Extinction Of Disease Research Articles

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Dynamics of a stochastic staged progression model of HIV transmission with a hybrid switch

In this paper, we investigate the dynamics behavior of stochastic hybrid switching staged progression model for the transmission of HIV. The method used in this paper is Lyapunov functions analysis. We establish sufficient conditions for extinction of disease. Especially, we conclude that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, some examples and their simulations are provided to illustrate the theoretical results.

Global behaviour of a class of discrete epidemiological SI models with constant recruitment of susceptibles

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number We establish the global asymptotic stability of the disease-free equilibrium (GAS) for We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for . For the transcritical bifurcation appears. For we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number .

Global Stability Analysis and Parameter Estimation for a Diphtheria Model: A Case Study of an Epidemic in Rohingya Refugee Camp in Bangladesh.

In this article, we have developed a deterministic Susceptible-Latent-Infectious-Recovered (SLIR) model for diphtheria outbreaks. Here, we have studied a case of the diphtheria outbreak in the Rohingya refugee camp in Bangladesh to trace the disease dynamics and find out the peak value of the infection. Both analytical and numerical investigations have been performed on the model to find several remarkable behaviors like the positive and bounded solution, basic reproductive ratio, and equilibria such as disease extinction equilibrium and disease persistence equilibrium which are characterized depending on the basic reproductive ratio and global stability of the model using Lyapunov function for both equilibria. Parameter estimation has been performed to determine the values of the parameter from the daily case data using numerical technique and determined the value of the basic reproductive number for the outbreak as ℛ0 = 5.86.

Strong Allee effect and basins of attraction in a discrete‐time zoonotic infectious disease model

AbstractMotivated by the Feline immunodeficiency virus, the virus that causes AIDS in cat populations, we use discrete‐time infectious disease models with demographic strong Allee effect to examine the impact of the fatal susceptible‐infected (SI) infections on two different types of growth functions: Holling type III or modified Beverton–Holt per‐capita growth function (compensatory density dependence), and Ricker per‐capita growth function with mating (overcompensatory density dependence). The occurrence of the strong Allee effect in the disease‐free equation renders the SI population model bistable, where the two coexisting locally asymptotically stable equilibrium points are either the origin (catastrophic extinction state) and the second fixed point (compensatory dynamics) or the origin and an intrinsically generated demographic period population cycle (overcompensatory dynamics). We use the basic reproduction number, , and the spectral radius, , to examine the structures of the coexisting attractors. In particular, we use MATLAB simulations to show that the fatal disease is not only capable of enlarging or shrinking the basin of attraction of the catastrophic extinction state, but it is also capable of fracturing the basins of attraction into several disjoint sets. Thus, making it difficult to specify the asymptotic zoonotic SI disease outcome in terms of all initial infections. The complexity of the basins of attractions appears to increase with an increase in the period of the intrinsically generated demographic population cycles.Recommendations for Resource Managers: Worldwide, disease agents which originate from animals or products of animal origin, such as Ebola virus, COVID‐19, West Nile virus, or malaria are responsible for millions of deaths each year. To help improve control and ultimately eradicate future pandemics in human populations, we use a Feline immunodeficiency virus disease model with the strong Allee effect to understand outbreaks of zoonotic diseases. In general, the strong Allee effect renders a population bistable with a catastrophic extinction state coexisting with a persistence state. We use the basic reproduction number and the spectral radius to obtain conditions for outbreaks of zoonotic diseases and conditions for zoonotic disease extinction. We show that a fatal zoonotic disease is not only capable of enlarging or shrinking the basin of attraction of the catastrophic extinction state, but it is also capable of fracturing the basins of attraction into several disjoint sets. Thus, making it difficult to specify the asymptotic zoonotic disease outcome in terms of all initial infections. The complexity of the basins of attractions appears to increase with an increase in the period of the demographic population cycles.

