Probability Of Disease Extinction Research Articles

The effect of demographic stochasticity on Zika virus transmission dynamics: Probability of disease extinction, sensitivity analysis, and mean first passage time.

Viral infections spread by mosquitoes are a growing threat to human health and welfare. Zika virus (ZIKV) is one of them and has become a global worry, particularly for women who are pregnant. To study ZIKV dynamics in the presence of demographic stochasticity, we consider an established ZIKV transmission model that takes into consideration the disease transmission from human to mosquito, mosquito to human, and human to human. In this study, we look at the local stability of the disease-free and endemic equilibriums. By conducting the sensitivity analysis both locally and globally, we assess the effect of the model parameters on the model outcomes. In this work, we use the continuous-time Markov chain (CTMC) process to develop and analyze a stochastic model. The main distinction between deterministic and stochastic models is that, in the absence of any preventive measures such as avoiding travel to infected areas, being careful from mosquito bites, taking precautions to reduce the risk of sexual transmission, and seeking medical care for any acute illness with a rash or fever, the stochastic model shows the possibility of disease extinction in a finite amount of time, unlike the deterministic model shows disease persistence. We found that the numerically estimated disease extinction probability agrees well with the analytical probability obtained from the Galton-Watson branching process approximation. We have discovered that the disease extinction probability is high if the disease emerges from infected mosquitoes rather than infected humans. In the context of the stochastic model, we derive the implicit equation of the mean first passage time, which computes the average amount of time needed for a system to undergo its first state transition.

Multipatch stochastic epidemic model for the dynamics of a tick-borne disease

Spatial heterogeneity and migration of hosts and ticks have an impact on the spread, extinction and persistence of tick-borne diseases. In this paper, we investigate the impact of between-patch migration of white-tailed deer and lone star ticks on the dynamics of a tick-borne disease with regard to disease extinction and persistence using a system of Itô stochastic differential equations model. It is shown that the disease-free equilibrium exists and is unique. The general formula for computing the basic reproduction number for all patches is derived. We show that for patches in isolation, the basic reproduction number is equal to the largest patch reproduction number and for connected patches it lies between the minimum and maximum of the patch reproduction numbers. Numerical simulations for a two-patch deterministic and stochastic differential equation models are performed to illustrate the dynamics of the disease for varying migration rates. Our results show that the probability of eliminating or minimizing the disease in both patches is high when there is no migration unlike when it is present. The results imply that the probability of disease extinction can be increased if deer and tick movement are controlled or even prohibited especially when there is an outbreak in one or both patches since movement can introduce a disease in an area that was initially disease-free. Thus, screening of infectives in protected areas such as deer farms, private game parks or reserves, etc. before they migrate to other areas can be one of the intervention strategies for controlling and preventing disease spread.

Probability of disease extinction and outbreak in a stochastic tuberculosis model with fast-slow progression and relapse

<abstract><p>A stochastic continuous-time Markov chain tuberculosis model with fast-slow progression and relapse is established to explore the impact of the demographic variation on TB transmission. At first, the extinction threshold and probability of the disease extinction and outbreak are obtained by applying the multitype Galton-Waston branching process for the stochastic model. In numerical simulations, the probability of the disease extinction and outbreak and expected epidemic duration of the disease are estimated. To see how demographic stochasticity affects TB dynamics, we compare dynamical behaviors of both stochastic and deterministic models, and these results show that the disease extinction in stochastic model would occur while the disease is persistent for the deterministic model. Our results suggest that minimizing the contact between the infectious and the susceptible, and detecting the latently infected as early as possible, etc., could effectively prevent the spread of tuberculosis.</p></abstract>

