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A significant proportion of malaria infections in humans exhibit no symptoms, but it is a reservoir for maintaining malaria transmission. A time periodic reaction-diffusion model for malaria spread is introduced in this paper, incorporating spatial heterogeneity, incubation periods, symptomatic and asymptomatic carriers. This paper introduces the concept of the basic reproduction number R 0 , which is defined as the spectral radius of the next generation operator, and we present some preliminary results by elementary analysis. The threshold dynamic behavior analysis shows that when R 0 < 1 , the disease is extinct, and when R 0 > 1 , the disease is persistent. We investigate the case of constant system parameters, focusing on the global asymptotic stability of the disease-free steady state when R 0 = 1 . In the numerical simulation section, we validate the theoretical results obtained, and then use elasticity analysis methods to explore the influence of parameters on the output solution. In addition, sensitivity analysis of the basic reproduction number under homogeneous conditions indicates direction of controlling malaria transmission. And several control measures are evaluated in the following steps. [ABSTRACT FROM AUTHOR]

This paper considers the wave propagation for a spatial discrete virus model with HIV viral load and 2-LTR dynamics during high active antiretroviral therapy. Applying Schauder fixed point theorem, technique of Lyapunov function and limiting arguments, we establish the existence of traveling wave solutions for the virus model when the basic reproduction number is larger than one and the wave speed is not less than a threshold speed. Then, using the methods of comparison principle and Laplace transform, we prove the nonexistence of traveling wave solutions when the basic reproduction number is either smaller than one; or larger than one but the wave speeds are less than the threshold speed. Indeed, the threshold speed is minimum wave speed for the propagation of traveling waves. We further show that the three classes of inhibitor can decrease the minimum wave speed; while the diffusion rate of visions will increase the minimum wave speed. [ABSTRACT FROM AUTHOR]

In control of wild mosquitoes to fight mosquito-borne diseases, release of mosquitoes with Wolbachia is one of the effective biological control methods. There are three release strategies, namely releasing both Wolbachia-infected females and males, only Wolbachia-infected females and only Wolbachia-infected males. All these three strategies have been confirmed to be capable of speeding up the Wolbachia persistence in mosquito populations. In this paper, we investigate how supplementary releases affect the Wolbachia spread dynamics in mosquito populations. Our aim is to compare the effectiveness among these three release strategies. We obtain theoretical results and provide numerical simulations that show that the first two strategies are more effective than the last strategy. For the first two strategies, the former strategy is either less effective than the latter strategy in each generation, or more effective than the latter strategy in previous generations, and then becomes less effective in late generations. [ABSTRACT FROM AUTHOR]

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number R 0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when R 0 = 1 . It is concluded that the endemic equilibrium is globally asymptotically stable when R 0 > 1 , by utilizing Dulac's criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained. [ABSTRACT FROM AUTHOR]

This paper explores a generalized nonlocal dispersion SIS epidemic model subject to the Neumann boundary conditions and spatial heterogeneity. We use a convolution operator to describe the nonlocal spatial movements of individuals. Our primary goal is to investigate this model, focusing on a generalized incidence function, which presents an additional challenge in the model analysis. This model’s basic reproduction number, R 0 , is identified, and it is proved that 1 - R 0 has the same sign as the principal eigenvalue of a generalized linear nonlocal operator. Furthermore, the asymptotic profiles of R 0 in terms of dispersion coefficients are also established. We also investigate the existence and uniqueness of an endemic steady state for R 0 > 1 , and we study the large dispersal rates effect on the asymptotic profiles of the steady endemic state. Finally, we discussed the global asymptotic behavior of the solution for different dispersal coefficients. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling wave solutions for a discrete diffusive mosquito-borne epidemic model with general incidence rate and constant recruitment. It is observed that whether the traveling wave solutions exist or not depend on the so-called basic reproduction ratio R 0 of the corresponding kinetic system and the critical wave speed c ∗ . More precisely, when R 0 > 1 and c ≥ c ∗ , the system admits a nontrivial traveling wave solution by constructing an invariant cone in a bounded domain with initial functions being defined on, and employing the method of upper and lower solution, Schauder’s fixed point theorem and a limiting approach. Moreover, the asymptotic behavior of traveling wave solutions at positive infinity is obtained by constructing a suitable Lyapunov functional. When 0 < c < c ∗ or R 0 ≤ 1 , the system has no nontrivial traveling wave solution by using a contradictory approach and two-sided Laplace transforms. [ABSTRACT FROM AUTHOR]

