Modeling and Dynamics of Infectious Diseases

The following sections are included:

• Preface

• Contents

The goal of this synthetic paper is to introduce a part of research directions on epidemic dynamics investigated by our group and our main results during the past several years. Before this, some basic knowledge on epidemic dynamics will be introduced which may be helpful to those readers who are not familiar with the mathematical modeling on Epidemiology.

Shortly after the 2002-03 Severe Acute Respiratory Syndrome (SARS) outbreak, a Canadian team on modeling communicable diseases was established and an adventure of interdisciplinary research involving close interaction and collaboration between modelers, epidemiologists, and public health policy makers started. This article provides a review of the history of this team, its collective efforts and long-term goal in modeling emerging/reemmerging communicable diseases of critical importance to Canada and the international community.

Metapopulation models consist of graphs, with systems of differential equations in each vertex. This modeling paradigm is appropriate for the description of the spatio-temporal spread of infectious diseases. In this paper, I present the setting of these models, and some of the mathematical techniques that can be used to study them. I conclude with a brief review of some models using this approach.

The following section are included:

• Introduction

• The network corresponding to a disease outbreak

• Transmissibility

• Some examples of contact networks

• Conclusions

• References

I provide a brief introduction to two complimentary mathematical approaches for incorporating evolution into epidemiological models. These are referred to as the invasion-analysis technique and the Price-equation technique.

Recently many mathematical models have been used to study the effectiveness of quarantine and isolation as control measures for the spread of infectious diseases. Most of the deterministic models have focused on the use of ordinary differential equation (ODE) models with the assumption of exponentially distributed disease stages. In this paper we demonstrate that some of these models may generate misleading predictions. We formulate a general integral equation model which assumes an arbitrarily distributed disease stage for both the latent and the infectious stages, and show that it reduces to the commonly used SEIR model (ODE) with quarantine and isolation when the stage distributions are exponential. The general model reduces to another ODE model when the disease stages are assumed to have a gamma distribution. The control reproductive number for each model is calculated. The effect of control strategies of these models is compared in terms of both and the final epidemic size. We demonstrate that the two ODE models can produce conflict pre-dictions regarding the effectiveness of disease control strategies via quarantine and isolation.

Congenital rubella syndrome (CRS) comprises a panoply of lesions, some fatal and others diagnosed too long after birth to be readily associated with first trimester maternal infections. Consequently, this syndrome is under-ascertained even by reviewing medical records for compatible conditions and confirming them serologically. Mathematical modeling provides an alternative to such studies, while also permitting health authorities to estimate the burden of this disease relatively easily and to evaluate means by which it might be mitigated. Our estimates of CRS from disease or serological surveillance and demographic information compare favorably to those from retrospective chart reviews in children's hospitals and clinics specializing on afflictions that may result from infection in utero. The CRS burden could be mitigated either by reducing susceptibility to infection on exposure among women of childbearing age (WCBA) or their risk of exposure. The first approach might involve vaccinating girls attending school or mothers delivering in hospital, neither of which however is universal. The second necessarily involves childhood vaccination, via routine age-appropriate doses, mass campaigns, or both. But if sufficient coverage is not sustained, the immunization of some children only temporarily protects others, whose susceptibility may persist into adulthood. Unless vaccinated, their own children may be infected while playing with neighbors or attending school, and infect them. The females among these susceptible parents may be pregnant. The potential of childhood vaccination to exacerbate the future burden of CRS has been known since the early 1980s, but casually dismissed for lack of evidence that could not be apparent for some time. However, examples have recently begun appearing in the literature. Romania exemplifies a newly-recognized hazard: rubella occurs annually, but major outbreaks typically occur every 5 years. The period of these multiannual cycles doubled as falling birth rates decreased the supply of susceptible children, increasing the mean age of infection, and concomitantly CRS, a phenomenon that may be anticipated in China due to efforts to curb population growth (Gao and Hethcote 2006). We describe models with which possible strategies for mitigating the burden of CRS have been evaluated in these and other developing countries. Given estimated costs of feasible tactics, policymakers could consider whether to devote their scarce resources to this or other health problems. We encourage monitoring of disease or serology to ensure policy objectives are attained, even if modeling aided in vaccination program design.

The question whether a vertically and a horizontally transmitted parasite strain can coexist under complete cross-protection is investigated in a host-parasite model with susceptibles and infectives only. It is shown that coexistence is possible even if the vertically transmitted strain would go extinct on its own provided that it is considerably less virulent than the horizontally transmitted strain. While the vertical transmitted strain is without benefit to the host as such, it protects the host against the more harmful horizontally transmitted strain. The coexistence is shown in the form of uniform strong persistence of the host and both parasite strains.

We propose to use Richards model, a logistic-type ordinary differential equation, to fit the daily cumulative case data from the 2003 severe acute respiratory syndrome outbreaks in Taiwan, Beijing, Hong Kong, Toronto, and Singapore. This model enabled us to estimate turning points and case numbers during each phases of an outbreak. The 3 estimated turning points are March 25, April 27, and May 24. Our modeling procedure provides insights into ongoing outbreaks that may facilitate real-time public health responses when faced with infectious disease outbreak in the future.

The following section are included;

• Introduction

• A general disease transmission model

• The basic reproduction number

• The final size of a simple epidemic

• A final size inequality for a general model

• Examples
  • The constant rate SIR epidemic model
  • Simple compartmental models
  • The SLIAR model for influenza
  • A discrete delay

• Further Reading

• References

A small reservoir is introduced into the classical epidemic models of SIR type. Since some of the models are structurally unstable (in the sense of dynamical systems theory), the dynamics may change completely and maximal prevalence may increase drastically. The effects of the reservoir on the time course of the disease and on endemic states are investigated by analytical methods and discussed in public health terms.

The question of the uniqueness and global stability of endemic equilibria of multigroup epidemic models of SEIR type is revisited. After a brief review of the literature, we prove that, for a general class of n-group models with bilinear incidence, the endemic equilibrium is unique and globally stable when the basic reproduction number is greater than 1. Our proof utilizes a global Lyapunov function well-known in the literature. The key to our proof is a complete description of the complex patterns exhibited in the derivatives if the Lyapunov function uses results from graph theory.

This article overviews mathematical approaches for analysis of epidemic models with time delays. We start by introducing the methodology of modeling epidemic diseases with time delays and illustrating the D-subdivision method for characteristic equations. Then we show the application of the Liapunov functional method. Lastly, we present the persistence techniques to determine conditions under which diseases are permanent in the population.

The model described here is developed for simulating an influenza pandemic. It intends to explain the manner in which the pandemic develops in a specific community. It shows how the longitudinal cases occurring pattern is built from inception and how this pattern is affected by various sizes of stockpiles of antiviral agents combining some preventing strategies. It assumes that potential inter-household and household-school contacts play very important role in the disease transmission. It uses statistical data about characteristics of households in the community and the parameters of disease transmission to generate repeated random trials of possible outcomes following the introduction of infective individuals into that community, then to track the statistically determined pathway of the infection and average the outcome results. It aims to describe the dynamics of pandemic itself, then the affection of antiviral treatment based on the available sizes of stockpile of the drugs can be obtained.

We carry out a preliminary mathematical modelling study of the West Nile Virus in the Peel region, Ontario Canada. The surveillance data from this region is used to show that the non-crow family birds which are susceptible to West Nile Virus are one of the key factors responsible for the endemic of the virus in the region. The density of dead crow is no longer an accurate indicator for the virus after the virus has sustained in the region, while the density of mosquitoes can give us better quantification of the risk level of the West Nile virus.