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<!DOCTYPE html>
<html lang="en"><head>
<script src="slides_files/libs/clipboard/clipboard.min.js"></script>
<script src="slides_files/libs/quarto-html/tabby.min.js"></script>
<script src="slides_files/libs/quarto-html/popper.min.js"></script>
<script src="slides_files/libs/quarto-html/tippy.umd.min.js"></script>
<link href="slides_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="slides_files/libs/quarto-html/light-border.css" rel="stylesheet">
<link href="slides_files/libs/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="slides_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
<script src="slides_files/libs/quarto-contrib/glightbox/glightbox.min.js"></script>
<link href="slides_files/libs/quarto-contrib/glightbox/glightbox.min.css" rel="stylesheet">
<link href="slides_files/libs/quarto-contrib/glightbox/lightbox.css" rel="stylesheet"><meta charset="utf-8">
<meta name="generator" content="quarto-1.5.43">
<meta name="author" content="James Mba Azam, PhD">
<meta name="dcterms.date" content="2024-10-12">
<title>Introduction to Infectious Disease Modelling</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="slides_files/libs/revealjs/dist/reset.css">
<link rel="stylesheet" href="slides_files/libs/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for citations */
div.csl-bib-body { }
div.csl-entry {
clear: both;
margin-bottom: 0em;
}
.hanging-indent div.csl-entry {
margin-left:2em;
text-indent:-2em;
}
div.csl-left-margin {
min-width:2em;
float:left;
}
div.csl-right-inline {
margin-left:2em;
padding-left:1em;
}
div.csl-indent {
margin-left: 2em;
} </style>
<link rel="stylesheet" href="slides_files/libs/revealjs/dist/theme/quarto.css">
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<link href="slides_files/libs/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
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padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
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border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
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flex-grow: 1;
}
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font-size: 1rem;
font-weight: 400;
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font-size: 0.9rem;
font-weight: 400;
}
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margin-top: 0.5em;
margin-bottom: 0.5em;
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margin-top: 0;
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margin-top: 0.7em;
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border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
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border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
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padding-left: 0.5em;
padding-right: 0.5em;
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height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
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width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
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display: flex
}
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padding-right: .5rem;
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margin-bottom: 0;
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margin-top: .5rem;
padding-right: .5rem;
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}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
}
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div.callout-important.callout-style-default .callout-title {
background-color: #f7dddc
}
div.callout-warning {
border-left-color: #f0ad4e !important;
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<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Introduction to Infectious Disease Modelling</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
James Mba Azam, PhD <a href="https://orcid.org/0000-0001-5782-7330" class="quarto-title-author-orcid"> <img src="data:image/png;base64,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"></a>
</div>