We consider a Markovian model of an SIR epidemic spreading on a contact graph that is drawn uniformly at random from the set of all graphs with n vertices and given vertex degrees. Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014) prove that the evolution of such an epidemic is well approximated by the solution to a simple set of differential equations, thus providing probabilistic underpinnings to the works of Miller (J Math Biol 62(3):349–358, 2011) and Volz (J Math Biol 56(3):293–310, 2008). The present paper provides an additional probabilistic interpretation of the limiting deterministic functions in Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014), thus clarifying further the connection between their results and the results of Miller and Volz. [ABSTRACT FROM AUTHOR]

This paper focuses on the characterization of viability zones in compartmental models with varying population size, due both to deaths caused by epidemics and to natural demography. This is achieved with the use of viscosity characterizations of viability and extensively illustrated on several models. An example taking into consideration real data is provided. The paper is completed with a viscosity approach to the optimality of minimal ("greedy") non-pharmaceutical interventions. [ABSTRACT FROM AUTHOR]

In order to reflect the dispersal of pollutants in non-adjacent areas and the large-scale movement of individuals, this paper proposes an epidemic model of nonlocal dispersal with air pollution, where the transmission rate is related to the concentration of pollutants. This paper checks the uniqueness and existence of the global positive solution and defines the basic reproduction number, R 0 . We simultaneously explore the global dynamics: when R 0 < 1 , the disease-free stable point is global asymptotic stability; when R 0 > 1 , the disease is uniformly persistent. Additionally, in order to approximate R 0 , a numerical method has been introduced. Illustrative examples are used to verify the theoretical outcomes and show the effect of the dispersal rate on the basic reproduction number R 0 . [ABSTRACT FROM AUTHOR]

This paper discusses a generalized SIR epidemic model that incorporates infectives with or without symptoms, allows arbitrary distributions for infectious periods and assumes infection rates depending on the current size of susceptibles. Our interest lies in the joint distribution of the state of the population and the severity of the disease at the end of infection. The approach is based on a formulation of the epidemic as a so-called collective model. First, a set of martingales is constructed which provides the distribution of this final epidemic outcome. Then, its corresponding distributional transform is expressed in terms of a family of pseudo-polynomials of Abel-Gontcharoff type. Some of the results obtained are illustrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive predator–prey model with two delays, i.e., a spatial memory delay and a toxic effect delay. Initially, we analyze the global existence of the solution of the system. We then analyze the equilibria’s local stability without delays and investigate the Hopf bifurcation induced by one delay. Subsequently, we establish an analytical framework for constructing the stability switching curve in the delay space. Finally, we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system. [ABSTRACT FROM AUTHOR]

In this paper, we establish a discrete model with periodic parameters to depict the Wolbachia spread dynamics in mosquito populations in cyclic environments. This work modifies the models established in the existing literature that did not take into account the variation of parameters with environmental periodic changes due to seasonality and other factors. When the parameters in our model are constants, it has been extensively studied and widely used. We present a conjecture about the existence of at most two periodic solutions worthy of further study, and show that the conjecture is true for the special case of 2-periodic parameters. Numerical simulations are also provided to illustrate the occurrence of periodic phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive susceptible-infected-susceptible epidemic model in a spatiotemporally heterogeneous environment. We consider a saturated incidence function of the form S I m + S + I , where m is a nonnegative function. When there is a positive disease-induced mortality rate everywhere, we demonstrate that the disease will always become extinct and the susceptible population will stabilize at zero or a positive constant. In the case where the disease-induced mortality rate is negligible, we examine the model in a time-periodic and spatially heterogeneous environment and establish the threshold dynamics between disease extinction and persistence in terms of the basic reproduction number. Under certain conditions, we show the global attractivity of both the disease-free equilibrium and endemic equilibrium by constructing appropriate Lyapunov functions. Moreover, we determine the spatial distribution of the disease when the diffusion rate of the susceptible or infected population (or both) is sufficiently small. Our findings suggest that the presence of a saturation effect reduces the transmission risk, allows for the total population size to have a substantial impact on disease dynamics, and may significantly alter the spatial distribution of the disease under certain circumstances. [ABSTRACT FROM AUTHOR]

