Outbreak properties of epidemic models: The roles of temporal forcing and stochasticity on pathogen invasion dynamics (original) (raw)

The role of initial inoculum on epidemic dynamics

Journal of Theoretical Biology, 2006

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.

Eight challenges for stochastic epidemic models involving global transmission

Epidemics, 2014

The most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. Simple as they are, there are still several open problems for such models. For example, when will such an epidemic go extinct and with what probability (questions depending on the population being fixed, changing or growing)? How can a model be defined explaining the sometimes observed scenario of frequent mid-sized epidemic outbreaks? How can evolution of the infectious agent transmission rates be modelled and fitted to data in a robust way?

Realistic Distributions of Infectious Periods in Epidemic Models: Changing Patterns of Persistence and Dynamics

Theoretical Population Biology, 2001

Most mathematical models used to study the epidemiology of childhood viral diseases, such as measles, describe the period of infectiousness by an exponential distribution. The effects of including more realistic descriptions of the infectious period within SIR (susceptible infec-tious recovered) models are studied. Less dispersed distributions are seen to have two important epidemiological consequences. First, less stable behaviour is seen within the model: incidence patterns become more complex. Second, disease persistence is diminished: in models with a finite population, the minimum population size needed to allow disease persistence increases. The assumption made concerning the infectious period distribution is of a kind routinely made in the formulation of mathematical models in population biology. Since it has a major effect on the central issues of population persistence and dynamics, the results of this study have broad implications for mathematical modellers of a wide range of biological systems.

Impact of heterogeneity on the dynamics of an SEIR epidemic model

Mathematical Biosciences and Engineering, 2012

An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number R 0 gives a sharp threshold. If R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If R 0 > 1, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.

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.

Characterizing the reproduction number of epidemics with early subexponential growth dynamics

Journal of the Royal Society, Interface, 2016

Early estimates of the transmission potential of emerging and re-emerging infections are increasingly used to inform public health authorities on the level of risk posed by outbreaks. Existing methods to estimate the reproduction number generally assume exponential growth in case incidence in the first few disease generations, before susceptible depletion sets in. In reality, outbreaks can display subexponential (i.e. polynomial) growth in the first few disease generations, owing to clustering in contact patterns, spatial effects, inhomogeneous mixing, reactive behaviour changes or other mechanisms. Here, we introduce the generalized growth model to characterize the early growth profile of outbreaks and estimate the effective reproduction number, with no need for explicit assumptions about the shape of epidemic growth. We demonstrate this phenomenological approach using analytical results and simulations from mechanistic models, and provide validation against a range of empirical di...

Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times

2012

Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, will also be discussed.

S-I-R Epidemic Models with Directed Diffusion

. A generalization of the Gurtin--MacCamy model for an S--I--R epidemic is described. The new model, which includes diffusion away from overcrowded regions, retains many of the interesting qualitative features of previous models. In addition, the global--in--time existence of solutions is proved for a special case. Qualitative properties of the solution are discussed and illustrated with a numerical example. 1. Introduction. Mathematical models for epidemics have been evolving in complexity and realism during the past 60 years. From the simple, unstructured model of Kermack and McKendrick of 1927 [7], many models incorporating, for example, age structure, time delays corresponding to incubation periods, spatial diffusion, and variable infectivity have been proposed (see, e.g., [2], [3], [4], [6]). The effects of dispersion on the spatial distribution of the subpopulations are of primary interest in this work. In addition to allowing for more realistic descriptions of the observed ph...

Multiple epidemic waves as the outcome of stochastic SIR epidemics with behavioral responses: a hybrid modeling approach

Nonlinear Dynamics, 2022

In the behavioral epidemiology (BE) of infectious diseases, little theoretical effort seems to have been devoted to understand the possible effects of individuals' behavioral responses during an epidemic outbreak in small populations. To fill this gap, here we first build general, behavior implicit, SIR epidemic models including behavioral responses and set them within the framework of nonlinear feedback control theory. Second, we provide a thorough investigation of the effects of different types of agents' behavioral responses for the dynamics of hybrid stochastic SIR outbreak models. In the proposed model, the stochas-A. d'Onofrio is no more affiliated at IPRI since 15 June 2020.

Biomathematical analysis and extension of the new class of epidemic models proposed by Satsuma et al. (2004)

Applied Mathematics and Computation, 2005

The aim of this paper is to discuss the new class of epidemic models proposed by Satsuma et al., which are characterized by incidence rates which are nonlinearly dependent on the number of susceptibles as follows: infection rate (S, I) = g(S)I. By adding the biologically plausible constraint g 0 (S) > 0, we study the SIR and the SEIR models with vital dynamics with such infection rate, and results are done on the global asymptotic stability of the disease free and of the endemic equilibria, similarly to the ones of the classical models, also in presence of traditional and pulse vaccination strategies. Relaxing the constraint g 0 (S) > 0, we show that the epidemic system may exhibit multiple endemic equilibria.