Approximating Time to Extinction for Endemic Infection Models

Abstract

Approximating the time to extinction of infection is an important problem in infection modelling. A variety of different approaches have been proposed in the literature. We study the performance of a number of such methods, and characterize their performance in terms of simplicity, accuracy, and generality. To this end, we consider first the classic stochastic susceptible-infected-susceptible (SIS) model, and then a multidimensional generalization of this which allows for Erlang distributed infectious periods. We find that (i) for a below-threshold infection initiated by a small number of infected individuals, approximation via a linear branching process works well; (ii) for an above-threshold infection initiated at endemic equilibrium, methods from Hamiltonian statistical mechanics yield correct asymptotic behaviour as population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck diffusion approximation gives a very poor approximation, but may retain some value for qualitative comparisons in certain cases; (iv) a more detailed diffusion approximation can give good numerical approximation in certain circumstances, but does not provide correct large population asymptotic behaviour, and cannot be relied upon without some form of external validation (eg simulation studies).

Estimating the distribution of time to extinction of infectious diseases in mean-field approaches

“…In this manuscript, we study the extinction times of diseases when R eff < 1 starting from a finite population infected, utilising results from a simplified birth-death model. We formulate the mean extinction time that is in agreement with the previous studies in the analogous limits [25,26,27,28]. Beyond that, our analysis provides a novel method to approximate the distribution of extinction times by calculating additional moments of extinction time.…”

Persistence time of SIS infections in heterogeneous populations and networks

“…This approach seems reasonable in terms of qualitative comparisons between infection models, and is common in the literature. However, the approach is known to give a very bad numerical approximation to mean persistence time, with incorrect leading-order asymptotic behaviour, due to the failure in the lower tail of the normal approximation to the quasi-stationary distribution [11,9]. The methods of the current paper, in contrast, deal directly with the expected persistence time and yield correct leading-order asymptotic formulae.…”

“…For the SIS model with Erlang-distributed infectious periods in a homogeneous population (k = 1), the solution U (θ) to the relevant Hamilton-Jacobi equation was found in [9] to be U (θ) = ln Taking the Legendre transform, we find that V (y) for this homogeneouspopulation model is given by…”

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations

“…For instance, Nåsell (1999) made use of this approach in studying an infection model incorporating demographic processes and disease-induced immunity (with exponentially distributed infectious periods), following on from which Andersson and Britton (2000) extended the model to include latency, with latent periods and infectious periods each being allowed to follow Erlang distributions. Unfortunately, while this approach can give some rough qualitative indication of the effect of model parameters upon persistence times, the numerical approximation to τ thus obtained is known to be extremely inaccurate (Clancy and Tjia 2018). Indeed, as pointed out in Nåsell (1999), this approximation does not yield correct N -dependence in the large population limit; specifically, the approximation which appears as equation (2.15) of Nåsell (1999) takes the form τ ≈ c √ N exp(a N ) for some constants a, c, in contrast to the asymptotic form (1) obtained in van Herwaarden and Grasman (1995).…”

“…Evaluation of A required numerical solution of a system of ordinary differential equations, while no method for evaluating C was given. It should be noted that van Herwaarden and Grasman (1995) studied a diffusion approximation to the infection model, but it is now well known (see, e.g., Clancy and Tjia 2018;Doering et al 2005) that such a diffusion approximation does not, in general, give correct leading-order asymptotics-that is, the value of the constant A computed from the diffusion approximation is not necessarily equal to the correct A value for the underlying discrete state-space model. Using rather different techniques, Andersson and Djehiche (1998) derived a result of the form (1), together with explicit expressions for the constants A, C, for the classic susceptible-infectious-susceptible (SIS) model of Weiss and Dishon (1971).…”