Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

SIS reaction–diffusion model with risk-induced dispersal under free boundary

A spatial susceptible–infected–susceptible epidemic model with a free boundary, where infected individuals disperse non-uniformly, is investigated in this study. Spatial heterogeneity and movement of individuals are essential factors that affect pandemics and the eradication of infectious diseases. Our goal is to investigate the effect of a dispersal strategy for infected individuals, known as risk-induced dispersal (RID), which represents the motility of infected individuals induced by risk depending on whether they are in a high- or a low-risk region. We first construct the basic reproduction number and then understand the manner in which a nonuniform movement of infected individuals affects the spreading–vanishing dichotomy of a disease in a one-dimensional domain. We conclude that even though the infected individuals reside in a high-risk initial domain, the disease can be eradicated from the region if the infected individuals move with a high sensitivity of RID as they disperse. Finally, we demonstrate our results via simulations for a one-dimensional case.     

Bifurcation analysis of an SIRS epidemic model with standard incidence rate and saturated treatment function

An epidemic model with standard incidence rate and saturated treatment function of infectious individuals is proposed to understand the effect of the capacity for treatment of infective individuals on the disease spread. The treatment function in this paper is a continuous and differential function which exhibits the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. It is proved that the existence and stability of the disease-free and endemic equilibria for the model are not only related to the basic reproduction number but also to the capacity for treatment of infective individuals. And a backward bifurcation is found when the capacity is not enough. By computing the first Lyapunov coefficient, we can determine the type of Hopf bifurcation, i.e., subcritical Hopf bifurcation or supercritical Hopf bifurcation. We also show that under some conditions the model undergoes Bogdanov-Takens bifurcation. Finally, numerical simulations are given to support some of the theoretical results.     

Fuzzy epidemic model for the transmission of worms in computer network

An e-epidemic SIRS (susceptible–infectious–recovered–susceptible) model for the fuzzy transmission of worms in computer network is formulated. We have analyzed the comparison between classical basic reproduction number and fuzzy basic reproduction number, that is, when both coincide and when both differ. The three cases of epidemic control strategies of worms in the computer network–low, medium, and, high–are analyzed, which may help us to understand the attacking behavior and also may lead to control of worms. Numerical methods are employed to solve and simulate the system of equations developed.     

Effects of information-dependent vaccination behavior on coronavirus outbreak: insights from a SIRI model

A mathematical model is proposed to assess the effects of a vaccine on the time evolution of a coronavirus outbreak. The model has the basic structure of SIRI compartments (susceptible–infectious–recovered–infectious) and is implemented by taking into account of the behavioral changes of individuals in response to the available information on the status of the disease in the community. We found that the cumulative incidence may be significantly reduced when the information coverage is high enough and/or the information delay is short, especially when the reinfection rate is high enough to sustain the presence of the disease in the community. This analysis is inspired by the ongoing outbreak of a respiratory illness caused by the novel coronavirus COVID-19.

     

Forces of Infection Allowing for Backward Bifurcation in an Epidemic Model with Vaccination and Treatment

We consider an epidemic model for the dynamics of a vaccine-preventable disease, which incorporates the treatment and an imperfect vaccine given to susceptible individuals. We show that in spite of the simple structure of the model, a backward bifurcation may always occur if the treatment rate is above a threshold value. This occurs regardless of the specific form of the force of infection, which is only required to be infinitesimal of the same order of the size of the infectious compartment I, as I→0. This includes many commonly used functionals, as the linear, the monotone saturating Michaelis-Menten and the non-monotone force of infection used to represent the ‘psychological effect’.     

Threshold behaviour of a SI epidemiological model with two structuring variables

In this article, we study a SI epidemic model describing the spread of a disease in a perfectly mixed managed population, representing an animal herd in a fattening farm. The epidemic process is characterized by a non-neglectable and variable incubation period, during which individuals are infectious but cannot be easily detected. The susceptible and infected populations are structured according to age and, for infected, to time remaining before the end of the incubation, where they show detectable clinical signs. We study the well posedness and the asymptotic behaviour of the problem and show that in some cases, even if the farm is fed with healthy animals, disease persistence can occur. We give an explicit formula for the basic reproduction number \({\mathcal{R}_0}\) and the biological interpretation of this threshold on a specific example. We finally illustrate the asymptotic behaviour of the model by numerical simulations.     