<div class="quarto-title-author-email">
<a href="mailto:james.azam@lshtm.ac.uk">james.azam@lshtm.ac.uk</a>
</div>
<p class="quarto-title-affiliation">
Epiverse-TRACE Initiative, London School of Hygiene and Tropical Medicine, UK
</p>
</div>
</div>
<p class="date">2024-10-12</p>
</section><section id="TOC">
<nav role="doc-toc">
<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#/an-overview-of-infectious-diseases" id="/toc-an-overview-of-infectious-diseases">An overview of infectious diseases</a></li>
<li><a href="#/infectious-disease-models" id="/toc-infectious-disease-models">Infectious disease models</a></li>
<li><a href="#/introduction-to-compartmental-models" id="/toc-introduction-to-compartmental-models">Introduction to compartmental models</a></li>
<li><a href="#/modelling-epidemic-control" id="/toc-modelling-epidemic-control">Modelling epidemic control</a></li>
<li><a href="#/brief-notes-on-modelling-host-heterogeneity" id="/toc-brief-notes-on-modelling-host-heterogeneity">Brief notes on modelling host heterogeneity</a></li>
<li><a href="#/final-remarks" id="/toc-final-remarks">Final Remarks</a></li>
<li><a href="#/list-of-resources" id="/toc-list-of-resources">List of Resources</a></li>
<li><a href="#/references" id="/toc-references">References</a></li>
</ul>
</nav>
</section>
<section>
<section id="an-overview-of-infectious-diseases" class="title-slide slide level1 center">
<h1>An overview of infectious diseases</h1>
</section>
<section id="what-are-infectious-diseases" class="slide level2">
<h2>What are infectious diseases</h2>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/disease_definition.png" class="lightbox" data-gallery="quarto-lightbox-gallery-1"><img data-src="images/disease_definition.png" class="quarto-figure quarto-figure-center" data-fig-cap="Definition of a disease according to the Merriam Webster dictionary"></a></p>
</figure>
</div>
</section>
<section class="slide level2">
<div class="columns">
<div class="column" style="width:45%;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/rotavirus.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-2" title="A 3D graphical representation of Rotavirus virions."><img data-src="images/rotavirus.jpeg" alt="A 3D graphical representation of Rotavirus virions."></a></p>
<figcaption>A 3D graphical representation of Rotavirus virions.</figcaption>
</figure>
</div>
</div><div class="column" style="width:55%;">
<p>Diseases can be classified according to:</p>
<ul>
<li><span style="color:tomato">Cause</span> (e.g., infectious, non-infectious)</li>
<li><span style="color:tomato">Duration</span> (e.g., acute, chronic)</li>
<li><span style="color:tomato">Mode of transmission</span> (direct or indirect)</li>
<li><span style="color:tomato">Impact</span> on the host (e.g., fatal, non-fatal)</li>
</ul>
</div>
</div>
</section>
<section class="slide level2">
<p>Note that these classifications are not mutually exclusive. Hence, a disease can be classified under more than one category at a time.</p>
</section>
<section id="sec-control-measures" class="slide level2">
<h2>How are infectious diseases controlled?</h2>
<ul>
<li>In general, infectious disease control aims to reduce disease transmission.</li>
<li>The type of control used depends on the disease and its characteristics.</li>
<li>Broadly, there are two main types of control measures:
<ul>
<li>Pharmaceutical interventions (PIs)</li>
<li>Non-pharmaceutical interventions (NPIs)</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<h3 id="pharmaceutical-interventions-pis">Pharmaceutical interventions (PIs)</h3>
<ul>
<li><p>Pharmaceutical Interventions are medical interventions that target the pathogen or the host.</p></li>
<li><p>Examples:</p>
<ul>
<li>Vaccines,</li>
<li>Antiviral drugs, and</li>
<li>Antibiotics.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<h4 id="sec-vaccination">Vaccination</h4>
<div class="columns">
<div class="column" style="width:40%;">
<p><a href="images/vaccine.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-3"><img data-src="images/vaccine.jpeg"></a></p>
</div><div class="column" style="width:60%;">
<ul>
