment and media awareness. By exploring the existence and the uniqueness of the optimal control solution, the optimal control strategies are given. Finally, we run numerical simulations to verify the theoretical analysis on actual data of China, and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period. It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals. [ABSTRACT FROM AUTHOR]

The pharmaceutical interventions of emerging infectious diseases are constrained by the available medical resources such as drugs, vaccines, hospital beds, isolation places and the efficiency of the treatment. The awareness of the population also plays an important role in reducing contacts and consequently, reducing the disease transmission rate. In this paper, we propose a multi-group Susceptible, Infected and Recovered (SIR) epidemic model incorporating the awareness of population and the saturated treatment function that describes the effects of the availability of medical resources for treatment. We assume that the treatment of the infected individuals of a group is affected by the medical resources for the treatment of each group. We calculate the basic reproduction number R 0 in the term of the awareness parameter using the next generation approach. We determine the local and global stabilities of equilibrium (disease free equilibrium and endemic equilibrium) in terms of R 0 and the availability of medical resources for treatment. We obtain that backward bifurcation occurs at R 0 = 1 along with the existence of multiple endemic equilibria when R 0 < 1. Further, we consider the special case with a single group epidemic system and ensure the existence of multiple endemic equilibria. We showed a necessary condition on the parameter related to the availability of medical resources when backward bifurcation occurs. This situation indicates that reducing the basic reproduction number below unity is not sufficient to remove the disease when the medical resources for treatment are scarce. We used numerical simulations to support and counterpart our theoretical results and discussed the impacts of the awareness of susceptible population and availability of medical resources for treatment in each group, on the epidemic size of each group. Our findings suggest that in the case of limited medical resources, the high treatment rate and awareness of the population are very helpful to control the disease (to reduce the prevalence of infection) and the eradication of disease also depends on initial population sizes. More importantly, it is also obtained that sufficient medical resources for every group are required to eradicate the disease from an entire population. [ABSTRACT FROM AUTHOR]