一类具有二次感染和接种的两病株流行病模型

本文考虑了一类具有二次感染和接种的两病株流行病模型，通过定义每一病株的基本再生数和侵入再生数，我们分析了非负平衡态的稳定性并获得了这样结论：对于较低的接种水平，病株一感染者处于支配地位而病株二感染者将从易感人群中消失，对于非常高的接种水平，疾病将均被消除。     

A generalized infectious model induced by the contacting distance (CTD)

The aim of this paper is to theoretically study the effect of the contacting distance (CTD) between the susceptible and infectious individuals in controlling infectious diseases. This paper formulates a generalized SEIR model incorporating the effect of the contacting distance (CTD). The dynamical behaviors of the proposed model are investigated and the controlling measures of the infectious diseases are developed. The results show that the contacting distance (CTD) between the susceptible and infectious individuals plays an important role in controlling infectious diseases. Some diseases will be globally controlled when the contacting distance (CTD) is larger than the threshold value. That is to say, the long contacting distance (CTD) implies the corresponding diseases will be controlled. However, for other diseases, the long or short contacting distance (CTD) will induce them to spread and be endemic. The moderate contacting distance (CTD) may be beneficial to control these diseases. Therefore, the appropriate contacting distance (CTD) should be selected for the given diseases in order to control the corresponding infectious diseases. Finally, a special numerical experiment is given to test our results. These results give some theoretical and experimental guides for the disease control authorities.     

Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model

This paper presents several simple linear vaccination-based control strategies for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The vaccination control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously that the remaining populations (i.e. susceptible plus infected plus infectious) tend asymptotically to zero.     

Backward bifurcation for some general recovery functions

We consider an epidemic model for the dynamics of an infectious disease that incorporates a nonlinear function h(I), which describes the recovery rate of infectious individuals. We show that in spite of the simple structure of the model, a backward bifurcation may occur if the recovery rate h(I) decreases and the velocity of the recovery rate is below a threshold value in the beginning of the epidemic. These functions would represent a weak reaction or slow treatment measures because, for instance, of limited allocation of resources o sparsely distributed populations. This includes commonly used functionals, as the monotone saturating Michaelis–Menten, and non monotone recovery rates, used to represent a recovery rate limited by the increasing number of infected individuals. We are especially interested in control policies that can lead to recovery functions that avoid backward bifurcation. Copyright © 2016 John Wiley & Sons, Ltd.     

Reducing mosquito-borne disease outbreak size: The relative importance of contact and transmissibility in a network model

The complex biological and environmental factors involved in the transmission of mosquito-borne diseases in humans have made their control elusive in many instances. Conceptual models contribute to gain insight and help to reduce the risk of taking poor managerial decisions. The focus of this paper is to compare, using a contact network model, the impact that perturbation of the number infectious contacts and of transmissibility have on the size of an outbreak. We illustrate the analysis on a contact network parametrized with data that associates humans and the mosquito Culex quinquefasciatus, a vector for lymphatic filariasis. The model suggests that, if the values corresponding to transmissibility and number of infectious contacts is relatively large, variations in the size of an outbreak are significantly in favor of control measures to reduce infectious contacts.     

Mathematical model on the transmission of worms in wireless sensor network

Wireless sensor networks (WSNs) have received extensive attention due to their great potential in civil and military applications. The sensor nodes have limited power and radio communication capabilities. As sensor nodes are resource constrained, they generally have weak defense capabilities and are attractive targets for software attacks. Cyber attack by worm presents one of the most dangerous threats to the security and integrity of the computer and WSN. In this paper, we study the attacking behavior of possible worms in WSN. Using compartmental epidemic model, we propose susceptible – exposed – infectious – recovered – susceptible with a vaccination compartment (SEIRS-V) to describe the dynamics of worm propagation with respect to time in WSN. The proposed model captures both the spatial and temporal dynamics of worms spread process. Reproduction number, equilibria, and their stability are also found. If reproduction number is less than one, the infected fraction of the sensor nodes disappears and if the reproduction number is greater than one, the infected fraction persists and the feasible region is asymptotically stable region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the systems of equations developed and also to validate our model. A critical analysis of vaccination class with respect to susceptible class and infectious class has been made for a positive impact of increasing security measures on worm propagation in WSN.     

