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This study investigates the utilization of various mathematical models for comprehending and managing outbreaks of infectious diseases, with a specific focus on how different distributions of incubation times influence predictions regarding epidemics. Two methodologies are examined: a compartmental SEnIR ODE model, which represents an enhanced version of the mean-field SEIR model, and a stochastic agent-based complex network model. Our findings demonstrate that the selection of diverse incubation time distributions can result in noteworthy discrepancies in critical epidemic forecasts, highlighting the crucial role of precise modeling in shaping effective public health interventions. The research underscores the necessity of integrating authentic distribution patterns into epidemic modeling to increase its reliability and applicability. • We show how key epidemiological predictions depend on the incubation time distribution. • We utilized two kinds of models: mean-field SEIR models, and agent-based models. • We emphasize that integrating authentic distributions increases the models' reliability and applicability. [ABSTRACT FROM AUTHOR]

A new epidemic S E I R model with discrete delay on complex network is discussed in this paper. The formula of the basic reproduction number R 0 for the model is given, the global dynamics of this model is fully determined by R 0 , and it is proved that the disease dies out when R 0 < 1 , while the endemic equilibrium is globally stable and the disease becomes endemic. • A new epidemic SEIR model with discrete latent period delay on complex network is proposed. • By using suitable Lyapunov function, global dynamics of the model is discussed. • Numerical simulations are given to demonstrate the main results, and the effect of time delay is analyzed. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a stochastic SIS epidemic model with vaccination and degenerate diffusion. We prove that such a stochastic system has a complete threshold R s. That is, when R s > 1 stochastic system is ergodic, which means the disease persists; when R s < 1 the disease I t becomes extinct exponentially. Highlights • The diffusion term of stochastic SISV model is degenerate. • We present a nearly sufficient and necessary condition for the ergodicity. • We determine a complete threshold for the disease to occur or not. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the dynamical behavior of a regime-switching SIRS epidemic model with a ratio-dependent incidence rate. We propose a stochastic reproduction number R 0 S which can be regarded as a threshold to use in identifying the stochastic extinction and persistence: if R 0 S < 1 , the disease dies out exponentially with probability one; while if R 0 S > 1 , there exists a unique ergodic stationary distribution of the positive solutions to the system which implies the stochastic persistence of the infectious disease. Highlights • A regime-switching SIRS epidemic model with a ratio-dependent incidence rate is studied. • A stochastic reproduction number R 0 S is proposed to be regarded as a threshold. • If R 0 S < 1 , the disease dies out exponentially with probability one. • If R 0 S < 1 , there is a unique ergodic stationary distribution of the system. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate a stochastic hepatitis B epidemic model driven by Lévy noise. We prove the existence of a unique global positive solution and we present sufficient conditions to derive extinction and persistence in mean of the disease. [ABSTRACT FROM AUTHOR]

Abstract We present the exact analytic solution to the nonlinear dynamic system describing the information propagation process in complex networks. The method is to switch to the dynamic system of equations for the new variables defined according to the nonlinear terms appearing in the original dynamic system, and then, after combining relevant consistent condition of solution to reduce the dynamic system of equations for the newly defined variables to the knownly solvable nonlinear Bernoulli differential equation. Numerical comparisons between the purely numerical solution and the exact analytic solution confirm the crosscheck and mutual proof for both of the solutions. The presented exact analytic solution, which does not restrict the coefficients in the original dynamic system to merely constants, is thus suitable for the study of information propagation in networks with evolving structure and changing properties. Highlights • The first reported exact analytic solution to the SEIR model of information propagation in complex networks with population dynamics. • That the presented exact analytic solution does not restrict the coefficients in the original dynamic system to merely constants makes it suitable for the study of information propagation in social networks with evolving structure and time varying properties. • The similarity between hydrodynamics and the mechanism of information propagation in complex networks may bring about a bridge for research of both subjects. [ABSTRACT FROM AUTHOR]