Modelling and stationary distribution of a stochastic citrus greening epidemic model with resistance

Citrus greening disease (also known as Huanglongbing (HLB)) is an insect-borne disease worldwide, and it is the greatest threat to the global citrus industry, with no resistant commercial citrus varieties and no curable chemicals. In this paper, a stochastic SEII-SI compartmental model for the spread of HLB between its vector (psyllids) and host (citrus trees) was considered. The effects of incubation and infectious periods and insecticides resistance were involved in the model. The existence of a unique global positive solution was discussed. Sufficient criteria were analyzed and established for the persistence and extinction in the mean. By using suitable Lyapunov functions, a unique stationary distribution was proven to exist, for the stochastic model. Model analysis indicated that increasing the intensity of the noises could benefit and accelerate the extinction of disease. However, insecticide resistance could increase the risk of epidemic prevalence and be detrimental to disease control. Some numerical simulations were presented to validate the theoretical analysis.

Stationary distribution and extinction of a stochastic influenza virus model with disease resistance.

Influenza is a respiratory infection caused influenza virus. To evaluate the effect of environment noise on the transmission of influenza, our study focuses on a stochastic influenza virus model with disease resistance. We first prove the existence and uniqueness of the global solution to the model. Then we obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Moreover, certain sufficient conditions are provided for the extinction of the influenza virus flu. Finally, several numerical simulations are revealed to illustrate our theoretical results. Conclusively, according to the results of numerical models, increasing disease resistance is favorable to disease control. Furthermore, a simple example demonstrates that white noise is favorable to the disease's extinction.

Threshold behaviour of a stochastic SIRS Le´vy jump model with saturated incidence and vaccination.

A stochastic SIRS system with $ \mathrm {L\acute{e}vy} $ process is formulated in this paper, and the model incorporates the saturated incidence and vaccination strategies. Due to the introduction of $ \mathrm {L\acute{e}vy} $ jump, the jump stochastic integral process is a discontinuous martingale. Then the Kunita's inequality is used to estimate the asymptotic pathwise of the solution for the proposed model, instead of Burkholder-Davis-Gundy inequality which is suitable for continuous martingales. The basic reproduction number $ R_{0}^{s} $ of the system is also derived, and the sufficient conditions are provided for the persistence and extinction of SIRS disease. In addition, the numerical simulations are carried out to illustrate the theoretical results. Theoretical and numerical results both show that $ \mathrm {L\acute{e}vy} $ process can suppress the outbreak of the disease.

Study on the Dynamics of an SIR Epidemic Model with Saturated Growth Rate

In this paper, we investigate the dynamic properties of an SIR epidemic model with saturated growth rate. Under the conditions of an arbitrary initial value, we prove that the system exists unique positive solution, and give the sufficient conditions caused by random environmental factors leading to the extinction of infectious diseases. Moreover, we verify the conditions for the persistence of infectious diseases in the mean sense. Finally, we provide the biology interpretation and some strategies to control the infectious diseases.

LONG-TIME BEHAVIOR OF STOCHASTIC STAGED PROGRESSION EPIDEMIC MODEL WITH HYBRID SWITCHING FOR THE TRANSMISSION OF HIV

<abstract> In this paper, one stochastic hybrid switching SP (staged progression) model for the transmission of HIV is proposed and investigated. The system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent. Furthermore, sufficient conditions for extinction of disease are established. At last, some examples and simulations are provided to illustrate our results. </abstract>

Deterministic and stochastic model for the hepatitis C with different types of virus genome

&lt;abstract&gt;&lt;p&gt;In this paper, a deterministic and stochastic model for hepatitis C with different types of virus genomes is proposed and analyzed. Some sufficient conditions are obtained to ensure the stability of the deterministic equilibrium points. We perform a stochastic extension of the deterministic model to study the fluctuation between environmental factors. Firstly, the existence of a unique global positive solution for the stochastic model is investigated. Secondly, sufficient conditions for the extinction of the hepatitis C virus from the stochastic system are obtained. Theoretical and numerical results show that the smaller white noise can ensure the persistence of susceptible and infected populations while the larger white noise can lead to the extinction of disease. By introducing the basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, the conditions that cause the disease to die out are indicated. The importance of environmental noise in the propagation of hepatitis C viruses is highlighted by these findings.&lt;/p&gt;&lt;/abstract&gt;

Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay

&lt;abstract&gt;&lt;p&gt;In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

A stochastic hybrid differential equation model: Analysis and application

Infectious diseases represent a real challenge to humanity and a true challenge for researchers to propose relevant solutions in order to reduce the number of infected individuals. Mathematical modeling of infectious diseases represents a better way to understand and control the spread of epidemics. In this work, we propose a stochastic SIQS epidemic model with a nonlinear incidence function and Markov switching. Firstly, we present our proposed stochastic model and its parameters. Secondly, we show the global existence and uniqueness of the positive solution. Then, we show a sufficient condition for the extinction of disease. Finally, we give numerical simulation to enrich our analytical results.