In this paper, two types impulsive delay stochastic budworm population models are proposed and investigated. Based on some standard assumptions, a series sufficient criteria about long time behavior of the solutions are given. For the unbounded positive impulse, we obtain a sufficient criteria that makes the solution weakly permanent, which is close to the necessary criterion. Then, we demonstrate the conditions that produce the solutions are stochastically permanent when positive impulse is bounded. Finally, numerical simulations of the model confirms our theoretical results. Our rsults improve and extend some previous results. [ABSTRACT FROM AUTHOR]

This paper studies a two-age structured SIRS epidemic model with logistic growth of susceptible population and two-time delays. We simultaneously introduce two-time delays, i.e., the immunity and incubation periods, into this dynamic system and investigate their impact on different dynamic behaviors for the model. By means of the C 0 -semigroup theory, the model is transformed into a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the endemic equilibrium is obtained. Following the spectral analysis, the characteristic equation technique, and the Hopf bifurcation theorem, we show that different combinations of the two delays perform a vital role in the instability/stability as well as the Hopf bifurcation results of equilibrium solutions. We numerically provide some graphical representations to check the main theoretical results and show the rich dynamics by varying the two delay parameters. [ABSTRACT FROM AUTHOR]

We develop, in this paper, a reaction–diffusion epidemic model with the general incidence rate and personal protection awareness, where, the direct transmission and indirect transmission are also introduced to describe the complexity of the spread of infectious diseases. The existence, uniqueness and boundedness of global solution are obtained first. And then, the basic reproduction number R 0 is derived to go as the threshold value for predicting the persistence and extinction of disease. To be more specific, model admits a globally asymptotically stable disease-free steady state if R 0 < 1 , while the disease is persistent for R 0 > 1 . Further, for R 0 = 1 , the global asymptotic stability of the disease-free steady state is demonstrated when all diffusion coefficients are constants. In addition, the global asymptotical stability of the endemic equilibrium of this model with spatially homogeneous is derived if the corresponding basic reproduction is great than 1. Particularly, we also discuss the existence of traveling wave solution and the minimum wave speed c ∗ of the spatial homology model. Numerical simulations are presented to validate the main results and provide some recommendations for the control of multi-transmission diseases. [ABSTRACT FROM AUTHOR]

In this paper, we will revisit the model studied in Lou and Zhao (J Math Biol 62:543–568, 2011), where the model takes the form of a nonlocal and time-delayed reaction–diffusion model arising from the fixed incubation period. We consider the infection age to be a continuous variable but without the limitation of the fixed incubation period, leading to an age-space structured malaria model in a bounded domain. By performing the elementary analysis, we investigate the well-posedness of the model by proving the global existence of the solution, define the explicit formula of basic reproduction number when all parameters remain constant. By analyzing the characteristic equations and designing suitable Lyapunov functions, we also establish the threshold dynamics of the constant disease-free and positive equilibria. Our theoretical results are also validated by numerical simulations for 1-dimensional and 2-dimensional domains. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global threshold-type result and traveling waves for a general HIV infection model involving CD 4 + T cell death caused by caspase-1-mediated pyroptosis of the predominance. We first give the well-posedness of the model and establish the existence of a global attractor. In a bounded domain, the basic reproduction number, denoted by ℜ 0 , is identified as a threshold parameter for indicating whether the infection occurs or not. Specifically, if ℜ 0 < 1 , then the system admits a globally asymptotically stable infection-free steady state; if ℜ 0 > 1 , the system is uniformly persistent. In an unbounded domain and homogeneous environment, we find that if ℜ 0 > 1 and wave speed is large enough, the system admits traveling wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we ameliorate the model proposed in [13], by incorporating the influence of hepatitis B e antigen (HBeAg) status of mothers on vertical transmission. We use the improved model to fit reported HBV new infections in the Zhejiang Province of China. Also to predict the course of the Hepatitis B (HBV) infection in this Chinese area, and in Tokombere, located in sub-Saharan Africa(SSA). Furthermore, we apply optimal control techniques in view to re-examine the effects of the newborn vaccination, the universal vaccination and the treatment of chronic carriers in preventing the HBV infection. Simulation results show that treatment slightly steps in the optimal strategy, while immunisation is an effective measure. On the other hand, they indicate that the control measures and immunization programs implemented in Zhejiang Province are effective. Besides, they suggest that in SSA, a package of several policies centred on birth dose vaccination should be implemented. [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]