Abstract Epidemic models have previously been employed to better understand the spread of internet content, such as individual viral videos. Motivated by geographical differences between viewing populations, as well as by multiple viral videos which are linked or related in some way, we propose a multi-group susceptible–exposed–infected–recovered–susceptible (SEIRS) delay differential equation epidemic model for the popularity evolution of viral videos. The model accounts for the multi-group environment via (in general, heterogeneous) networks of single-group SEIRS equations. Existence and stability conditions for disease-free and endemic equilibrium states are obtained, while numerical simulations demonstrate the emergence of non-symmetric endemic equilibrium states across the network — in the case of heterogeneous parameters. In order to demonstrate the effectiveness of our modeling approach, we compare solutions to our model with data obtained from coupled YouTube videos. Our results demonstrate that the inclusion of coupling between such videos allows for better agreement with data than considering each video in isolation. Such results suggest that the multi-group epidemic modeling approach including time delays and coupling between videos is a useful tool for understanding the dynamics of viral videos which are connected in some manner. While the proposed network SEIRS differential delay equation epidemic model was applied to viral video data, it may be applicable to a wide variety of internet content and, more generally, may be used to model information propagation within and between communities under the assumption of time delays in information propagation. Highlights • We formulate a network delay differential equation model of linked viral videos. • Time-delay describes temporary immunity of individuals after repeated viewing. • Edges in the network model represent links between individual videos. • The approach allows for an understanding of the popularity evolution of linked videos. • Model fit to YouTube music video data with reasonable agreement. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the dynamics of a network-based SIRS epidemic model with vaccination and a nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger. Using the analytical method, epidemic threshold R 0 is obtained. When R 0 is less than one, we prove the disease-free equilibrium is globally asymptotically stable and the disease dies out, while R 0 is greater than one, there exists a unique endemic equilibrium. By constructing a suitable Lyapunov function, we also prove the endemic equilibrium is globally asymptotically stable if the inhibitory factor α is sufficiently large. Numerical experiments are also given to support the theoretical results. It is shown both theoretically and numerically a larger α can accelerate the extinction of the disease and reduce the level of disease. Highlights • A network-based SIRS epidemic model with vaccination and a nonmonotone incidence rate is proposed. • Epidemic threshold is obtained. • Criteria of global stability of the equilibrium are established. • Numerical experiments for a finite scale-free network are presented. [ABSTRACT FROM AUTHOR]

Abstract This study investigates the disease dynamics of a SIRS epidemic model as affected by environment fluctuations and media coverage. A unique global positive solution for the epidemic model is obtained. The stochastic endemic dynamics of the system is discussed by constructing suitable Lyapunov functions, and the ergodic stationary distribution of the stochastic model is established based on the method of Khasminskii. The extinction of the epidemic disease is also analyzed. Highlights • A stochastic SIRS epidemic model incorporating media coverage and saturated incidence rate is investigated. • The stochastic endemic dynamics and ergodic stationary distribution of the stochastic model are discussed. • Sufficient conditions for the extinction of the disease are established. • The results reveal that the maximum reduced contact rate, due to media coverage, will accelerates the extinction of infective populations and reduces the risk of epidemic prevalence. [ABSTRACT FROM AUTHOR]

Investigations of possible links between population density and the propagation and magnitude of epidemics have so far proved inconclusive. There are three possible reasons (i) A lack of focus on appropriate density intervals. (ii) For the density to be a meaningful variable the population must be distributed as uniformly as possible. If an area has towns and cities where a majority of the population is concentrated its average density is meaningless. (iii) In propagation of an epidemic the initial proportion of susceptibles (persons who have not developed an immunity) is an essential, yet usually unknown, factor. The assumption that most of the population is susceptible holds only for new strains of diseases. Here we show that when these requirements are properly accounted for, the size of epidemics is indeed closely connected with the population density. This empirical observation comes as a welcome confirmation of the classical KMK (Kermack–McKendrick, 1927) model. Indeed, one of its key predictions is that the size of the epidemic increases strongly (and in a non linear way) with the initial density of susceptibles. An interesting consequence is that, contrary to common beliefs, in sparsely populated territories, like Alaska, Australia or the west coast of the United states the size of epidemics among native populations must have been limited by the low density even for diseases for which natives had no immunity (i.e., were susceptibles). [ABSTRACT FROM AUTHOR]