EXTINCTION THRESHOLD IN A STOCHASTIC EPIDEMIC MODEL FOR ROTAVIRUS DYNAMICS WITH CONTAMINATED ENVIRONMENT

Mathematical models for the transmission dynamics of infectious diseases have aided our understanding of the important factors that drive epidemic patterns. In this paper, we formulate and analyze a stochastic epidemic model, a continuous-time Markov chain, in order to understand rotavirus dynamics with a contaminated environment. The assumptions of the deterministic model are utilized in the formulation of the corresponding stochastic model. We perform both local and global stability analyses of the equilibria of the deterministic model with respect to the basic reproduction number. The extinction threshold for the stochastic model and conditions for either disease extinction or persistence are derived by employing the branching process to the infectious classes only. It is shown that the probability of rotavirus extinction obtained from the branching process is in excellent agreement with the numerically approximated probability. Numerical results indicate that the probability of rotavirus extinction is the highest if the contaminated environment introduces the virus into a totally susceptible population at the beginning of the epidemic process. Thus, a major rotavirus outbreak is likely if the virus emanates from infectious children at the onset of the epidemic. Results of sensitivity analysis showed that shedding of the virus into the environment by infectious children is the most sensitive parameter of the model. Further, it is shown that decreasing the shedding rate leads to an increase in the probability of disease extinction and vice versa. This, therefore, implies that disposal of stool of infectious children should be well managed if efforts to curb further spread of the disease or even eliminating it are to bear desirable fruits.

Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds.

Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.

Local approximation of Markov chains in time and space

ABSTRACTIn epidemic modelling, the emergence of a disease is characterized by the low numbers of infectious individuals. Environmental randomness impacts outcomes such as the outbreak or extinction of the disease in this case. This randomness can be accounted for by modelling the system as a continuous time Markov chain, . The probability of extinction given some initial state is the probability of hitting a subset of the state space associated with extinction for the initial state. This hitting probability can be studied by passing to the discrete time Markov chain (DTMC), . An approach is presented to approximate a DTMC on a countably infinite state space by a DTMC on a finite state space for the purpose of solving general hitting problems. This approach is applied to approximate the probability of disease extinction in an epidemic model. It is also applied to evaluate a heterogeneous disease control strategy in a metapopulation.

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence.

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton-Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.

Discrete stochastic metapopulation model with arbitrarily distributed infectious period

In this study, a stochastic discrete-time model is developed to study the spread of an infectious disease in an n-patch environment. The model includes an arbitrary distribution of the (random) infectious period T, and the results are used to investigate how the distribution of T may influence the model outcomes. General results are applied to specific distributions including Geometric, Negative Binomial, Poisson and Uniform. The model outcomes are contrasted both numerically and analytically by comparing the corresponding basic reproduction numbers R0 and probability of a minor epidemic (or probability of disease extinction) P0. It is shown analytically that for n = 2 the reproduction numbers corresponding to different distributions of T can be ordered based on the probability generating function ϕT of T. In addition, numerical simulations are carried out to examine the final epidemic size F and duration of the epidemic D of a two-patch model.

Discrete Stochastic Metapopulation Model with Arbitrarily Distributed Infectious Period

In this talk, a stochastic discrete-time model is introduced to study the spread of an infectious disease in an n-patch environment. The model includes an arbitrary distribution of the (random) infectious period T, and the results are used to investigate how the distribution of T may influence the model outcomes. Specific distributions including Geometric, Negative Binomial, Poisson and Uniform distributions are compared both numerically and analytically. The measures used in model comparisons include (i) the basic reproduction number ; (ii) probability of a minor epidemic (or probability of disease extinction) ; (iii) final epidemic size; and (iv) duration of the epidemic. The results are illustrated for the case of two patches. It is also shown analytically that the reproduction numbers corresponding to different distributions of T can be ordered based on the probability generating function of T.

Development and Application of a Stochastic Epidemic Model for the Transmission of Salmonella Typhimurium at the Farm Level of the Pork Production Chain

In previous work a deterministic model for the compartment level was built, taking into account the two different syndromes with which Salmonella Typhimurium appears at pig farms. Based on this model, a stochastic one was built in this work that simulated different compartmental sizes, taking into account compartments of 200 to 400 pigs. Multiple scenarios of starting conditions of infection (SCI) ranging from 0.25 to 100% were tested for each population size. The effect of each of these two factors on the probability of disease extinctions and the prevalence of each of the classes of the model and the risk groups of pigs were estimated. The results showed that the compartment population had an inverse effect on the probability of disease extinction. On the other hand, low SCI resulted in high levels of early extinctions reaching 45%, while higher SCI led to high levels of late extinctions. Early extinctions resulted in the absence of the pathogen from the compartment, while late extinctions did not assure it. This effect shows that reducing the population of the compartment combined with appropriate cleaning and good farming practices could have a positive effect in the reduction of the risk of introducing S. Typhimurium into the slaughtering procedure. On the other hand, the profile of seroprevalence at slaughter age allows for risk characterization of the farm, given the relative stability and the small variation for higher SCI.