SDE SIS epidemic model with demographic stochasticity and varying population size

In this paper we look at the two dimensional stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stochasticity where births and deaths are regarded as stochastic processes with per capita disease contact rate depending on the population size. First we look at the SDE model for the total population size and show that there exists a unique non-negative solution. Then we look at the two dimensional SDE SIS model and show that there exists a unique non-negative solution which is bounded above given the total population size. Furthermore we show that the number of infecteds and the number of susceptibles become extinct in finite time almost surely. Lastly, we support our analytical results with numerical simulations using theoretical and realistic disease parameter values

A stochastic differential equation SIS epidemic model with two independent Brownian motions

In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory

Bayesian inference for indirectly observed stochastic processes, applications to epidemic modelling

Stochastic processes are mathematical objects that offer a probabilistic representation of how some quantities evolve in time. In this thesis we focus on estimating the trajectory and parameters of dynamical systems in cases where only indirect observations of the driving stochastic process are available. We have first explored means to use weekly recorded numbers of cases of Influenza to capture how the frequency and nature of contacts made with infected individuals evolved in time. The latter was modelled with diffusions and can be used to quantify the impact of varying drivers of epidemics as holidays, climate, or prevention interventions. Following this idea, we have estimated how the frequency of condom use has evolved during the intervention of the Gates Foundation against HIV in India. In this setting, the available estimates of the proportion of individuals infected with HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a fractional stochastic volatility model, indirectly observed through market prices. The intractability of the likelihood function, requiring augmentation of the parameter space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods robust to refinements in time discretisations, made necessary to enforce accuracy of Euler schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh free property. We propose the use of fast approximate filters as a pre-exploration tool to estimate the shape of the target density, for a quicker and more robust adaptation phase of the asymptotically exact algorithm. The fBM problem could not be treated with the PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also be applicable in the Markovian setting

Fitting stochastic epidemic models to gene genealogies using linear noise approximation

Phylodynamics is a set of population genetics tools that aim at reconstructing demographic history of a population based on molecular sequences of individuals sampled from the population of interest. One important task in phylodynamics is to estimate changes in (effective) population size. When applied to infectious disease sequences such estimation of population size trajectories can provide information about changes in the number of infections. To model changes in the number of infected individuals, current phylodynamic methods use non-parametric approaches, parametric approaches, and stochastic modeling in conjunction with likelihood-free Bayesian methods. The first class of methods yields results that are hard-to-interpret epidemiologically. The second class of methods provides estimates of important epidemiological parameters, such as infection and removal/recovery rates, but ignores variation in the dynamics of infectious disease spread. The third class of methods is the most advantageous statistically, but relies on computationally intensive particle filtering techniques that limits its applications. We propose a Bayesian model that combines phylodynamic inference and stochastic epidemic models, and achieves computational tractability by using a linear noise approximation (LNA) --- a technique that allows us to approximate probability densities of stochastic epidemic model trajectories. LNA opens the door for using modern Markov chain Monte Carlo tools to approximate the joint posterior distribution of the disease transmission parameters and of high dimensional vectors describing unobserved changes in the stochastic epidemic model compartment sizes (e.g., numbers of infectious and susceptible individuals). We apply our estimation technique to Ebola genealogies estimated using viral genetic data from the 2014 epidemic in Sierra Leone and Liberia.Comment: 43 pages, 6 figures in the main tex

Stochastic differential equation for two-phase growth model

Most mathematical models to describe natural phenomena in ecology are models with single-phase. The models are created as such to represent the phenomena as realistic as possible such as logistic models with different types. However, several phenomena in population growth such as embryos, cells and human are better approximated by two-phase models because their growth can be divided into two phases, even more, each phase requires different growth models. Most two-phase models are presented in the form of deterministic models, since two-phase models using stochastic approach have not been extensively studied. In previous study, Zheng’s two-phase growth model had been implemented in continuous time Markov chain (CTMC). It assumes that the population growth follows Yule process before the critical size, and the Prendiville process after that. In this research, Zheng’s two-phase growth model has been modified into two new models. Generally, probability distribution of birth and death processes (BDPs) of CTMC is intractable; and even if its first–passage time distribution can be obtained, the conditional distribution for the second-phase is complicated to be determined. Thus, two-phase growth models are often difficult to build. To overcome this problem, stochastic differential equation (SDE) for two-phase growth model is proposed in this study. The SDE for BDPs is derived from CTMC for each phase, via Fokker-Planck equations. The SDE for twophase population growth model developed in this study is intended to be an alternative to the two-phase models of CTMC population model, since the significance of the SDE model is simpler to construct, and it gives closer approximation to real data

A stochastic differential equation SIS epidemic model with two correlated Brownian motions

In this paper, we introduce two perturbations in the classical deterministic susceptible-infected-susceptible epidemic model with two correlated Brownian Motions. We consider two perturbations in the deterministic SIS model and formulate the original model as a stochastic differential equation (SDE) with two correlated Brownian Motions for the number of infected population, based on previous work from Gray et al. in 2011 and Hening’s work in 2017. Conditions for the solution to become extinction and persistence are then stated, followed by computer simulation to illustrate the results

Modelling the effects of ecology on wildlife disease surveillance

Surveillance is the first line of defence against disease, whether to monitor endemic cycles or to detect emergent epidemics. Knowledge of disease in wildlife is of considerable importance for managing risks to humans, livestock and wildlife species. Recent public health concerns (e.g. Highly Pathogenic Avian Influenza, West Nile Virus, Ebola) have increased interest in wildlife disease surveillance. However, current practice is based on protocols developed for livestock systems that do not account for the potentially large fluctuations in host population density and disease prevalence seen in wildlife. A generic stochastic modelling framework was developed where surveillance of wildlife disease systems are characterised in terms of key demographic, epidemiological and surveillance parameters. Discrete and continuous state-space representations respectively, are simulated using the Gillespie algorithm and numerical solution of stochastic differential equations. Mathematical analysis and these simulation tools are deployed to show that demographic fluctuations and stochasticity in transmission dynamics can reduce disease detection probabilities and lead to bias and reduced precision in the estimates of prevalence obtained from wildlife disease surveillance. This suggests that surveillance designs based on current practice may lead to underpowered studies and provide poor characterisations of the risks posed by disease in wildlife populations. By parameterising the framework for specific wildlife host species these generic conclusions are shown to be relevant to disease systems of current interest. The generic framework was extended to incorporate spatial heterogeneity. The impact of design on the ability of spatially distributed surveillance networks to detect emergent disease at a regional scale was then assessed. Results show dynamic spatial reallocation of a fixed level of surveillance effort led to more rapid detection of disease than static designs. This thesis has shown that spatio-temporal heterogeneities impact on the efficacy of surveillance and should therefore be considered when undertaking surveillance of wildlife disease systems

Early warning signs for saddle-escape transitions in complex networks

Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.Comment: revised versio