Endemic Equilibrium State Research Articles

Threshold dynamics of an SEIR epidemic model with a nonlinear incidence rate and a discontinuous treatment function

We consider a susceptible-exposed-infected-removed (SEIR) epidemic model with a nonlinear incidence rate and a discontinuous treatment function. By applying the LaSalle-type invariance principle for differential inclusions and the Lyapunov theory for discontinuous differential equations, we show that the threshold dynamics are completely described by the basic reproduction number $$R_0$$, under some mild assumptions on the incidence rates and the treatment functions. The disease-free equilibrium state is globally asymptotically stable when $$R_0\le 1$$ and there is endemic equilibrium state which is globally asymptotically stable when $$R_0>1$$. Furthermore, we prove that the disease described by the SEIR epidemic model will die out in finite time. Finally, four illustrative numerical examples are given to support our theoretical results.

A mathematical model on HIV/AIDS with fusion effect: Analysis and homotopy solution

In this study, first we analyzed the dynamical behaviour of a deterministic dynamic model for HIV-infected CD4+ T cells in a population of three classes of cell compartments: uninfected, infected and virus, with fusion effect. The qualitative analysis of the model, i.e., positivity and boundedness of the solutions have been discussed. For the proposed model, the non-infected and endemic equilibrium points are recognized and their local stability examined by the Jacobian matrix. The Lyapunov functional and geometric approaches are discussed in detail in order to show the global stability of the non-infected and endemic equilibrium states, respectively. Additional to this qualitative analysis, the approximate analytical solution was obtained for the proposed model with the help of the homotopy analysis method (HAM), and we demonstrated the convergence region by the ℏ curve. The residual error is also calculated for the HAM solution. We have drawn the numerical solutions to confirm all analytical and HAM solution, which reflects the reliability of all solutions.

Modeling multi-group dynamics of related viral videos with delay differential equations

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.

Analysis of HIV Transmission of Commercial Sex Wokers and Their Clients with Condom Use Treatment

A model of the HIV/AIDS epidemic among sex workers and their clients is discussed to study the effects of condom use in the prevention of HIV transmission. The model is addressed to determine the existence of equilibrium states, and then analyze the global stability of disease free and endemic equilibrium states. The global stability of equilibria depends on the vales of the basic reproduction ratio derived from the next generation matrix of the model. The endemic equilibrium state is globally stable when the ratio exceeds unity. The simulation results are presented to discuss the effect of condom use treatment in preventing the spread of HIV/AIDS among sex workers and their clients. The results show that the effectiveness level in using condoms in sexual intercourse corresponds to the decreasing level of the spread of HIV/AIDS. We use Maple and Matlab software to simulate the impact of condom use.

Modelling the Effects of Vertical Transmission in Mosquito and the Use of Imperfect Vaccine on Chikungunya Virus Transmission Dynamics

In this paper, a deterministic mathematical model for Chikungunya virus (Chikv) transmission and control is developed and analyzed to underscore the effect of vaccinating a proportion of the susceptible human, and vertical transmission in mosquito population. The disease free, and endemic equilibrium states were obtained and the conditions for the local and global stability or otherwise were given. Sensitivity analysis of the effective reproductive number, Rc (the number of secondary infections resulting from the introduction of a single infected individual into a population where a proportion is fairly protected) shows that the recruitment rate of susceptible mosquito (ΛM) and the proportion of infectious new births from infected mosquito (β) are the most sensitive parameters. Bifurcation analysis of the model using center manifold theory reveals that the model undergoes backward bifurcation (coexistence of disease free and endemic equilibrium when Rc ＜ 1 ). Numerical simulation of the model shows that vaccination of susceptible human population with imperfect vaccine will have a positive impact and that vertical transmission in mosquito population has a negligible effect. To the best of our knowledge, our model is the first to incorporate vaccinated human compartment and vertical transmission in (Chikv) model.

Global stability for a [formula omitted] dimensional HIV/AIDS epidemic model with treatments

In this work, we derive and analyze a 2n+1-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R0, n, Rt, n and Rt,n as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R0, n, Rt, n and Rt,n. We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if Rt,n=1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if Rt, n < 1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing Rt,n>1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.