The effect of population dispersal on the spread of a disease

The effect of population dispersal among n patches on the spread of a disease is investigated. Population dispersal does not destroy the uniqueness of a disease free equilibrium and its attractivity when the basic reproduction number of a disease R₀<1. When R₀>1, the uniqueness and global attractivity of the endemic equilibrium can be obtained if dispersal rates of susceptible individuals and infective individuals are the same or very close in each patch. However, numerical calculations show that population dispersal may result in multiple endemic equilibria and even multi-stable equilibria among patches, and also may result in the extinction of a disease, even though it cannot be eradicated in each isolated patch, provided the basic reproduction numbers of isolated patches are not very large.     

Optimal two-phase vaccine allocation to geographically different regions under uncertainty

In this article, we consider a decision process in which vaccination is performed in two phases to contain the outbreak of an infectious disease in a set of geographic regions. In the first phase, a limited number of vaccine doses are allocated to each region; in the second phase, additional doses may be allocated to regions in which the epidemic has not been contained. We develop a simulation model to capture the epidemic dynamics in each region for different vaccination levels. We formulate the vaccine allocation problem as a two-stage stochastic linear program (2-SLP) and use the special problem structure to reduce it to a linear program with a similar size to that of the first stage problem. We also present a Newsvendor model formulation of the problem which provides a closed form solution for the optimal allocation. We construct test cases motivated by vaccine planning for seasonal influenza in the state of North Carolina. Using the 2-SLP formulation, we estimate the value of the stochastic solution and the expected value of perfect information. We also propose and test an easy to implement heuristic for vaccine allocation. We show that our proposed two-phase vaccination policy potentially results in a lower attack rate and a considerable saving in vaccine production and administration cost.     

Approximating individual risk of infection in a Markov chain epidemic network model with a deterministic system

In the Markov chain model of infectious diseases in a connected network of heterogeneous individuals, the computation of the risk of infection for each individual and the expected size of the infected population over time is an NP-hard problem. We show that the individual risk of infection over time can be approximated by orbits of a nonlinear discrete dynamical system on a phase space of dimension equal to the number of individuals in the network. An upper bound for the eradication rate of the infectious disease in the network is also obtained.     

Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases

In this paper, a non-linear mathematical model for the effects of awareness programs on the spread of infectious diseases such as flu has been proposed and analyzed. In the modeling process it is assumed that disease spreads due to the contact between susceptibles and infectives only. The growth rate of awareness programs impacting the population is assumed to be proportional to the number of infective individuals. It is further assumed that due to the effect of media, susceptible individuals form a separate class and avoid contact with the infectives. The model is analyzed by using stability theory of differential equations. The model analysis shows that the spread of an infectious disease can be controlled by using awareness programs but the disease remains endemic due to immigration. The simulation analysis of the model confirms the analytical results.     

Modeling pulse infectious events irrupting into a controlled context: A SIS disease with almost periodic parameters

A new epidemic model of seasonal/cyclical pulse contagions of an infectious disease is introduced: a population with a controlled infectious disease is perturbed by a sequence of pulse infectious events arising from the specific features of the population’s behavior. The purpose of this article is to obtain an epidemic threshold which allows us to decide how a sequence of epidemic events could destabilize the previous controlled scenario and how a new endemic equilibrium appears. A threshold is obtained when supposing a set of almost periodic properties in the model.     

一类带有非线性传染率的SEIR传染病模型的全局分析

通过假设被传染的易感者一部分经过一段潜伏期后才具有传染性,而另一部分被感染的易感者直接成为传染者,建立了一类带有非线性传染率的SEIR传染病模型,得到了确定疾病是否成为地方病的基本再生数以及无病平衡点和地方病平衡点的全局稳定性.