This paper focuses on a cholera model with temporary immunity and recovery-age. We first investigate the existence of unique solution to the model by reformulating the system as an abstract Cauchy problem, and present the basic reproduction number. By analyzing the characteristic equation, we discuss the stability of the steady states and further verify the existence of Hopf bifurcation. The theoretical results show that the temporary immunity effect takes a crucial effect on the perturbation around the endemic equilibrium. Numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the existence and nonexistence of traveling wave solution for a diffusive vector-borne disease model with nonlocal reaction and distributed delays. We demonstrate that the basic reproduction number R 0 of the corresponding ordinary differential equation system as a threshold determines whether the model admits traveling waves or not and there exists a critical wave speed c m ∗ > 0 when R 0 > 1 . Specifically, (i) As R 0 > 1 and the wave speed c > c m ∗ , the existence of traveling waves for the system is established with the aid of a perturbed system; (ii) As R 0 > 1 and 0 < c < c m ∗ , the nonexistence of traveling waves is proved via the two-sided Laplace transform; (iii) As R 0 ≤ 1 and c > 0 , the nonexistence is obtained by utilizing the comparison principle. The theoretical results are applied to dengue fever epidemics. We study the effects of geographical movement, nonlocal interaction, incubation period and R 0 on the threshold speed c m ∗ for dengue fever. [ABSTRACT FROM AUTHOR]

Bayesian phylogenetic inference is powerful but computationally intensive. Researchers may find themselves with two phylogenetic posteriors on overlapping data sets and may wish to approximate a combined result without having to re-run potentially expensive Markov chains on the combined data set. This raises the question: given overlapping subsets of a set of taxa (e.g. species or virus samples), and given posterior distributions on phylogenetic tree topologies for each of these taxon sets, how can we optimize a probability distribution on phylogenetic tree topologies for the entire taxon set? In this paper we develop a variational approach to this problem and demonstrate its effectiveness. Specifically, we develop an algorithm to find a suitable support of the variational tree topology distribution on the entire taxon set, as well as a gradient-descent algorithm to minimize the divergence from the restrictions of the variational distribution to each of the given per-subset probability distributions, in an effort to approximate the posterior distribution on the entire taxon set. [ABSTRACT FROM AUTHOR]

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number R 0 and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of R 0 is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established. [ABSTRACT FROM AUTHOR]

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein–Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus. [ABSTRACT FROM AUTHOR]

The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014–2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled. [ABSTRACT FROM AUTHOR]

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, R 0 , which completely determines the global dynamics of the model system. Namely, if R 0 ≤ 1 , then the disease–free equilibrium is globally asymptotically stable, and if R 0 > 1 , then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, R 0 can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton–Jacobi characterization of the value function, via the verification argument and explicit trajectory-based computations. Aside from providing an alternative to the Pontryagin complex arguments in Avram et al. (Appl Math Comput 418:126816, 2022) (see also Avram et al. in Appl Math Comput 423:127012, 2022), this method allows one to consider more general classes of costs; in particular state-dependent ones. On the other hand, the paper is completed by linear programming methods allowing one to deal with possibly discontinuous costs. In particular, we propose a brief exposition of classes of linearized dynamic programming principles based on our previous work and ensuing dual linear programming algorithms. We emphasize the particularities of our state space and possible generations of forward scenarios using the description of reachable sets. [ABSTRACT FROM AUTHOR]