An agent model of epidemic spreading is proposed. When a susceptible agent is exposed to pathogen levels above its specified infection-tolerance threshold F , its state changes to infected, where a pathogen is a virus, microparasite, or any other disease-causing organism or material. The spreading sources of pathogens are the agents that are in the infected state. The growth and decay of a pathogen in an infected host obeys a given function, f t . This function increases linearly in the early period of infection t 1 , and then decreases linearly to zero in the latter period t 2 . The simulation results show that as the agent density increases, the model undergoes a phase transition from a local epidemic phase to a pandemic phase. For immobile agents, transition density ρ C equals transition density ρ P of the corresponding site percolation model. For random-walking agents, the transition density decreases as ρ C U ≈ ρ P × U − 0 . 3 , where U denotes the average path length of the walking agent during the period t 1 + t 2 . This model provides a reliable alternative to the standard SIR model, which is a simple compartmental model of susceptibility, infection, and pathogen removal. Moreover, it can predict epidemic phenomena using fact-based parameters. [ABSTRACT FROM AUTHOR]

Being able to theoretically predict the outbreak threshold of an epidemic is essential in disease control. Real-world correlated networks are ubiquitous and their topological structure strongly affects the prediction accuracy of present-day theoretical methods. Quantifying their accuracy and fitness in predicting outbreak thresholds of correlated networks is thus essential. We use a susceptible–infected–removed (SIR) model to examine four widely-used theoretical methods – the heterogeneous mean-field ( H M F ), quenched mean-field ( Q M F ), dynamical message passing ( D M P ), and connectivity matrix ( C M ) methods – to predict the outbreak threshold of a correlated network. The potential topological structure of correlated network impacts on prediction accuracy of these four methods. We emphasize that the quantitative changes of degree correlation, degree distribution exponent and network size strongly affect the outbreak of SIR spreading dynamics, and compare the simulation results with the theoretical ones obtained from these four methods. The extensive experiments in synthetic networks show that (a) the increasing degree correlation coefficients reduce outbreak threshold and suppress outbreak size; (b) the increasing degree distribution exponents raise outbreak threshold but suppress outbreak size; (c) the increasing network sizes decrease outbreak threshold but do not affect outbreak size; (d) as for four theoretical methods, C M and D M P are more likely to precisely predict outbreak threshold because they to some extent incorporate network topology with dynamical correlations. The experimental results in 50 real-world networks also prove the above conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we consider an impulsive SIR model with incidence rate stochastically perturbed. First, we prove the existence and uniqueness of the global positive solution by constructing the equivalent system without impulse. Second, we obtain a sufficient condition which determines epidemic to extinct. Then we demonstrate the existence and global attraction of the boundary periodic solution under the certain condition. Finally, we present a sufficient condition which allows the existence of a positive periodic solution. [ABSTRACT FROM AUTHOR]

Taking into account of both white and colored noises, a stochastic epidemic model with nonlinear incident rate under regime switching is formulated. Based on this model, we investigate the dynamic behaviors such as ergodicity and extinction of the SIR model with Beddington–DeAngelis incidence rate and Markov switching. First, we study the existence of the unique positive solution of system (1.3). Secondly, by using Lyapunov functions, we prove that the system has a ergodic stationary distribution under certain sufficient conditions. Then, we obtain the conditions for extinction. Finally, numerical simulations are employed to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic SIS epidemic model with vertical infection. By constructing suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a stationary probability measure of the model. In addition, we also establish sufficient conditions for extinction and persistence of the disease. [ABSTRACT FROM AUTHOR]

Abstract For understanding the influence of user security awareness on the long-term spreading behavior of malware over mobile networks, in this paper, we intensively study the global dynamics of a novel network-based epidemic model with weakly-protected and strongly-protected susceptible nodes. Both analytical and numerical results show that the global dynamics of the model is completely governed by a threshold value. Specifically, we prove that when the value is lower than one, the malware-free equilibrium is globally asymptotically stable and mobile malware will disappear. When the value is greater than one, mobile malware will persist on the network, and in the meantime there exists a unique malware equilibrium which is globally asymptotically stable under certain conditions. The obtained results improve and enrich some known ones. Interestingly, increasing the recovery rate of infected nodes can result in the increase of strongly-protected susceptible nodes and the decrease of the threshold value. The study has valuable guiding significance in effectively controlling mobile malware spread. Highlights • Dynamics of a novel network-based WSIS epidemic model is further investigated. • Criteria of global asymptotical stability of the equilibria is given • The permanence of mobile malware over complex networks is studied. • Numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an SIR model with birth and death on complex networks is analyzed, where infected individuals are divided into m groups according to their infection and contact between human is treated as a scale-free social network. We obtain the basic reproduction number R 0 as well as the effects of various immunization schemes. The results indicate that the disease-free equilibrium is locally and globally asymptotically stable in some conditions, otherwise disease-free equilibrium is unstable and exists an unique endemic equilibrium that is globally asymptotically stable. Our theoretical results are confirmed by numerical simulations and a promising way for infectious diseases control is suggested. [ABSTRACT FROM AUTHOR]

