Eight challenges for stochastic epidemic models involving global transmission (original) (raw)
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Nine challenges for deterministic epidemic models
Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where superinfection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.
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...
Comparison and Assessment of Epidemic Models
Statistical Science, 2018
Model criticism is a growing focus of research in stochastic epidemic modelling, following the successful addressing of model fitting and parameter estimation via powerful computationally intensive statistical methods in recent decades. In this paper, we consider a variety of stochastic representations of epidemic outbreaks, with emphasis on individual-based continuous-time models, and review the range of model comparison and assessment approaches currently applied. We highlight some of the factors that can serve to impede checking and criticism of epidemic models such as lack of replication, partial observation of processes, lack of prior knowledge on parameters in competing models, the nonnested nature of models to be compared, and computational challenges. Based on a wide selection of approaches as reported in the literature, we argue that assessment and comparison of stochastic epidemic models is complex and often, by necessity, idiosyncratic to specific applications. We particularly advocate following the advice of Box [J. Amer. Statist. Assoc. 71 (1976) 791-799] to be selective regarding the model inadequacies for which one tests and, moreover, to be open to the blending of classical and Bayesian ideas in epidemic model criticism, rather than adhering to a single philosophy.
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...
Journal of Theoretical Biology, 2011
Despite temporally-forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number , while the same properties are highly sensitive to small amplitude temporal forcing, particularly when is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.
A SCALABLE DISCRETE EVENT STOCHASTIC AGENT-BASED MODEL OF INFECTIOUS DISEASE PROPAGATION
We propose a new stochastic model of infectious disease propagation. This model tracks individual outcomes, but does so without needing to create connectivity graphs for all members of the population. This makes the model scalable to much larger populations than traditional agent-based models have been able to cope with, while preserving the impact of variability during the critical early stages of an outbreak. This contrasts favorably with aggregate deterministic models, which ignore variability, and negates the requirement to assume " convenient " but potentially unrealistic distribution choices which aggregate stochastic models need in order to be analytically tractable. Initial explorations with our new model show behaviors similar to the observed course of Ebola outbreaks over the past 30+ years—while many outbreaks will fizzle out relatively quickly, some appear to reach a critical mass threshold and can turn into widespread epidemics.
Effects of distribution of infection rate on epidemic models
Physical Review E
A goal of many epidemic models is to compute the outcome of the epidemics from the observed infected early dynamics. However, often, the total number of infected individuals at the end of the epidemics is much lower than predicted from the early dynamics. This discrepancy is argued to result from human intervention or nonlinear dynamics not incorporated in standard models. We show that when variability in infection rates is included in standard susciptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models the total number of infected individuals in the late dynamics can be orders lower than predicted from the early dynamics. This discrepancy holds for SIS and SIR models, where the assumption that all individuals have the same sensitivity is eliminated. In contrast with network models, fixed partnerships are not assumed. We derive a moment closure scheme capturing the distribution of sensitivities. We find that the shape of the sensitivity distribution does not affect R 0 or the number of infected individuals in the early phases of the epidemics. However, a wide distribution of sensitivities reduces the total number of removed individuals in the SIR model and the steady-state infected fraction in the SIS model. The difference between the early and late dynamics implies that in order to extrapolate the expected effect of the epidemics from the initial phase of the epidemics, the rate of change in the average infectivity should be computed. These results are supported by a comparison of the theoretical model to the Ebola epidemics and by numerical simulation.
On Deterministic and Stochastic Multiple Pathogen Epidemic Models
Epidemiologia, 2021
In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result.
Stochastic effects on endemic infection levels of disseminating versus local contacts
Mathematical Biosciences, 2002
The effects of two levels of mixing on endemic infection levels are shown to differ for identically conformed deterministic compartmental (DC) and stochastic compartmental (SC) models. Both DC and SC models give similar endemic levels when populations are large, immunity is short lived, and mixing is universal. But local transmissions and/or transient immunity decrease overall population infection levels in SC but not in DC models. DC models also fail to detect the greater effects of eliminating disseminating transmissions in comparison to eliminating local transmissions shown by SC models. These differences in model behavior arise because localities that encounter few infections from distant sites and that have stochastically low infection levels have decreased infection rates while localities with stochastically high levels of infection do not decrease the rate at which they lose infection. At the extreme this generates local stochastic die out with subsequent build up of susceptibility in SC but not DC models. This phenomenon should act upon all endemic infections that have changing geographic or social foci of infection. Neither standard epidemiological investigations nor sufficient-component cause models can capture these effects because they occur in the absence of differences between individuals.