Background

From the Imperial College COVID-19 Response Team (Ferguson et al. 2020):

Ferguson et al. (2020) used an individual-based model, to investigate the effects of various mitigation and suppression strategies on human fatalities in the US and Great Britain. An individual-based model actually models individual agents (in this case humans). All of them. Each a little bit different.

Their model relies on assumes:

SIR

The simplest epidemiological model is the “S-I-R” model. The letters refer to the three types of individuals we model: Susceptible, Infected, and Resistant. It assumes a constant population size, and is most useful for understanding disease outbreaks, where a disease enters a susceptible population, spreads through the population, and individuals become resistant or die.

\[\frac{dS}{dt} = - \beta SI\] \[\frac{dI}{dt} = \beta SI - \gamma I\] \[\frac{dR}{dt} = \gamma I\]

The rate at which susceptible individuals become infected depends on the density of infected individuals in the population. We refer to this as density-dependent transmission.

In this model, we assume that any susceptible individual could become infected by any infectious individual. This requires a well-mixed population. This can be a very useful approximation in some cases, such as a single cattle herd, or a school, and where the disease is spread via aerosol droplets and contact surfaces such as doorknobs.

The SIR model may be a less useful approximation in other cases, such as

Often the spread of diseases, such as STDs, depend on disease prevalence. Prevalence is the proportion or frequency of the population that is infected. In these cases, we refer to this situations as exhibiting frequency-dependent transmission. In these situations, transmission is driven by frequency, \(I/N\), rather than density, \(I\).

The SIR model with frequency dependent transmission is thus

\[\frac{dS}{dt} = - \beta \frac{SI}{N}\] \[\frac{dI}{dt} = \beta \frac{SI}{N} - \gamma I\] \[\frac{dR}{dt} = \gamma I\]

Scientistics often think of social structure as a network, in which individuals within groups are connected very tightly and interact frequently, while individuals in different groups interact only weakly or infrequently. In the case of global and regional disease transmission, much of this social structure has an important spatial component. For instance, I interact with strangers in the gorcery store or gas station when we stand in line, use the same doorknob or the same gas pump. I don’t interact with people in different states unless people travel.

Social distancing is the practice of weakening network connections by increasing physical space between people and reducing the frequency of close encounters. Quaratines are ways to create extreme social distancing between groups or individuals (self-quearatine). Most often, quarantines separate those who are sick from those who are healthy.

Separating those who are sick from those who are healthy is the most important thing we can do to reduce the transmission rate. However, we often do not know who is sick and who is not. This is because people can be infectious and never show symptoms themselves, and also because even when we do get symptoms, we will be infectious for some time prior to the onset of those symptoms. Social distancing is a way of separating the infected from the uninfected even when we don’t know who is infected and who is not infected.

A simple way of building network sturcture into our model is to moderate the magnitude of the frequency dependence. We will make call this our quasi-frequency-dependent transmission model, where transmission depends on \(I/N^z\) where \(0\le z \le 1\). When \(z=0\), this is density-dependent transmission and when \(z=1\) this is pure frequency-dependent tramission. When \(z\) is between these, it represents some degree of spatial, social, or network structure, and mechanism of spread.

\[\frac{dS}{dt} = - \beta \frac{SI}{N^z}\] \[\frac{dI}{dt} = \beta \frac{SI}{N^z} - \gamma I\] \[\frac{dR}{dt} = \gamma I\] #### R0 One number that epidemiologists worry about a lot is the basic reproduction number of a disease, \(R_0\). We refer to this number as “R-naught”, and it is the number of new infections caused by a single infected individual during their infectious period. It is a maximum rate, if an individual were dropped into the middle of an entirely susceptible population.

\(R_0\) depends on

In addition to \(R_0\), epidemiologists worry about the effective reproduction number, \(R_e\). The effective reproductive number takes other factors into account, including the fraction of the population that is already resistant, or otherwise isolated from infectious individuals.

\[R_0 = \frac{\beta}{\gamma}\]

Investigate variation in frequency-dependence and R0

Use the COVID-19 spreadsheet simulation to investigate of range of transmission scenarios.

Using statistitics assumed by Ferguson et al. (2020), we can start with a mean disease duration of 6.5, and \(R_0 = 2.4\), and a fatality rate 0f 0.9%. (This means \(\gamma = 1/6.5\), and \(\beta=\gamma R_0 = 0.369\)). Try \(z = 0.95\). Note that Riou et al. assumed a fatalitiy rate of 1.6% (Riou et al. 2020).

This is what that would look like in a population of 330 million people.