Influential spreaders are the key players in maximizing or controlling the spreading in a complex network. Identifying the influential spreaders using kshell decomposition method has become very popular in the recent time. In the literature, the core nodes i.e. with the largest kshell index of a network are considered as the most influential spreaders. We have studied the kshell method and spreading dynamics of nodes using Susceptible–Infected–Recovered (SIR) epidemic model to understand the behavior of influential spreaders in terms of its topological location in the network. From the study, we have found that every node in the core area is not the most influential spreader. Even a strategically placed lower shell node can also be a most influential spreader. Moreover, the core area can also be situated at the periphery of the network. The existing indexing methods are only designed to identify the most influential spreaders from core nodes and not from lower shells. In this work, we propose a kshell hybrid method to identify highly influential spreaders not only from the core but also from lower shells. The proposed method comprises the parameters such as kshell power, node’s degree, contact distance, and many levels of neighbors’ influence potential. The proposed method is evaluated using nine real world network datasets. In terms of the spreading dynamics, the experimental results show the superiority of the proposed method over the other existing indexing methods such as the kshell method, the neighborhood coreness centrality, the mixed degree decomposition, etc. Furthermore, the proposed method can also be applied to large-scale networks by considering the three levels of neighbors’ influence potential. [ABSTRACT FROM AUTHOR]

We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number R 0 = 1 . Our results demonstrate that the disease transmission rate β and exogenous reinfection rate α plays an important role to change the qualitative dynamics of TB. The disease transmission rate β give rises to the possibility of backward bifurcation for R 0 < 1 , and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that R 0 < 1 may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate β and the exogenous reinfection rate α . We conducted some numerical simulations to support our analytical findings. [ABSTRACT FROM AUTHOR]

Epidemiological models for the spread of pathogens in a population are usually only able to describe a single pathogen. This makes their application unrealistic in cases where multiple pathogens with similar symptoms are spreading concurrently within the same population. Here we describe a method which makes possible the application of multiple single-strain models under minimal conditions. As such, our method provides a bridge between theoretical models of epidemiology and data-driven approaches for modeling of influenza and other similar viruses. Our model extends the Susceptible–Infected–Recovered model to higher dimensions, allowing the modeling of a population infected by multiple viruses. We further provide a method, based on an overcomplete dictionary of feasible realizations of SIR solutions, to blindly partition the time series representing the number of infected people in a population into individual components, each representing the effect of a single pathogen. We demonstrate the applicability of our proposed method on five years of seasonal influenza-like illness (ILI) rates, estimated from Twitter data. We demonstrate that our method describes, on average, 44% of the variance in the ILI time series. The individual infectious components derived from our model are matched to known viral profiles in the populations, which we demonstrate matches that of independently collected epidemiological data. We further show that the basic reproductive numbers ( R 0 ) of the matched components are in range known for these pathogens. Our results suggest that the proposed method can be applied to other pathogens and geographies, providing a simple method for estimating the parameters of epidemics in a population. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior for a stochastic SIS epidemic model with isolation which is as an important strategy for the elimination of infectious diseases. It is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient, the death rate and the proportional coefficient of the isolation of infective. It is shown that the extinction and persistence in the mean of the model are determined by a threshold value R 0 S . That is, if R 0 S < 1 , then disease dies out with probability one, and if R 0 S > 1 , then the disease is stochastic persistent in the means with probability one. Furthermore, the existence of a unique stationary distribution is discussed, and the sufficient conditions are established by using the Lyapunov function method. Finally, some numerical examples are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a stochastic delayed SIR epidemic model with vaccination and double diseases which make the research more complex. The environment variability in this paper is characterized by white noise and Lévy noise. We establish sufficient conditions for extinction and persistence in the mean of the two epidemic diseases. It is shown that: (i) time delay and Lévy noise have important effects on the persistence and extinction of epidemic diseases; (ii) two diseases can coexist under certain conditions. [ABSTRACT FROM AUTHOR]

