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3.9 Modeling and Control of a Campus Outbreak of Coronavirus COVID-19

This Jupyter/Python notebook presents models for the outbreak of an infectious disease into a susceptible population using standard epidemiological models. Model parameters are taken from a rapidly evolving scientific literature documenting the global COVID-19 outbreak. A control policy based on 'social distancing' is included in the model.

The notebook is organized as follows:

  1. Brief Background on the SARS-CoV-2 Coronavirus and COVID-19
  2. Model 1. SIR Model for an Infectious Disease
  3. Model 2. SEIR Model
  4. Model 3. SEIR Model with Control
  5. Model 4. An Improved SEIR model.
  6. Ideas for student projects.

The executable Python code in this notebook can be edited and run Google's cloud servers by clicking on the "Open in Colab" button located in the header. Use the interactive sliders to adjust model parameters and perform 'what if' simulations.

3.9.1 1. Brief background on the SARS-CoV-2 Coronavirus and COVID-19

COVID-19 is caused by the human coronavirus SARS-CoV-2. First identified in the 1960's, there are currently four human coronaviruses endemic to populations around the world:

These four common coronaviruses cause an upper respiratory disease that can progress to pneumonia. These endemic viruses cause about a quarter of all common colds. Most people will suffer from at least one during their lifetimes.

In recent decades, three additional coronaviruses that normally infect animals have evolved to infect humans. These include:

The last of these, now called SARS-CoV-2, first appeared in December, 2019, at a seafood market in Wuhan (Hubei, China). The rapid spread of SARS-CoV-2 in Wuhan, and subsequent appearance in other locations around the globe, has resulted in declaration of a global health emergency by the World Health Organization (WHO). Most countries are mobilizing to track the virus and control new outbreaks. At this stage, it is too early to know if efforts to contain the mitigate transmission of the virus will be successful in preventing COVID-19 from becoming a pandemic, or later an endemic disease with a global footprint.

The latest status on the global outbreak of COVID-19 can be found at the following links:

The purpose of this notebook is to demonstrate the modeling of an infectious epidemic using the latest available data for COVID-19, and to provide a framework for evaluating the performance of 'social distancing' and other mitigation strategies. The models and data used in this notebook have been extracted from a rapidly emerging and changing literature. Recent papers on COVID-19 can be found at the following links.

3.9.2 Model 1. SIR model for an infectious disease

3.9.2.1 Readings

3.9.2.2 Introduction to SIR models

The SIR model is deterministic compartment model for the spread of an infectious disease that describes key phenomena encountered in epidemiology. In the SIR model, a population is broken into three non-overlapping groups corresponding to stages of the disease:

Neglecting demographic processes of birth and death from other causes, and assuming a negligible death rate due to infectious disease at issue, the progression of an epidemic can be modeled by rate processesl

$$\text{Susceptible} \xrightarrow{\frac{\beta S I}{N}} \text{Infectious} \xrightarrow{\gamma I} \text{Recovered} $$

The rate processes are modeled as follows.

A model for the spread of an infectious disease in a uniform population is given by the deterministic SIR equations

\begin{align*} \frac{dS}{dt} & = -\frac{\beta S I}{N} \\ \frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I \\ \frac{dR}{dt} & = \gamma I \end{align*}

The model becomes more generic by working with population fractions rather than raw population counts. To this end, define

\begin{align} s = \frac{S}{N} \qquad i = \frac{I}{N} \qquad r = \frac{R}{N} \end{align}

After substitution, this results in a system of four equations.

\begin{align*} \frac{ds}{dt} & = -\beta s i \\ \frac{di}{dt} & = \beta s i - \gamma i \\ \frac{dr}{dt} & = \gamma i \end{align*}

where $s + i + r = 1$ is an invariant.

3.9.2.3 Key Properties of the SIR Model

The SIR model describes key epidemiological phenemena. Here is a brief synposis of the relevant results.