<li>Activates the host’s immune system to produce antibodies against the pathogen.</li>
<li>Generally applied to reduce the risk of infection and disease.</li>
<li>The most effective way to prevent infectious diseases.</li>
</ul>
</div>
</div>
</section>
<section class="slide level2">
<h5 id="challenges-with-vaccination">Challenges with vaccination</h5>
<ul>
<li>Take time to develop for new pathogens</li>
<li>Never 100% effective and limited duration of protection</li>
<li>Adverse side effects</li>
<li>Some individuals cannot be vaccinated or refuce vaccination</li>
<li>Some pathogens mutate rapidly (e.g., influenza virus)</li>
<li>Logistical challenges (e.g., cold chain requirements)</li>
</ul>
</section>
<section class="slide level2">
<h3 id="sec-npi">Non-pharmaceutical interventions (NPIs)</h3>
<div class="columns">
<div class="column" style="width:50%;">
<p><a href="images/mask.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-4"><img data-src="images/mask.jpeg" style="width:70.0%"></a></p>
<p><a href="images/screening.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-5"><img data-src="images/screening.jpeg" style="width:70.0%"></a></p>
</div><div class="column" style="width:50%;">
<ul>
<li><p>Non-pharmaceutical interventions are measures that do not involve medical interventions.</p></li>
<li><p>Examples:</p>
<ul>
<li>Quarantine,</li>
<li>Physical/social distancing, and</li>
<li>Mask-wearing.</li>
</ul></li>
</ul>
</div>
</div>
</section>
<section class="slide level2">
<h4 id="quarantine">Quarantine</h4>
<ul>
<li><em>Isolation of individuals who may have been exposed to a contagious disease</em>.</li>
<li>Advantage is that it’s simple and its effectiveness does not depend on the disease.</li>
<li>Disadvantages include:
<ul>
<li>Infringement on individual rights</li>
<li>Can be difficult to enforce</li>
<li>Can be costly</li>
<li>Can lead to social stigma</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<h4 id="contact-tracing">Contact tracing</h4>
<ul>
<li>Contact tracing is used to identify exposed individuals, i.e., individuals who might been in contact with an infected/infectious individual.</li>
<li>It involves identifying, assessing, and managing people who have been exposed to a contagious disease to prevent further transmission.</li>
<li>It is a critical component of infectious disease surveillance and is often used in combination with other control measures.</li>
</ul>
</section>
<section class="slide level2">
<div class="callout callout-caution no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Discussion</strong></p>
</div>
<div class="callout-content">
<ul>
<li>How can we quantify the impact of a control measure?</li>
</ul>
</div>
</div>
</div>
</section></section>
<section>
<section id="infectious-disease-models" class="title-slide slide level1 center">
<h1>Infectious disease models</h1>
</section>
<section id="what-are-infectious-disease-models" class="slide level2">
<h2>What are infectious disease models?</h2>
<ul>
<li><span style="color:tomato;"><em>Models</em></span> generally refer to conceptual representations of an object or system.</li>
<li><span style="color:tomato;"><em>Mathematical models</em></span> use mathematics to describe the system. For example, the famous <span class="math inline">\(E = mc^2\)</span> is a mathematical model that describes the relationship between mass and energy.</li>
<li><span style="color:tomato;"><em>Infectious disease models</em></span> use mathematics/statistics to represent dynamics/spread of infectious diseases.</li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>Mathematical models can be used to link the biological process of disease transmission and the emergent dynamics of infection at the population level.</li>
<li>Models require making some assumptions and abstractions.</li>
</ul>
</section>
<section class="slide level2">
<div class="columns">
<div class="column" style="width:60%;">
<ul>
<li>By definition, <span style="color:tomato">“all models are wrong, but some are useful”</span> <span class="citation" data-cites="Box1979">(<a href="#/references" role="doc-biblioref" onclick="">Box 1979</a>)</span>.
