Epidemic modelling with compartmental models using R

[After reading through this module you should have an intuitive understanding of how infectious disease spreads in the population, and how that process can be described using a compartmental model with flow between the compartments.  You should be able to write down the differential equations of a simple disease model, and you will learn in this module how to numerically solve those differential equations in R to obtain the model estimate of the epidemic curve]

An excellent reference book with background material related to these lectures is Mathematical Epidemiology by Brauer et al. 

Contents:

Introduction

Models of disease spread can yield insights into the mechanisms and dynamics most important to the spread of disease (especially when the models are compared to epidemic data).  With this improved understanding, more effective disease intervention strategies can potentially be developed. Sometimes disease models are also used to forecast the course of an epidemic, and doing exactly that for the 2009 pandemic was my introduction to the field of computational epidemiology.

There are lots of different ways to model epidemics, and there are several modules on this site on the topic, but let’s begin with one of the simplest epidemic models for an infectious disease like influenza: the Susceptible, Infected, Recovered (SIR) model.

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Compartmental modelling without calculus

In another module on this site I describe how an epidemic for certain kinds of infectious diseases (like influenza) can be modelled with a simple Susceptible, Infectious, Recovered (SIR) model. Readers who have not yet been exposed to calculus (such as junior or senior high school students) may have been daunted by the system of differential equations shown in that post.  However, with only a small amount of programming experience in R, students without calculus can still easily model epidemics, or any other system that can be described with a compartmental model.  In this post I will show how that is done.
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SIR infectious disease model with age classes

[After reading through this module, students should have an understanding of contact dynamics in a population with age structure (eg; kids and adults). You should understand how population age structure can affect the spread of infectious disease. You should be able to write down the differential equations of a simple SIR disease model with age structure, and you will learn in this module how to solve those differential equations in R to obtain the model estimate of the epidemic curve]

Contents:

Introduction

In a previous module I discussed epidemic modelling with a simple Susceptible, Infected, Recovered (SIR) compartmental model.  The model presented had only a single age class (ie; it was homogenous with respect to age).  But in reality, when we consider disease transmission, age likely does matter because kids usually make more contacts during the day than adults. The differences in contact patterns between age groups can have quite a profound impact on the model estimate of the epidemic curve, and also have implications for development of optimal disease intervention strategies (like age-targeted vaccination, social distancing, or closing schools).
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SIR modelling of influenza with a periodic transmission rate

[After going through this module, students will be familiar with time-dependent transmission rates in a compartmental SIR model, will have explored some of  the complex dynamics that can be created when the transmission is not constant, and will understand applications to the modelling of influenza pandemics.]

Contents:

 

Introduction

Influenza is a seasonal disease in temperate climates, usually peaking in the winter.  This implies that the transmission of influenza is greater in the winter (whether this is due to increased crowding and higher contact rates in winter, and/or due to higher transmissibility of the virus due to favorable environmental conditions in the winter is still being discussed in the literature).  What is very interesting about influenza is that sometimes summer epidemic waves can be seen with pandemic strains (followed by a larger autumn wave).  An SIR model with a constant transmission rate simply cannot replicate the annual dual wave nature of an influenza pandemic.

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ASU AML 610 Module VIII: Fitting to initial exponential rise of epidemic curves

In this module students will compare the performance of several fitting methods (Least squares, Pearson chi-squared, and likelihood fitting methods) in estimating the rate of exponential rise in initial epidemic incidence data.  Students will learn about the properties of good estimators (bias and efficiency).

A good reference source for this material is Statistical Data Analysis, by G.Cowan

Another good reference source (in a very condensed format) for statistical data analysis methods can be found here.

Contents:
Introduction
Properties of good estimators
Generating simulated exponential rise data
Estimation of the rate of exponential rise: Least Squares
Estimation of the rate of exponential rise: Pearson chi-squared
The Poisson maximum likelihood method
Estimation of parameter confidence intervals: any maximum likelihood method
Estimation of the rate of exponential rise: Poisson maximum likelihood method
Testing for over- or under-dispersion.
Correcting for over- or under-dispersion
Better method for determination of parameter estimates and their covariance when using the Pearson chi-squared method

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