This paper aims to study an SIS epidemic model with media coverage from a general deterministic model to a stochastic differential equation with environment fluctuation. Mathematically, we use the Markov semigroup theory to prove that the basic reproduction number R 0 s can be used to control the dynamics of stochastic system. Epidemiologically, we show that environment fluctuation can inhibit the occurrence of the disease, namely, in the case of disease persistence for the deterministic model, the disease still dies out with probability one for the stochastic model. So to a great extent the stochastic perturbation under media coverage affects the outbreak of the disease. [ABSTRACT FROM AUTHOR]

Comparing with single networks, the multiplex networks bring two main effects on the spreading process among individuals. First, the pathogen or information can be transmitted to more individuals through different layers at one time, which enlarges the spreading scope. Second, through different layers, an individual can also transmit the pathogen or information to the same individuals more than once at one time, which makes the spreading more effective. To understand the different roles of the spreading scope and effectiveness, we propose an epidemic model on multiplex networks with link overlapping, where the spreading effectiveness of each interaction as well as the variety of channels (spreading scope) can be controlled by the number of overlapping links. We find that for Poisson degree distribution, increasing the epidemic scope (the first effect) is more efficient than enhancing epidemic probability (the second effect) to facilitate the spreading process. However, for power-law degree distribution, the effects of the two factors on the spreading dynamics become complicated. Enhancing epidemic probability makes pathogen or rumor easier to outbreak in a finite system. But after that increasing epidemic scopes is still more effective for a wide spreading. Theoretical results along with reasonable explanation for these phenomena are all given in this paper, which indicates that the epidemic scope could play an important role in the spreading dynamics. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic avian–human influenza epidemic model with psychological effect in human population and saturation effect within avian population is investigated. This model describes the transmission of avian influenza among avian population and human population in random environments. For stochastic avian-only system, persistence in the mean and extinction of the infected avian population are studied. For the avian–human influenza epidemic system, sufficient conditions for the existence of an ergodic stationary distribution are obtained. Furthermore, a threshold of this stochastic model which determines the outcome of the disease is obtained. Finally, numerical simulations are given to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic nonautonomous SIRI epidemic model is formulated. Given that the incidence rates of diseases may change with the environment, we propose a novel type of transmission function. The main aim of this paper is to obtain the thresholds of the stochastic SIRI epidemic model. To this end, we investigate the dynamics of the stochastic system and establish the conditions for extinction and persistence in mean of the disease by constructing some suitable Lyapunov functions and using stochastic analysis technique. Furthermore, we show that the stochastic system has at least one nontrivial positive periodic solution. Finally, numerical simulations are introduced to illustrate our results. [ABSTRACT FROM AUTHOR]

We study susceptible–infected–recovered–susceptible epidemic model in weighted, regular, and random complex networks. We institute a pairwise-type mathematical model with a general transmission rate to evaluate the influence of the link-weight distribution on the spreading process. Furthermore, we develop a dimensionality reduction approach to derive the condition for the contagion outbreak. Finally, we analyze the influence of the heterogeneity of weight distribution on the outbreak condition for the scenario with a linear transmission rate. Our theoretical analysis is in agreement with stochastic simulations, showing that the heterogeneity of link-weight distribution can have a significant effect on the epidemic dynamics. [ABSTRACT FROM AUTHOR]