Stability Analysis and Stochastic SI Modelling of Endemic Diseases

In this paper, we study a stochastic epidemic model in Meta-population setting. The stochastic model is obtained from the deterministic model by set up random perturbations about the endemic equilibrium state. The outcome of random perturbations on the stability actions of endemic equilibrium is discussed. Stability of the two equilibriums is studied using the Lyapunov function.

The SEIR Dynamical Transmission Model of Dengue Disease with and Without the Vertical Transmission of the Virus

The transmission of dengue disease when there is a possibility of Vertical Transmission (VT) is studied using mathematical modeling. In the normal case, the mosquito is infected by the dengue virus when it bites an infectious human being. Evidence is gathering that the mosquitoes can also be infected through sexual contact with infected male mosquito. To see the possible consequence of having this addition mode of transmission, a SEIR, Susceptible-Exposed-Infected-Recovery, model is constructed. The Routh – Hurwitz criteria are applied to the model in order to establish the stability of the infection. It is seen that the model without the VT model has 2 equilibrium points, a disease free equilibrium point and an endemic equilibrium point, while the model with the VT has only an endemic equilibrium point. The numerical solutions of differential equations of the model without the VT mode exhibit dynamical behaviors that converges to the disease free equilibrium state when basic reproduction time R0 is less than 1 and converges to endemic equilibrium state when R0>1. The trajectories of the numerical solutions for all possibilities (with and without VT mode) projected onto various 2D planes and 3D spaces are presented.

Generalized reproduction numbers, sensitivity analysis and critical immunity levels of an SEQIJR disease model with immunization and varying total population size

An SEQIJR model of epidemic disease transmission which includes immunization and a varying population size is studied. The model includes immunization of susceptible people (S), quarantine (Q) of exposed people (E), isolation (J) of infectious people (I), a recovered population (R), and variation in population size due to natural births and deaths and deaths of infected people. It is shown analytically that the model has a disease-free equilibrium state which always exists and an endemic equilibrium state which exists if and only if the disease-free state is unstable. A simple formula is obtained for a generalized reproduction number Rg where, for any given initial population, Rg<1 means that the initial population is locally asymptotically stable and Rg>1 means that the initial population is unstable. As special cases, simple formulas are given for the basic reproduction number R0, a disease-free reproduction number Rdf, and an endemic reproduction number Ren. Formulas are derived for the sensitivity indices for variations in model parameters of the disease-free reproduction number Rdf and for the infected populations in the endemic equilibrium state. A simple formula in terms of the basic reproduction number R0 is derived for the critical immunization level required to prevent the spread of disease in an initially disease-free population. Numerical simulations are carried out using the Matlab program for parameters corresponding to the outbreaks of severe acute respiratory syndrome (SARS) in Beijing, Hong Kong, Canada and Singapore in 2002 and 2003. From the sensitivity analyses for these four regions, the parameters are identified that are the most important for preventing the spread of disease in a disease-free population or for reducing infection in an infected population. The results support the importance of isolating infectious individuals in an epidemic and in maintaining a critical level of immunity in a population to prevent a disease from occurring.

Dynamics of Amoebiasis Transmission: Stability and Sensitivity Analysis

Compartmental epidemic models are intriguing in the sense that the generic model may explain different kinds of infectious diseases with minor modifications. However, there may exist some ailments that may not fit the generic capsule. Amoebiasis is one such example where transmission through the population demands a more detailed and sophisticated approach, both mathematical and numerical. The manuscript engages in a deep analytical study of the compartmental epidemic model; susceptible-exposed-infectious-carrier-recovered-susceptible (SEICRS), formulated for Amoebiasis. We have shown that the model allows the single disease-free equilibrium (DFE) state if R 0 , the basic reproduction number, is less than unity and the unique endemic equilibrium (EE) state if R 0 is greater than unity. Furthermore, the basic reproduction number depends uniquely on the input parameters and constitutes a key threshold indicator to portray the general trends of the dynamics of Amoebiasis transmission. We have also shown that R 0 is highly sensitive to the changes in values of the direct transmission rate in contrast to the change in values of the rate of transfer from latent infection to the infectious state. Using the Routh–Hurwitz criterion and Lyapunov direct method, we have proven the conditions for the disease-free equilibrium and the endemic equilibrium states to be locally and globally asymptotically stable. In other words, the conditions for Amoebiasis “die-out” and “infection propagation” are presented.

