Predicting extinction rates in stochastic epidemic models (original) (raw)
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Enhancement of epidemic extinction by random vaccination
2008
Submitted for the MAR08 Meeting of The American Physical Society Enhancement of epidemic extinction by random vaccination 1 IRA SCHWARTZ, Naval Research Laboratory, MARK DYKMAN, Michigan State University-We study the probability of epidemic extinction in large populations. We use the susceptible-infected-susceptible (SIS) model since it forms the foundation of many epidemic processes. Fluctuations in the SIS system have two sources. The major source is the randomness of the "reactions" in which the number of susceptibles and/or infected changes. In addition, we assume that vaccination is done at random, leading to the decrease of the number of susceptibles. The vaccination is modeled by a Poisson process. The probability distribution is found from the master equation, which is solved in the eikonal approximation. It is shown that, even in the absence of vaccination, the logarithm of the extinction rate displays scaling dependence on the parameters. It scales as the square of the distance to the parameter value where the average number of infected vanishes. This is very different from the familiar 3/2 scaling law for saddle-node bifurcations. Finally, we show that even weak vaccination can dramatically increase the extinction probability. The correction to the logarithm of the probability becomes exponential in the vaccination rate when this rate is not too small.
Disease Extinction in the Presence of Random Vaccination
Physical Review Letters, 2008
We investigate disease extinction in an epidemic model described by a birth-death process. We show that, in the absence of vaccination, the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point, with an unusual critical exponent. Even a comparatively weak Poissondistributed random vaccination leads to an exponential increase in the extinction rate, with the exponent that strongly depends on the vaccination parameters.
Vaccine enhanced extinction in stochastic epidemic models
2012
We study vaccine control for disease spread on an adaptive network modeling disease avoidance behavior. Control is implemented by adding Poisson-distributed vaccination of susceptibles. We show that vaccine control is much more effective in adaptive networks than in static networks due to feedback interaction between the adaptive network rewiring and the vaccine application. When compared to extinction rates in static social networks, we find that the amount of vaccine resources required to sustain similar rates of extinction are as much as two orders of magnitude lower in adaptive networks.
Journal of Advances in Mathematics and Computer Science
In this paper, we include stochastic perturbation into SIRS epidemic model incorporating media coverage and study their dynamics. Our model is obtained by taking into account both for demographic stochasticity and environmental fluctuations on contact rate before alert media β1. First, we show that the model is biologically well-posed by proving the global existence, positivity and boundedness of solution . Then, sufficient conditions for the extinction of infectious diseaseis proved. We also established sufficient conditions for the existence of an ergodic stationary distribution to the model. Finally, the theoretical results are illustrated by numerical simulations; in addition we show that the media coverage can reduce the peak of infective individuals via numerical simulations.
On the stochastic engine of transmittable diseases in exponentially growing populations
2021
The purpose of this paper is to analyze the interplay of deterministic and stochastic models for epidemic diseases. Deterministic models for epidemic diseases are prone to predict global stability. If the natural birth and death rates are assumed small in comparison to disease parameters like the contact rate and the recovery rate, then the globally stable endemic equilibrium corresponds to a tiny proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results. Diffusion effects force frequency functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model are, however, oscillatory and predict oscillatory patterns. Smaller or isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction. We prove that evolution maximizes the infectiousness of the disease as measured by th...
Extinction times for closed epidemics: the effects of host spatial structure
Ecology Letters, 2002
Although of practical importance, the relationship between the duration of an epidemic and host spatial structure is poorly understood. Here we use a stochastic metapopulation model for the transmission of infection in a spatially structured host population. There are three qualitatively different regimes for the extinction time, which depend on patch population size, the within-patch basic reproductive number and the strength of coupling between patches. In the first regime, the extinction time for the metapopulation (i.e. from all patches) is approximately equal to the extinction time for a single patch. In the second regime, the metapopulation extinction time is maximal but also highly variable. In the third regime, the extinction time for the metapopulation (T E) is given by T E ¼ a + bn 1 ⁄ 2 where a is the local extinction time (i.e. from last patch), b is the transit time (i.e. the time taken for infection to spread from one patch to another) and n is the total number of patches.
Permanence and extinction for the stochastic SIR epidemic model
Journal of Differential Equations, 2020
The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for SIR epidemic system with multi noises dS(t) = a 1 − b 1 S(t) − I(t)f (S(t), I(t)) dt + σ 1 S(t)dB 1 (t) − I(t)g(S(t), I(t))dB 3 (t), dI(t) = − b 2 I(t) + I(t)f (S(t), I(t)) dt + σ 2 I(t)dB 2 (t) + I(t)g(S(t), I(t))dB 3 (t). Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results.
The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate
Bulletin of mathematical biology, 2015
A stochastic SIRS epidemic model with nonlinear incidence rate and varying population size is formulated to investigate the effect of stochastic environmental variability on inter-pandemic transmission dynamics of influenza A. Sufficient conditions for extinction and persistence of the disease are established. In the case of persistence, the existence of endemic stationary distribution is proved and the distance between stochastic solutions and the endemic equilibrium of the corresponding deterministic system in the time mean sense is estimated. Based on realistic parameters of influenza A in humans, numerical simulations have been performed to verify/extend our analytical results. It is found that: (i) the deterministic threshold of the influenza A extinction [Formula: see text] may exist and the threshold parameter will be overestimated in case of neglecting the impaction of environmental noises; (ii) the presence of environmental noises is capable of supporting the irregular recu...
Disease Extinction Versus Persistence in Discrete-Time Epidemic Models
Bulletin of Mathematical Biology
We focus on discrete-time infectious disease models in populations that are governed by constant, geometric, Beverton-Holt or Ricker demographic equations, and give a method for computing the basic reproduction number, R 0. When R 0 < 1 and the demographic population dynamics are asymptotically constant or under geometric growth (non-oscillatory), we prove global asymptotic stability of the disease-free equilibrium of the disease models. Under the same demographic assumption, when R 0 > 1, we prove uniform persistence of the disease. We apply our theoretical results to specific discrete-time epidemic models that are formulated for SEIR infections, cholera in humans and anthrax in animals. Our simulations show that a unique endemic equilibrium of each of the three specific disease models is asymptotically stable whenever R 0 > 1.