In this paper we elaborate on homogeneous and heterogeneous SIR-type epidemiological models. We find an unexpected correspondence between the epidemic trajectory of a transmissible disease in a homogeneous SIR-type model and radial null geodesics in the Schwarzschild spacetime. We also discuss modeling of population heterogeneity effects by considering both a one- and two-parameter gamma-distributed function for the initial susceptibility distribution, and deriving the associated herd immunity threshold. We furthermore describe how mitigation measures can be taken into account by model fitting. • We elaborate on homogeneous and heterogeneous SIR-type epidemiological models; • discuss modeling of population heterogeneity effects; • consider a 1-parameter gamma-distributed function for the initial susceptibility; • then consider a 2-parameter gamma-distributed function; • derive the associated herd immunity threshold; • describe how mitigation measures can be taken into account by model fitting; • find a correspondence between epidemic trajectory and Schwarzschild null geodesics. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the SIR model with random vertex weights on Erdős–Rényi graph G ( n , p ) . The Erdős–Rényi graph G ( n , p ) is generated from the complete graph C n with n vertices through independently deleting each edge with probability ( 1 − p ) . We assign i. i. d. copies of a positive r. v. ρ on each vertex as the vertex weights. For the SIR model, each vertex is in one of the three states ‘susceptible’, ‘infective’ and ‘removed’. An infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices. An infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at t = 0 there is no removed vertex and the number of infective vertices follows a Bernoulli distribution B ( n , θ ) . Our main result is a law of large numbers of the model. We give two deterministic functions H S ( ψ t ) , H V ( ψ t ) for t ≥ 0 and show that for any t ≥ 0 , H S ( ψ t ) is the limit proportion of susceptible vertices and H V ( ψ t ) is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment t as n grows to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we present the threshold of a stochastic SIQS epidemic model which determines the extinction and persistence of the disease. Furthermore, we find that noise can suppress the disease outbreak. Numerical simulations are also carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two stochastic SIR epidemic models with higher order perturbation. For the nonautonomous periodic case of the model, by using Has’minskii’s theory of periodic solution, we show that the system has at least one nontrivial positive T -periodic solution. For the system disturbed by both the white noise and telephone noise, we establish sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the positive solution. [ABSTRACT FROM AUTHOR]

A class of SIR epidemic model with generalized nonlinear incidence rate is presented in this paper. Temporary immunity and stochastic perturbation are also considered. The existence and uniqueness of the global positive solution is achieved. Sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established. Moreover, the threshold behavior is discussed, and the threshold value R 0 is obtained. We show that if R 0 < 1 , the disease eventually becomes extinct with probability one, whereas if R 0 > 1 , then the system remains permanent in the mean. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a class of periodic stochastic SIS epidemic models with general nonlinear incidence f ( S , I ) . Some sufficient conditions on the permanence in the mean and extinction of positive solutions with probability one are established. By using the Khasminskii’s boundary periodic Markov processes, the existence of stochastic nontrivial periodic solution for the models is also obtained. The numerical simulations are given to illustrate the main theoretical results and some interesting conjectures are presented. [ABSTRACT FROM AUTHOR]

For the long term population behavior, demography is a very important factor impacting the disease spread. Infection age also helps us better understand the process of the disease transmission. In this paper, we incorporate these two factors in an SIR epidemic model on complex networks. Through mathematical analysis, the basic reproduction number R 0 is shown to be a sharp threshold to determine whether or not the disease spreads. Precisely, if R 0 is less than 1 then the disease-free equilibrium is globally asymptotically stable while if R 0 is larger than 1 then the endemic equilibrium is globally asymptotically stable. Some simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We study global stability of endemic equilibrium of an epidemic model with birth and death on complex networks. Under some conditions, the local asymptotic stability of the endemic equilibrium was established by Zhang and Jin (2011) for correlated networks, and the global asymptotic stability was obtained by Chen and Sun (2014) for uncorrelated networks. In this work, we remove those conditions, and prove by constructing a Lyapunov function that the endemic equilibrium is globally asymptotically stable. Numerical simulations are also presented to illustrate the feasibility of the result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SEIR epidemic model with standard incidence. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution to the model. Our result does not depend on the positive endemic equilibrium E ∗ of the deterministic system, which improves the previous result to a greater extent. Moreover, we also establish sufficient conditions for extinction of the disease. [ABSTRACT FROM AUTHOR]

The delay of an infectious disease is significant when aiming to predict its strength and spreading patterns. In this paper the SEIR ​(susceptible–exposed–infected–recovered) epidemic spread with time delay is analyzed through a two-dimensional cellular automata model. The time delay corresponding to the infectious span, predominantly, includes death during the latency period in due course of infection. The advancement of whole system is described by SEIR transition function complemented with crucial factors like inhomogeneous population distribution, birth and disease independent mortality. Moreover, to reflect more realistic population dynamics some stochastic parameters like population movement and connections at local level are also considered. The existence and stability of disease free equilibrium is investigated. Two prime behavioral patterns of disease dynamics is found depending on delay. The critical value of delay, beyond which there are notable variations in spread patterns, is computed. The influence of important parameters affecting the disease dynamics on basic reproduction number is also examined. The results obtained show that delay plays an affirmative role to control disease progression in an infected host. [ABSTRACT FROM AUTHOR]