<ul>
<li>Good enough models are those that capture the <span style="color:tomato">essential features</span> of the system being studied.</li>
</ul></li>
</ul>
</div><div class="column" style="width:40%;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/GeorgeEPBox.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-6" title="George Box"><img data-src="images/GeorgeEPBox.jpeg" alt="George Box"></a></p>
<figcaption>George Box</figcaption>
</figure>
</div>
</div>
</div>
</section>
<section class="slide level2">
<ul>
<li>What makes models “wrong” by definition?:
<ul>
<li>Simplifications of reality; not capturing all the complexities of the system being studied.</li>
</ul></li>
</ul>
</section>
<section id="factors-that-influence-model-formulationchoice" class="slide level2">
<h2>Factors that influence model formulation/choice</h2>
<ul>
<li><span style="color:tomato">Accuracy</span>: how well does the model to reproduce observed data and predict future outcomes?</li>
<li><span style="color:tomato">Transparency</span>: is it easy to understand and interpret the model and its outputs? (This is affected by the model’s complexity)</li>
<li><span style="color:tomato">Flexibility</span>: the ability of the model to be adapted to different scenarios.</li>
</ul>
<aside class="notes">
<p>These will be touched on in the scenario modelling lectures.</p>
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</section>
<section id="what-are-models-used-for" class="slide level2">
<h2>What are models used for?</h2>
<ul>
<li>Generally, models can be used to predict and understand/explain the dynamics of infectious diseases.</li>
</ul>
<div class="callout callout-caution no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Discussion</strong></p>
</div>
<div class="callout-content">
<p>How are these two uses impacted by accuracy, transparency, and flexibility?</p>
</div>
</div>
</div>
</section>
<section class="slide level2">
<h3 id="prediction-of-the-future-course">Prediction of the future course</h3>
<div class="columns">
<div class="column" style="width:40%;">
<ul>
<li>Must be <span style="color:tomato;">accurate</span>.</li>
<li>“But the estimate proved to be off. Way, way off. Like, 65 times worse than what ended up happening.”</li>
</ul>
</div><div class="column" style="width:60%;">
<p><a href="images/wrong_ebola_deaths_estimate.png" class="lightbox" data-gallery="quarto-lightbox-gallery-7"><img data-src="images/wrong_ebola_deaths_estimate.png" data-fig-cap="[CDC’s top modeler courts controversy with disease estimate](https://apnews.com/domestic-news-domestic-news-fbb4fc8921d54201a1c5ca91e5b601f5)"></a></p>
</div>
</div>
</section>
<section class="slide level2">
<!-- \< Insert examples of models that have made accurate predictions of the future course of an outbreak \> -->
<!-- - See <Justin Lessler, Derek A. T. Cummings, Mechanistic Models of Infectious Disease and Their Impact on Public Health, American Journal of Epidemiology, Volume 183, Issue 5, 1 March 2016, Pages 415–422, https://doi.org/10.1093/aje/kww021> -->
<!-- ------------------------------------------------------------------------ -->
<h3 id="understanding-or-explaining-disease-dynamics">Understanding or explaining disease dynamics</h3>
<ul>
<li>Models can be used to understand how a disease spreads and how its spread can be controlled.</li>
<li>The insights gained from models can be used to:
<ul>
<li>inform public health policy and interventions.</li>
<li>design interventions to control the spread of the disease, for example, randomised controlled trials.</li>
<li>collect new data.</li>
<li>build predictive models.</li>
</ul></li>
</ul>
<!-- \< Insert examples of models that explain the spread and dynamics of infectious diseases \> -->
<!-- - See <Justin Lessler, Derek A. T. Cummings, Mechanistic Models of Infectious Disease and Their Impact on Public Health, American Journal of Epidemiology, Volume 183, Issue 5, 1 March 2016, Pages 415–422, https://doi.org/10.1093/aje/kww021> -->
<!-- ------------------------------------------------------------------------ -->
<!-- \< Insert examples of models that evaluate the impact of interventions and determine the next course of action \> -->
<!-- ------------------------------------------------------------------------ -->
</section>
<section id="limitations-of-infectious-disease-models" class="slide level2">
<h2>Limitations of infectious disease models</h2>
<ul>
<li>Host behaviour is often difficult to predict.</li>