Lyapunov Function for a Dengue Transmission Model where two Species of Mosquitoes are Present: Global Stability

In this study, we are interested in the newly observed endemic due to the ZIKA virus. In the absence of a full knowledge of the dynamics of the infection due to this virus, we consider a model of the closely related dengue virus when there are two species of mosquitoes, A. aegypti and A. albopictus mosquitoes who are both capable of transmitting the two viruses. We use a Susceptible-Infected-Recovered (SIR) model to describe the infection of the human populations. We obtain the basic reproduction ratio (R0) and show that if R0 is less than 1, the disease free equilibrium state is global asymptotically stable. If R0 is greater than 1, we use the Lyapunov function approach to find the conditions for the unique dengue endemic equilibrium state to be globally stable. We then point out the insights into the global stability of the ZIKA epidemic that can be gained by looking at the global stability of a model for the dengue infection in the presence of two species of mosquitoes that can transmit the disease.

Stability Analysis of the Endemic Equilibrium State on the Spread of Malaria Using Bellman and Cooke’s Theorem

Stability Analysis of the Endemic Equilibrium State on the Spread of Malaria Using Bellman and Cooke’s Theorem

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

This paper studies the nonnegativity and local and global stability properties of the solutions of a newly proposed SEIADR model which incorporates asymptomatic and dead-infective subpopulations into the standard SEIR model and, in parallel, it incorporates feedback vaccination plus a constant term on the susceptible and feedback antiviral treatment controls on the symptomatic infectious subpopulation. A third control action of impulsive type (or “culling”) consists of the periodic retirement of all or a fraction of the lying corpses which can become infective in certain diseases, for instance, the Ebola infection. The three controls are allowed to be eventually time varying and contain a total of four design control gains. The local stability analysis around both the disease-free and endemic equilibrium points is performed by the investigation of the eigenvalues of the corresponding Jacobian matrices. The global stability is formally discussed by using tools of qualitative theory of differential equations by using Gauss-Stokes and Bendixson theorems so that neither Lyapunov equation candidates nor the explicit solutions are used. It is proved that stability holds as a parallel property to positivity and that disease-free and the endemic equilibrium states cannot be simultaneously either stable or unstable. The periodic limit solution trajectories and equilibrium points are analyzed in a combined fashion in the sense that the endemic periodic solutions become, in particular, equilibrium points if the control gains converge to constant values and the control gain for culling the infective corpses is asymptotically zeroed.

Effect of Rainfall for the Dynamical Transmission Model of the Dengue Disease in Thailand.

The SEIR (Susceptible-Exposed-Infected-Recovered) model is used to describe the transmission of dengue virus. The main contribution is determining the role of the rainfall in Thailand in the model. The transmission of dengue disease is assumed to depend on the nature of the rainfall in Thailand. We analyze the dynamic transmission of dengue disease. The stability of the solution of the model is analyzed. It is investigated by using the Routh-Hurwitz criteria. We find two equilibrium states: a disease-free state and an endemic equilibrium state. The basic reproductive number (R0) is obtained, which indicates the stability of each equilibrium state. Numerical results taking into account the rainfall are obtained and they are seen to correspond to the analytical results.

Mathematical Modeling and Simulation Study for the Control and Transmission Dynamics of Measles

In this paper, a five compartmental model has been considered and investigated the transmission dynamics of measles disease in the human populations. The only one infected compartment in the standard model has been split into two: Infected catarrh, and infected eruption. Measles is a deadly disease that is very common and contagious in the world. However, if enough care is taken one can survive easily against Measles disease. The Measles disease has no specific treatment but vaccination is available. It has been shown that the model has a positive solution and is bounded. The basic reproduction number is derived using the next generation matrix method. The disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Numerical simulation study is conducted using ode 45 of MATLAB. The results and interpretations are elaborated and included in the text. Description of the model, Mathematical analysis, stability analysis, and simulation studies are conducted and the results are included. The standard model and the proposed models have been compared and the observations are presented in a tabular form.

The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss

In this paper, a compartment model has been built, presented and investigated the dynamics and spread of zika virus in both human and mosquito populations. It is focused to study the impact of symptomatic and asymptomatic infective immigrants on the spread of zika virus. A new mathematical model SI1I2R for human and SI model for vector population has been designed and presented. Here I1 is symptomatic infective and I2 is asymptomatic infective human populations. The present model is developed making some reasonable modifications in the corresponding epidemic SIR model by considering symptomatic and asymptomatic infective immigrants. Susceptible vectors get infection either from symptomatic or asymptomatic infected human populations. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Simulation study is conducted using MATLAB ode45.