Considering that the exposed nodes may become the removed nodes at a rate, we present a novel rumor propagation S E I R model on heterogeneous network. We give formula of the rumor spreading threshold for the model and analyze the globally dynamical behaviors of the rumor free equilibrium set. Meanwhile, we discuss two immunization strategies for the rumor spreading. Numerical simulations are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the global stability and attractivity of the endemic equilibrium for a network-based SIS epidemic model with nonmonotone incidence rate. The model was introduced in Li (2015). We prove that the endemic equilibrium is globally asymptotically stable if α (a parameter of this model) is sufficiently large, and is globally attractive if the transmission rate λ satisfies λ λ c ∈ ( 1 , 2 ] , where λ c is the epidemic threshold. Some numerical experiments are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

A new epidemic model with two infection periods is developed to account for the human behavior in social network, where newly infected individuals gradually restrict most of future contacts or are quarantined, causing infectivity change from a degree-dependent form to a constant. The corresponding dynamics are formulated by a set of ordinary differential equations (ODEs) via mean-field approximation. The effects of diverse infectivity on the epidemic dynamics ​are examined, with a behavioral interpretation of the basic reproduction number. Results show that such simple adaptive reactions largely determine the impact of network structure on epidemics. Particularly, a theorem proposed by Lajmanovich and Yorke in 1976 is generalized, so that it can be applied for the analysis of the epidemic models with multi-compartments especially network-coupled ODE systems. [ABSTRACT FROM AUTHOR]

Cooperative learning is one of the most effective teaching methods, which has been widely used. Students’ mutual contact forms a cooperative learning network in this process. Our previous research demonstrated that the cooperative learning network has complex characteristics. This study aims to investigating the dynamic spreading process of the knowledge in the cooperative learning network and the inspiration of leaders in this process. To this end, complex network transmission dynamics theory is utilized to construct the knowledge dissemination model of a cooperative learning network. Based on the existing epidemic models, we propose a new susceptible–infected–susceptible-leader (SISL) model that considers both students’ forgetting and leaders’ inspiration, and a susceptible–infected–removed-leader (SIRL) model that considers students’ interest in spreading and leaders’ inspiration. The spreading threshold λ c and its impact factors are analyzed. Then, numerical simulation and analysis are delivered to reveal the dynamic transmission mechanism of knowledge and leaders’ role. This work is of great significance to cooperative learning theory and teaching practice. It also enriches the theory of complex network transmission dynamics. [ABSTRACT FROM AUTHOR]

We introduce the stochasticity into a deterministic model which has state variables susceptible–exposed–infected with varying population size in this paper. The infected individuals could return into susceptible compartment after recovering. We show that the stochastic model possesses a unique global solution under building up a suitable Lyapunov function and using generalized Itô’s formula. The densities of the exposed and infected tend to extinction when some conditions are being valid. Moreover, the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria. The stochastic model admits a stationary distribution around the endemic equilibrium, which means that the disease will prevail. To check the validity of the main results, numerical simulations are demonstrated as end of this contribution. [ABSTRACT FROM AUTHOR]

In heterogeneous wireless sensor network (WSN) each data-packet traverses through multiple hops over restricted communication range before it reaches the sink. The amount of energy required to transmit a data-packet is directly proportional to the number of hops. To balance the energy costs across the entire network and to enhance the robustness in order to improve the lifetime of WSN becomes a key issue of researchers. Due to high dimensionality of an epidemic model of WSN over a general scale free network, it is quite difficult to have close study of network dynamics. To overcome this complexity, we simplify a general scale free network by partitioning all of its motes into two classes: higher-degree motes and lower-degree motes, and equating the degrees of all higher-degree motes with lower-degree motes, yielding a reduced scale free network. We develop an epidemic model of WSN based on reduced scale free network. The existence of unique positive equilibrium is determined with some restrictions. Stability of the system is proved. Furthermore, simulation results show improvements made in this paper have made the entire network have a better robustness to the network failure and the balanced energy costs. This reduced model based on scale free network theory proves more applicable to the research of WSN. [ABSTRACT FROM AUTHOR]