Infectious diseases have a devastating impact on individual health and social development. Different external environmental factors will affect the scale and the speed of disease outbreaks, such as sanitary conditions and policy interventions. In this paper, we attempt to establish a dual-system susceptible-infectious-quarantine-recovered model with different environmental impacts to explore it. For the deterministic model, we calculate the basic reproduction number, simultaneously, and investigate the local asymptotic stability of the disease-free equilibrium and the endemic equilibrium. Numerical simulations and theoretical analyses verify the conclusion of this paper. It is a surprise that there is a great probability of finding a quarantine inflection point, which can effectively control the scale of infection outbreak when two different systems of infection rate and recovery rate are determined. [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]

In this paper, a six-neuron incommensurate fractional order BAM neural network model with multi-delays is considered. We demonstrate that the equilibrium point of the system loses its stability and Hopf bifurcation emerges when the delay passes through a critical value. And the relationship between the critical delay of Hopf bifurcation and size of fractional orders is found. Finally, some numerical simulations are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we construct a general four-dimensional delayed HIV infection model with virus-to-cell infection, cell-to-cell transmission, CTL immune response and parameter perturbations. By substitution, the four-dimensional delayed stochastic differential equations can be transformed into a degenerate eight-dimensional stochastic differential equations. The existence of the global positive solution of the system is obtained rigorously. By constructing appropriate Lyapunov functions, the existence of a stationary Markov process is derived when the stochastic reproduction number R 0 s is greater than one. Finally, we investigate the effects of noise level and the cell-to-cell transmission on the dynamic behavior of the model. Our model is a general model of the existing stochastic virus infection model, and the theoretical results improve and generalize the existing conclusion. [ABSTRACT FROM AUTHOR]

In this paper, considering the inevitable effects of environmental perturbations on disease transmission, we mainly study a stochastic SVEIR epidemic model in which the transmission rate satisfies the log-normal Ornstein–Uhlenbeck process and the incidence rate is general. To analyze the dynamic properties of the stochastic model, we firstly verify that there is a unique positive global solution. By constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. The sufficient condition for disease extinction is also given. Next, as a special case, we investigate the asymptotic stability of equilibria for the deterministic model and establish the exact expression of the probability density function of stationary distribution for the stochastic model. Finally, we calculate the mean first passage time from the initial value to the stationary state or extinction state to study the influence of environmental perturbations; meanwhile, some numerical simulations are carried out to demonstrate theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper explores the effect of immigration in a generalized model that considers susceptibles, infected, chronic carriers, and recovered, the incidence rate is considered as a general function and the immigration as a constant in all its populations. This model has the characteristic that carriers and infected can transmit the disease, besides it has not a disease-free equilibrium point and no basic reproductive number when the immigration is considered. Using an appropriate Lyapunov function and with suitable conditions on the functions involved in the general incidence, we show that the endemic equilibrium point is globally asymptotically stable. When the immigration is not considered the model has a disease-free equilibrium point, endemic equilibrium and basic reproductive number which are globally asymptotically stable depending on the magnitude of this threshold. Through numerical simulations we show that even having a good vaccination rate, recovery rate, diagnosis rate cannot stop the transmission of the disease if migration or immigration is considered. [ABSTRACT FROM AUTHOR]

Malaria is preventable and curable but critical disease caused by parasites that are transmitted to people through the bites of female Anopheles mosquitoes. There were an estimated 228 million cases of malaria globally and its mortality remained at 405,000 in 2018. There are many models that have been developed but the aim of this paper is to analyse the potential impact of multiple current interventions in communities with limited resources. The authors in their previous work, developed a population-based model of malaria transmission dynamics to investigate the effectiveness of five different interventions. This model captured both the human and the mosquito compartments and considered 5 control interventions. Namely it was: educational campaigns to mobilise people for diagnostic test and treatment and to sleep under bed nets; treatment through mass drug administration; indoor residual spraying with insecticide to reduce malaria transmission; insecticide treated net to reduce morbidity; and regular destruction of mosquito breeding sites to reduce the number of new mosquito and bites/contact at dusks and dawn. In the present work we carried out basic mathematical analysis of the model, simulate the different scenarios developed and optimise the control interventions with optimal control. The potential of the control interventions to reduce transmission within 120 days was observed. The numerical experiments showed that the optimal strategy to effectively control malaria was through the combinations of controls in the models. The developed malaria model predicted the reduction, control and/or elimination of malaria threats through incorporating multiple control interventions. Therefore, multiple control measures should be adopted for malaria but in areas of limited resources, we can make use of strategy E and others in places where there are more resources. [ABSTRACT FROM AUTHOR]