3.9.2.4 Simulation

The following Python code implements a simulation of the SIR model. The parameter values were selected from the recent survey by Boldog, et al. (2020).

3.9.2.5 How many people will be infected following an outbreak?

Given an outbreak in a susceptible population, the final state is reached when $i$ returns to a near zero value. A formula for the final state can be found by taking the ratio

\begin{align} \frac{di}{ds} & = \frac{\frac{di}{dt}}{\frac{ds}{dt}} = -1 + \frac{1}{s R_0} & \\ \end{align}

Integrating,

\begin{align} \int_{i_0}^{i_f} di & = \int_{s_0}^{s_f} (-1 + \frac{1}{s R_0}) ds & \\ \end{align}\begin{align} i_f - i_0 & = -(s_f - s_0) + \frac{1}{R_0} \ln\frac{s_f}{s_0} & \\ \end{align}

The starting point of an outbreak begins with a very small value $i_0 \approx 0$ and ends with a very small value $i_f \approx 0$. Setting $i_0 = i_f = 0$ gives

\begin{align} s_f - \frac{1}{R_0}\ln s_f & = s_0 - \frac{1}{R_0} \ln{s_0} & \\ \end{align}

For the special case of an initially susceptible population, $s_0 = 1$ which gives

\begin{align} s_f - \frac{1}{R_0} \ln s_f & = 1 \end{align}

3.9.3 Model 2. SEIR model

The SEIR model extends the SIR model by adding an additional population compartment containing those individuals who have been exposed to the virus but not yet infective.

The compartment model can be diagrammed as follows.

$$\text{Susceptible} \xrightarrow{\frac{\beta S I}{N}} \text{Exposed} \xrightarrow{\alpha E} \text{Infectious} \xrightarrow{\gamma I} \text{Recovered} $$

The rate processes are modeled as follows.

An elementary model for the spread of an infectious disease in a uniform population is given by the deterministic SEIR equations}

After substitution, this results in a system of four equations.

\begin{align*} \frac{ds}{dt} & = -\beta s i \\ \frac{de}{dt} & = \beta s i - \alpha e \\ \frac{di}{dt} & = \alpha e - \gamma i \\ \frac{dr}{dt} & = \gamma i \end{align*}

where $s + e + i + r = 1$ is an invariant.

3.9.4 Model 3. Mitigation and Social Distancing

The lack of a vaccine reduces the options for controlling the COVID-19 outbreak. Current efforts are focused on 'social distancing' designed to reduce transmission of the virus from individuals in the infective state to susceptible individuals.

For the purposes of modeling, we introduce a control parameter $u$ indicating the effectiveness of these efforts. $u=0$ corresponds to no controls, $u=1$ corresponds to perfect isolation of infective individuals. The purpose of this model is to explore how a social distancing stragtegy affects the outcome of an epidemic.

The compartment model can be diagrammed as follows.

$$\text{Susceptible} \xrightarrow{(1-u)\frac{\beta S I}{N}} \text{Exposed} \xrightarrow{\alpha E} \text{Infectious} \xrightarrow{\gamma I} \text{Recovered} $$

The rate processes are modeled as follows.

After substitution, this results in a system of four equations.

\begin{align*} \frac{ds}{dt} & = -(1-u)\beta s i \\ \frac{de}{dt} & = (1-u)\beta s i - \alpha e \\ \frac{di}{dt} & = \alpha e - \gamma i \\ \frac{dr}{dt} & = \gamma i \end{align*}

where $s + e + i + r = 1$ is an invariant.

Social distancing has several beneficial effects:

3.9.4.1 Social Distancing Strategies for the Campus

The basic strategy is to slow transmission through 'social distancing' with the following goals:

CDC Social Distancing

3.9.5 Model 4. Improving the fidelity of the model.

Boldog, et al.

3.9.6 Additional Modeling Opportunities. Transportation, power law kinetics, etc.

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