<li>The pathogen often has unknown characteristics or known characteristics that are difficult to model.</li>
<li>Data is often not available or is of poor quality.</li>
</ul>
</section>
<section id="summary" class="slide level2">
<h2>Summary</h2>
<ul>
<li>Models:
<ul>
<li><span style="color:tomato">Simplifications of reality</span> and do not capture all the <span style="color:tomato">complexities</span> of the system being studied.</li>
<li>Only as good as the <span style="color:tomato">data</span> used to parameterize them.</li>
<li>Sensitive to the <span style="color:tomato">assumptions</span> made during their formulation.</li>
<li><span style="color:tomato">Computationally expensive</span> and require a lot of data to run.</li>
<li><span style="color:tomato">Difficult to interpret</span> and communicate to non-experts.</li>
</ul></li>
</ul>
</section></section>
<section>
<section id="introduction-to-compartmental-models" class="title-slide slide level1 center">
<h1>Introduction to compartmental models</h1>
</section>
<section id="what-are-compartmental-models" class="slide level2">
<h2>What are compartmental models?</h2>
<ul>
<li>Compartmental models:
<ul>
<li>divide populations into compartments (or groups) based on the individual’s infection status and track them through time <span class="citation" data-cites="Blackwood2018a">(<a href="#/references" role="doc-biblioref" onclick="">Blackwood and Childs 2018</a>)</span>.</li>
<li>are mechanistic, meaning they describe processes such the interaction between hosts, biological processes of pathogen, host immune response, and so forth.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<p>Compartmental models are different from statistical models, which are used to describe the relationship between variables.</p>
</section>
<section class="slide level2">
<ul>
<li>Individuals in a compartment:
<ul>
<li>are assumed to have the same features (disease state, age, location, etc)</li>
<li>can only be in one compartment at a time.</li>
<li>move between compartments based on defined transition rates.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<div class="columns">
<div class="column" style="width:60%;">
<ul>
<li>Common compartments:
<ul>
<li>Susceptible (S) - hosts are not infected but can be infected</li>
<li>Infected (I) - hosts are infected (and can infect others)</li>
<li>Removed (R) - hosts are no longer infected and cannot be re-infected</li>
</ul></li>
</ul>
</div><div class="column" style="width:40%;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/infection_timeline.png" class="lightbox" data-gallery="quarto-lightbox-gallery-8" title="Infection timeline illustrating how a pathogen in a host interacts with the host’s immune system (Source: Modelling Infectious Diseases of Humans and Animals)"><img data-src="images/infection_timeline.png" alt="Infection timeline illustrating how a pathogen in a host interacts with the host’s immune system (Source: Modelling Infectious Diseases of Humans and Animals)"></a></p>
<figcaption>Infection timeline illustrating how a pathogen in a host interacts with the host’s immune system (Source: Modelling Infectious Diseases of Humans and Animals)</figcaption>
</figure>
</div>
</div>
</div>
</section>
<section class="slide level2">
<ul>
<li><p>Other compartments can be added to the model to account for important events or processes (e.g., exposed, recovered, vaccinated, etc.)</p></li>
<li><p>It is, however, important to keep the model simple, less computationally intensive, and interpretable.</p></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>Compartmental models either have <em>discrete</em> or <em>continuous</em> time scales:
<ul>
<li><span style="color:tomato;">Discrete time scales</span>: time is divided into discrete intervals (e.g., days, weeks, months).</li>
<li><span style="color:tomato;">Continuous time scales</span>: time is continuous and the model is described using differential equations.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>Compartmental models can be <em>deterministic</em> or <em>stochastic</em>:
<ul>
<li><span style="color:tomato;">Deterministic</span> models always return the same output for the same input.</li>
<li><span style="color:tomato;">Stochastic</span> models account for randomness in the system and model output always varies. Hence, they are often run multiple times to get an average output.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>The choice of model type depends on:
<ul>
<li>the research <span style="color:tomato">question</span>,</li>
<li><span style="color:tomato">data</span> availability,</li>
<li><span style="color:tomato">computational resources</span>,</li>
<li>modeller <span style="color:tomato">skillset</span>.</li>
</ul></li>
<li>In this introduction, we will focus on <span style="color:tomato;">deterministic compartmental models with continuous time scales</span>.</li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>Now, back to the models, we are going to consider infections that either confer immunity after recovery or not.</li>
<li>The simplest compartmental models for capturing this is the SIR model.</li>
</ul>
</section>
<section id="the-susceptible-infected-recovered-sir-model" class="slide level2">
<h2>The Susceptible-Infected-Recovered (SIR) model</h2>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="./images/model_diagrams/model_diagrams.001.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-9"><img data-src="./images/model_diagrams/model_diagrams.001.jpeg" class="quarto-figure quarto-figure-center" style="width:40.0%"></a></p>
</figure>
</div>
<p>This model groups individuals into three <em>disease states</em>:</p>
<ul>
<li><p><span style="color:green;">Susceptible (S)</span>: not infected but can be.</p></li>
<li><p><span style="color:tomato;">Infected (I)</span>: infected & infectious.</p></li>
<li><p><span style="color:blue;">Recovered/removed (R)</span>: recovered & immune.</p></li>
</ul>
</section>
<section class="slide level2">
<h3 id="how-do-individuals-move-between-compartments">How do individuals move between compartments?</h3>
<h4 id="process-1-transmission">Process 1: Transmission</h4>
<p><a href="images/model_diagrams/model_diagrams.002.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-10"><img data-src="images/model_diagrams/model_diagrams.002.jpeg" style="width:70.0%"></a></p>
<p><span style="color:red;">What drives transmission?</span></p>
</section>
<section class="slide level2">
<ul>
<li>Transmission is driven by several factors, including:
<ul>
<li>Disease prevalence, <span style="color:blue;"><span class="math inline">\(I\)</span></span>, i.e., number of infected individuals at the time.</li>
<li>The number of contacts, <span style="color:blue;"><span class="math inline">\(C\)</span></span>, susceptible individuals have with infected individual.</li>
<li>The probability, <span style="color:blue;"><span class="math inline">\(p\)</span></span>, a susceptible individual will become infected when they contact an infected individual.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<p><a href="images/model_diagrams/model_diagrams.002.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-11"><img data-src="images/model_diagrams/model_diagrams.002.jpeg" style="width:70.0%"></a></p>
<p>The transmission term is often defined through the <span style="color:tomato;">force of infection (FOI), <span class="math inline">\(\lambda\)</span></span>.</p>
</section>
<section class="slide level2">
<h3 id="a-tour-of-the-force-of-infection-foi">A tour of the force of infection (FOI)</h3>
<ul>
<li>FOI, <span class="math inline">\(\lambda\)</span>, is the <span style="color:tomato;">per capita rate</span> at which susceptible individuals become infected.</li>
</ul>
<div class="callout callout-note no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Note</strong></p>
</div>
<div class="callout-content">
<p>“Per capita” means the rate of an event occurring per individual in the population per unit of time.</p>
</div>
</div>
</div>
<ul>
<li>Given the rate per individual per time, FOI, the rate at which new infecteds are generated is given by <span style="color:blue;"><span class="math inline">\(\lambda \times S\)</span></span>, where <span class="math inline">\(S\)</span> is the number of susceptible individuals.</li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>The force of infection is made up of the probabilities/rates that:
<ul>
<li>contacts happen, <span style="color:blue"><span class="math inline">\(c\)</span></span>,</li>
<li>a given contact is with an infected individual, <span style="color:blue"><span class="math inline">\(p\)</span></span>, and</li>
<li>a contact results in successful transmission, <span style="color:blue"><span class="math inline">\(v\)</span></span>.</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>The FOI can be formulated in two ways, depending on how the contact rate is expected to change with the population size:
<ul>
<li>Frequency-dependent/mass action transmission</li>
<li>Density-dependent transmission</li>
</ul></li>
</ul>
</section>
<section class="slide level2">
<h4 id="frequency-dependentmass-action-transmission">Frequency-dependent/mass action transmission</h4>
<p>The rate of contact between individuals is constant irrespective of the population density, <span style="color:blue"><span class="math inline">\(\dfrac{N}{A}\)</span></span>, where <span style="color:blue"><span class="math inline">\(N\)</span></span> is the population size and <span style="color:blue"><span class="math inline">\(A\)</span></span> is the area occupied by the population.</p>
</section>
<section class="slide level2">
<ul>
<li><p>Recall that transmission also depends on the probability of contact with an infected host, <span style="color:blue"><span class="math inline">\(p\)</span></span>, which is assumed to be <span style="color:blue"><span class="math inline">\(\dfrac{I}{N}\)</span></span>.</p></li>
<li><p>Hence, the frequency-dependent mass action is given by <span style="color:blue"><span class="math inline">\(\lambda = \beta \times \dfrac{I}{N}\)</span></span>, where <span style="color:blue"><span class="math inline">\(\beta\)</span></span> is the transmission rate.</p></li>
</ul>
</section>
<section class="slide level2">
<div class="callout callout-caution no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Question</strong></p>
</div>
<div class="callout-content">
<p>Why does the frequency-dependent transmission contain <span style="color:blue"><span class="math inline">\(\dfrac{1}{N}\)</span></span> if it does not depend on the population density?</p>
</div>
</div>
</div>
</section>
<section class="slide level2">
<ul>
<li>Assume that the rate of new infections is given by <span style="color:blue"><span class="math inline">\(\dfrac{dI}{dt} = S \times c \times p \times v\)</span></span> where <span style="color:blue"><span class="math inline">\(S\)</span></span> is the number of susceptible hosts, <span style="color:blue"><span class="math inline">\(c\)</span></span> is the contact rate, and <span style="color:blue"><span class="math inline">\(p\)</span></span> is the probability of contact with an infected host, and <span style="color:blue"><span class="math inline">\(v\)</span></span> is the probability of transmission per contact.</li>
</ul>
</section>
<section class="slide level2">
<ul>
<li><p><span style="color:blue"><span class="math inline">\(p\)</span></span> is usually assumed to be the disease prevalence, <span style="color:blue"><span class="math inline">\(\dfrac{I}{N}\)</span></span>.</p></li>
<li><p>Hence, the rate of new infections, <span style="color:blue;"><span class="math inline">\(\dfrac{dI}{dt} = S \times c \times \dfrac{I}{N} \times v\)</span></span>.</p></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li>In frequency dependent transmission, the contact rate <span style="color:blue"><span class="math inline">\(c\)</span></span> is also assumed to be constant, say <span style="color:blue"><span class="math inline">\(c = \eta\)</span></span> irrespective of population density, <span style="color:blue"><span class="math inline">\(\dfrac{N}{A}\)</span></span>, where <span style="color:blue"><span class="math inline">\(N\)</span></span> is the population size and <span style="color:blue"><span class="math inline">\(A\)</span></span> is the area occupied by the population.</li>
<li>Hence, <span style="color:blue"><span class="math inline">\(\dfrac{dI}{dt} = S \times \eta \times v \times \dfrac{I}{N}\)</span></span></li>
<li>Therefore, <span style="color:blue"><span class="math inline">\(\dfrac{dI}{dt} = \beta S \times \dfrac{I}{N}\)</span></span>, where <span style="color:blue"><span class="math inline">\(\beta = \eta \times v\)</span></span>, and <span style="color:blue"><span class="math inline">\(\lambda = \beta \times \dfrac{I}{N}\)</span></span>.</li>
</ul>
</section>
<section class="slide level2">
<ul>
<li><span style="color:tomato;"><em>Frequency-dependent/mass action transmission</em></span> is often used to model sexually-transmitted diseases and diseases with heterogeneity in contact rates.</li>
<li>Sexual transmission in this case does not depend on how many infected individuals are in the population.</li>
</ul>
</section>
<section class="slide level2">
<h4 id="density-dependent-transmission">Density dependent transmission</h4>