To achieve the aim of immediately halting spread of COVID-19 it is essential to know the dynamic behavior of the virus of intensive level of replication. Simply analyzing experimental data to learn about this disease consumes a lot of effort and cost. Mathematical models may be able to assist in this regard. Through integrating the mathematical frameworks with the accessible disease data it will be useful and outlay to comprehend the primary components involved in the spreading of COVID-19. There are so many techniques to formulate the impact of disease on the population mathematically, including deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional derivative modeling is one of the essential techniques for analyzing real-world issues and making accurate assessments of situations. In this paper, a fractional order epidemic model that represents the transmission of COVID-19 using seven compartments of population susceptible, exposed, infective, recovered, the quarantine population, recovered–exposed, and dead population is provided. The fractional order derivative is considered in the Caputo sense. In order to determine the epidemic forecast and persistence, we calculate the reproduction number R 0. Applying fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied. Moreover, we implement the generalized Adams–Bashforth–Moulton method to get an approximate solution of the fractional-order COVID-19 model. Finally, numerical result and an outstanding graphic simulation are presented. • Caputo fractional order derivative modeling is one of the most effective approaches to comprehend real-world issues and make an accurate analysis of the situation. • In this work we describe the fractional order epidemic model for the COVID-19 outbreak. • The population is divided into seven compartments namely susceptible, exposed, infected, and recovered, the quarantine population, recovered–exposed population, and dead population. • The existence and uniqueness of the solutions to the fractional order derivative have been discussed using fixed point theory. • Generalized Adams–Bashforth–Moulton method is used to achieve an approximate solution for the COVID-19 fractional order model. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic delayed SEIR epidemic model with nonlinear incidence. Firstly we verify that there exists a unique global positive solution of the system. Then by constructing some suitable Lyapunov functions, we study the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic non-autonomous SISV epidemic model. For the non-autonomous periodic system, firstly, we get the threshold of the system which determines whether the epidemic occurs or not. Then in the case of persistence, we show that there exists at least one nontrivial positive periodic solution of the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, Brownian motion and L e ́ vy jumps are introduced to a SIS type epidemic model with nonlinear incidence rate. The dynamical behavior of the considered model is investigated. In order to reveal the extinction and permanence of the disease, two threshold values R ˜ 0 , R ̄ 0 are showed. We find that if R ˜ 0 < 1 , the disease may die out, and when R ̄ 0 > 1 , the disease may be persistent. Finally, the numerical simulations are presented to illustrate our mathematical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the dynamics of a stochastic nonautonomous SIR epidemic model, in which population growth is subject to logistic growth in absence of disease. For the periodic system, we present sufficient conditions for persistence of the epidemic and in the case of persistence, by constructing some suitable Lyapunov functions, we show that there is at least one nontrivial positive periodic solution. One of the most important findings is that random perturbations may be beneficial to formate the periodic solution to the stochastic nonautonomous SIR epidemic model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic extinction and persistence of a parasite–host epidemiological model. We show that the global dynamics of the stochastic model can be governed by the basic reproduction number R 0 S : if R 0 S < 1 , under mild extra conditions, the disease goes to extinction with probability one and the disease-free dynamics occurs; while R 0 S > 1 , under mild extra conditions, the disease persists and endemic dynamics occurs almost surely, the solutions of the stochastic model fluctuate around the steady state of the deterministic model, and a unique stationary distribution can be found. Based on realistic parameters of Daphnia -microparasite system, numerical simulations have been performed to verify/extend our analytical results. Epidemiologically, we find that: (1) Large environment fluctuations can suppress the outbreak of disease; (2) The distributions are governed by R 0 S ; (3) The noise perturbations can be beneficial to control the spread of disease on average. [ABSTRACT FROM AUTHOR]

Message propagation in social networks is becoming a popular topic in complex networks. One of the message types in social networks is called broadcast message. It refers to a type of message which has a unique and unknown destination for the publisher, such as ‘lost and found’. Its propagation always has two stages. Due to this feature, rumor propagation model and epidemic propagation model have difficulty in describing this message’s propagation accurately. In this paper, an improved two-stage susceptible–infected–removed model is proposed. We come up with the concept of the first forwarding probability and the second forwarding probability. Another part of our work is figuring out the influence to the successful message transmission chance in each level resulting from multiple reasons, including the topology of the network, the receiving probability, the first stage forwarding probability, the second stage forwarding probability as well as the length of the shortest path between the publisher and the relevant destination. The proposed model has been simulated on real networks and the results proved the model’s effectiveness. [ABSTRACT FROM AUTHOR]