The Impact of Infective Immigrants on the Spread of Dog Rabies

In this paper, it is proposed and analyzed a new mathematical model and that is developed on the basis of some reasonable modification made to the standard epidemic model. The impact of immigration, treatment and the effect of vaccination are included in the model. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. Numerical simulation study is conducted using <i>ode 45</i> of MATLAB. It has been shown that the solution is positive and bounded. Algebraic expression for the reproduction number is constructed. Equilibrium points are identified and their stability analysis is carried out. It is pointed out that the disease dies out if the immigration of the infected dogs is controlled and the vaccination and the treatments are improved. Otherwise, the disease spreads rapidly in the dog population and it becomes an epidemic. Further, it is also pointed out that the impact of infective immigrants on the spread of dog rabies is positive and additive. The details are presented and discussed in the text.

Stability of epidemic models over directed graphs: A positive systems approach

We study the stability properties of a susceptible–infected–susceptible (SIS) diffusion model, so-called the n-intertwined Markov model, over arbitrary directed network topologies. As in the majority of the work on infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the disease-free equilibrium is the unique equilibrium over the network. Otherwise, an endemic equilibrium state emerges, where some infection remains within the network. Using notions from positive systems theory, we provide novel proofs for the global asymptotic stability of the equilibrium points in both cases over strongly connected networks based on the value of the basic reproduction number, a fundamental quantity in the study of epidemics. When the network topology is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of an endemic state, and study the stability of the disease-free equilibrium. Finally, we demonstrate that the n-intertwined Markov model can be viewed as a best-response dynamical system of a concave game among the nodes. This characterization allows us to cast new infection spread dynamics; additionally, we provide a sufficient condition for global convergence to the disease-free equilibrium, which can be checked in a distributed fashion.

Stability analysis of the endemic equilibrium state of an infection age-structured HIV/AIDS disease pandemic

In this work we present an infection-age-structured mathematical model of AIDS disease dynamics and examine the endemic equilibrium state for stability. An explicit formula for the basic reproduction number R0 was obtained in terms of the demographic and epidemiological parameters of the model. The endemic equilibrium state was found to be locally asymptotically stable under certain conditions. Furthermore, by constructing a suitable Lyapunov functional, the endemic equilibrium state was found to be globally asymptotically stable under certain conditions prescribed on the model parameters.Keywords: Basic reproduction number, HIV/AIDS, Lyapunov functional

A dynamical model for bark beetle outbreaks

Tree-killing bark beetles are major disturbance agents affecting coniferous forest ecosystems. The role of environmental conditions on driving beetle outbreaks is becoming increasingly important as global climatic change alters environmental factors, such as drought stress, that, in turn, govern tree resistance. Furthermore, dynamics between beetles and trees are highly nonlinear, due to complex aggregation behaviors exhibited by beetles attacking trees. Models have a role to play in helping unravel the effects of variable tree resistance and beetle aggregation on bark beetle outbreaks. In this article we develop a new mathematical model for bark beetle outbreaks using an analogy with epidemiological models. Because the model operates on several distinct time scales, singular perturbation methods are used to simplify the model. The result is a dynamical system that tracks populations of uninfested and infested trees. A limiting case of the model is a discontinuous function of state variables, leading to solutions in the Filippov sense. The model assumes an extensive seed-bank so that tree recruitment is possible even if trees go extinct. Two scenarios are considered for immigration of new beetles. The first is a single tree stand with beetles immigrating from outside while the second considers two forest stands with beetle dispersal between them. For the seed-bank driven recruitment rate, when beetle immigration is low, the forest stand recovers to a beetle-free state. At high beetle immigration rates beetle populations approach an endemic equilibrium state. At intermediate immigration rates, the model predicts bistability as the forest can be in either of the two equilibrium states: a healthy forest, or a forest with an endemic beetle population. The model bistability leads to hysteresis. Interactions between two stands show how a less resistant stand of trees may provide an initial toe-hold for the invasion, which later leads to a regional beetle outbreak in the resistant stand.