<ul>
<li><p>The rate of contact between individuals depends on the population density, <span style="color:blue"><span class="math inline">\(\dfrac{N}{A}\)</span></span>.</p></li>
<li><p>Transmission also depends on <span style="color:blue"><span class="math inline">\(p\)</span></span> - the probability that a given contact is with an infected individual, often taken to be <span style="color:blue"><span class="math inline">\(\dfrac{I}{N}\)</span></span>.</p></li>
</ul>
</section>
<section class="slide level2">
<ul>
<li><p>The density-dependent transmission is therefore given as <span style="color:blue"><span class="math inline">\(\lambda = \beta \times \dfrac{I}{A}\)</span></span>.</p></li>
<li><p>Here, because transmission increases with the density of infected individuals, it is called density-dependent transmission.</p></li>
</ul>
<div class="callout callout-note no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Note</strong></p>
</div>
<div class="callout-content">
<ul>
<li><p>Notice that <span style="color:blue"><span class="math inline">\(\lambda = \beta \times \dfrac{I}{N} \times \dfrac{N}{A}\)</span></span> and the <span class="math inline">\(N's\)</span> cancel out.</p></li>
<li><p><span style="color:blue"><span class="math inline">\(A\)</span></span> is often ignored.</p></li>
</ul>
</div>
</div>
</div>
</section>
<section class="slide level2">
<ul>
<li><span style="color:tomato;"><em>Density dependent transmission</em></span> can be used to model airborne and directly transmitted diseases, for example, measles.</li>
</ul>
</section>
<section class="slide level2">
<div class="callout callout-caution no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Density-dependent vs frequency dependent transmission</strong></p>
</div>
<div class="callout-content">
<p>This is one of the most confused and debated concepts in disease modelling. Several studies have attempted to clarify it, including the brilliant work by <span class="citation" data-cites="begon2002clarification">Begon et al. (<a href="#/references" role="doc-biblioref" onclick="">2002</a>)</span>. Most of the clarifications provided here are based on this paper.</p>
</div>
</div>
</div>
<ul>
<li>We will only use the <span style="color:tomato;">density-dependent</span> formulation in this course.</li>
</ul>
</section>
<section class="slide level2">
<p><a href="images/model_diagrams/model_diagrams.003.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-12"><img data-src="images/model_diagrams/model_diagrams.003.jpeg"></a></p>
</section>
<section class="slide level2">
<h4 id="process-2-recovery">Process 2: Recovery</h4>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/model_diagrams/model_diagrams.005.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-13"><img data-src="images/model_diagrams/model_diagrams.005.jpeg" class="quarto-figure quarto-figure-center" style="width:60.0%"></a></p>
</figure>
</div>
<p><span style="color:tomato;">Recovery</span>, governed by the recovery rate, <span class="math inline">\(\gamma\)</span> (rate at which infected individuals recover and become immune).</p>
</section>
<section class="slide level2">
<h5 id="some-notes-on-the-recovery-process">Some notes on the recovery process</h5>
<ul>
<li><p>If the duration of infection is <span style="color:blue"><span class="math inline">\(\dfrac{1}{\gamma}\)</span></span>, then the rate at which infected individuals recover is <span style="color:blue"><span class="math inline">\(\gamma\)</span></span>.</p></li>
<li><p>The average infectious period is often <span style="color:tomato;">estimated from epidemiological data</span>.</p></li>
</ul>
<div class="callout callout-note no-icon callout-titled callout-style-default">
<div class="callout-body">
<div class="callout-title">
<p><strong>Note</strong></p>
</div>
<div class="callout-content">
<p>You will learn about parameter estimation in the model fitting and calibration lectures.</p>
</div>
</div>
</div>
</section>
<section class="slide level2">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/epi_parameters.png" class="lightbox" data-gallery="quarto-lightbox-gallery-14" title="Source: @anderson1982directly"><img data-src="images/epi_parameters.png" alt="Source: Anderson and May (1982)"></a></p>
<figcaption>Source: <span class="citation" data-cites="anderson1982directly">Anderson and May (<a href="#/references" role="doc-biblioref" onclick="">1982</a>)</span></figcaption>
</figure>
</div>
</section>