Comparison Between Deterministic and Stochastic Model for Interaction (COVID-19) With Host Cells in Humans

In this paper, the deterministic and the stochastic models are proposed to study the interaction of the Coronavirus (COVID-19) with host cells inside the human body. In the deterministic model, the value of the basic reproduction number determines the persistence or extinction of the COVID-19. If , one infected cell will transmit the virus to less than one cell, as a result, the person carrying the Coronavirus will get rid of the disease .If the infected cell will be able to infect all cells that contain ACE receptors. The stochastic model proves that if are sufficiently large then maybe give us ultimate disease extinction although , and this facts also proved by computer simulation.

Dynamics study of a stochastic SIQR epidemic model with vaccination and saturated incidence

The paper considers a stochastic SIQR epidemic model with vaccination and saturated incidence. First, the existence and boundedness of the global positive solution are given. Second, the threshold for disease extinction is established. Then, through building proper Lyapunov function, the existence of an ergodic stationary distribution is obtained. At last, the main results are confirmed by numerical simulations, which manifests that the large white noise gets to repress the occurrence of disease.

Dynamic Behavior of a Stochastic Tungiasis Model for Public Health Education

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using It formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.

Long time behavior of an two diffusion stochastic sir epidemic model with nonlinear incidence and treatment

In this paper we propose a stochastic SIR epidemic model to evaluate effect of the randomness on treatment and nonlinear incidence rate. More precisely, we perturb both nonlinear incidence and treatment rates in deterministic SIR model with Gaussian white noise and obtain two diffusion stochastic model. For the model, we theoretically prove that it?s solution is positive and global, and then, we obtain the conditions under which we can claim the existence of the stationary distribution. Also, by constructing suitable Lyapunov functions, weestablish sufficient conditions for p-th moment and almost sure exponential stability of disease-free equilibrium. Conditions for disease extinction are obtained, as well. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use real-life data for spread of cholera in the Department of Artibonite in Haiti, as well as for influenza A H1N1 in Guangdong Province, China.

Stochastic dynamical behavior of COVID-19 model based on secondary vaccination.

This paper mainly studies the dynamical behavior of a stochastic COVID-19 model. First, the stochastic COVID-19 model is built based on random perturbations, secondary vaccination and bilinear incidence. Second, in the proposed model, we prove the existence and uniqueness of the global positive solution using random Lyapunov function theory, and the sufficient conditions for disease extinction are obtained. It is analyzed that secondary vaccination can effectively control the spread of COVID-19 and the intensity of the random disturbance can promote the extinction of the infected population. Finally, the theoretical results are verified by numerical simulations.

Classifying the level set of principal eigenvalue for time-periodic parabolic operators and applications

We study the effect of diffusion and frequency on the principal eigenvalue of linear time-periodic parabolic operators with zero Neumann boundary conditions. Monotonocity of the principal eigenvalue and its asymptotic behavior, as diffusion rate and frequency approach zero or infinity, are established. This leads to a classification of the topological structures of level sets for the principal eigenvalue, as a function of diffusion rate and frequency. As applications, we investigate a susceptible-infected-susceptible reaction-diffusion model in spatially heterogeneous and time-periodic environment. We characterize the parameter regions for the persistence and extinction of infectious disease by the basic reproduction number. The asymptotic profiles of endemic periodic solutions are determined when the diffusion rate of susceptible population is small. Our results suggest that fast movement of infected populations and high frequency of oscillation tend to eliminate the disease. Even if the disease persists, it can be controlled by limiting the movement of susceptible populations.

Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.