In this paper we consider an age-structured SIS epidemic patch model. We define the principal eigenvalue λ 0 d , basic reproduction number R 0 d and analyze the effects of the diffusion rate d on λ 0 d and R 0 d . More precisely, we investigate their asymptotic behavior for small and large diffusion rates. Finally we study the global behavior of this age-structured SIS epidemic patch model. [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]

Multi-patch and multi-city models have been proposed to investigate the dynamics of disease propagation in discrete regions characterized by varying connectivity and movement rates. This paper aims to comprehensively explore the impact of network topology and travel rates on disease spread within a population represented by random networks of interconnected cities. This approach enables a more comprehensive analysis of disease outbreaks across spatial domains compared to conventional multi-patch models, which typically consider a limited number of homogeneous links among subpopulations within distinct spatial units (i.e., patches). In this study, cities are represented using a probabilistic cellular automaton model that incorporates local interactions among individuals, while the network’s edges represent the travel rates between pairs of cities (i.e., nodes). Two types of complex networks are considered, namely small-world and Barabási–Albert networks. By employing a flexible numerical model, this study surpasses previous models in its capacity to accommodate a larger number of patches or cities. The primary findings of this research reveal that reducing the travel rates within the network can potentially “flatten the curve” of infected cases; however, it does not impact the population’s basic reproduction number. To effectively reduce the reproduction number, localized interventions targeting disease transmissibility are required. Additionally, concentrating control efforts on “central” cities within the network may prove crucial in impeding the rapid propagation of the disease. In summary, this study employs a rigorous framework to investigate the influence of network topology and travel rates on disease spreading within a population represented by random networks of interconnected cities. The findings contribute to a comprehensive understanding of disease dynamics in complex spatial settings and inform targeted intervention strategies for controlling and mitigating infectious disease outbreaks. [ABSTRACT FROM AUTHOR]

The SIR (susceptible–infected–recovered) is one of the simplest models for epidemic outbreaks. The present paper demonstrates the parametric solution of the model in terms of quadratures and derives a double exponential analytical asymptotic solution for the I-variable, which is valid on the entire real line. Moreover, the double exponential solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the parametric solution. A second, refined, asymptotic solution involving exponential gamma kernels was also demonstrated. The approach was applied to the coronavirus disease 2019 (COVID-19) pandemic in six European countries—Belgium, Italy, Sweden, France, Spain and Bulgaria in the period 2020-2021. [ABSTRACT FROM AUTHOR]

Age and spatial heterogeneities play a significant role in the prediction of transmission patterns of the cholera prevalence. In this paper, we propose a diffusive cholera epidemic model with chronological age and infection age structures. First, we adopt some embedding tricks and transform the model into a non-densely Cauchy problem. Then we establish the existence, uniqueness and dissipativity of the model by the integral semigroup and the Volterra integral equation theory. Finally, we show that the global dynamics of each feasible equilibrium determined by the basic reproduction number R 0 . If R 0 < 1 , the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , the endemic equilibrium E ∗ is globally attractive as long as the shedding rate of an infected individual is a function of chronological age or infection age. [ABSTRACT FROM AUTHOR]

This paper establishes the global attractivity of a positive constant equilibrium of a nonlocal and time-delayed diffusive malaria model in a homogeneous case. The same problem was achieved in a recent paper (Lou and Zhao in J Math Biol 62:543–568, 2011) by using the fluctuation method, but with a sufficient condition that the disease will become stable requires a sufficiently large basic reproduction number ℜ 0 . The present study is devoted to remove the sufficient condition by utilizing an appropriate Lyapunov functional and shows that the disease will become stable when ℜ 0 is exactly greater than one, which remarkably improves the known results in Lou and Zhao (2011). [ABSTRACT FROM AUTHOR]