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_04_sir_model.qmd
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_04_sir_model.qmd
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{width="40%" fig-align="center"}
This model groups individuals into three *disease states*:
- [Susceptible (S)]{style="color:green;"}: not infected but can be.
- [Infected (I)]{style="color:tomato;"}: infected & infectious.
- [Recovered/removed (R)]{style="color:blue;"}: recovered & immune.
------------------------------------------------------------------------
### How do individuals move between compartments?
#### Process 1: Transmission
{width="70%"}
[What drives transmission?]{style="color:red;"}
------------------------------------------------------------------------
- Transmission is driven by several factors, including:
- Disease prevalence, [$I$]{style="color:blue;"}, i.e., number of infected individuals at the time.
- The number of contacts, [$C$]{style="color:blue;"}, susceptible individuals have with infected individual.
- The probability, [$p$]{style="color:blue;"}, a susceptible individual will become infected when they contact an infected individual.
------------------------------------------------------------------------
{width="70%"}
The transmission term is often defined through the [force of infection (FOI), $\lambda$]{style="color:tomato;"}.
------------------------------------------------------------------------
### A tour of the force of infection (FOI)
- FOI, $\lambda$, is the [per capita rate]{style="color:tomato;"} at which susceptible individuals become infected.
::: {.callout-note collapse="true" icon="false"}
"Per capita" means the rate of an event occurring per individual in the population per unit of time.
:::
- Given the rate per individual per time, FOI, the rate at which new infecteds are generated is given by [$\lambda \times S$]{style="color:blue;"}, where $S$ is the number of susceptible individuals.
------------------------------------------------------------------------
- The force of infection is made up of the probabilities/rates that:
- contacts happen, [$c$]{style="color:blue"},
- a given contact is with an infected individual, [$p$]{style="color:blue"}, and
- a contact results in successful transmission, [$v$]{style="color:blue"}.
------------------------------------------------------------------------
- The FOI can be formulated in two ways, depending on how the contact rate is expected to change with the population size:
- Frequency-dependent/mass action transmission
- Density-dependent transmission
------------------------------------------------------------------------
#### Frequency-dependent/mass action transmission
The rate of contact between individuals is constant irrespective of the population density, [$\dfrac{N}{A}$]{style="color:blue"}, where [$N$]{style="color:blue"} is the population size and [$A$]{style="color:blue"} is the area occupied by the population.
------------------------------------------------------------------------
- Recall that transmission also depends on the probability of contact with an infected host, [$p$]{style="color:blue"}, which is assumed to be [$\dfrac{I}{N}$]{style="color:blue"}.
- Hence, the frequency-dependent mass action is given by [$\lambda = \beta \times \dfrac{I}{N}$]{style="color:blue"}, where [$\beta$]{style="color:blue"} is the transmission rate.
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Question
Why does the frequency-dependent transmission contain [$\dfrac{1}{N}$]{style="color:blue"} if it does not depend on the population density?
:::
------------------------------------------------------------------------
- Assume that the rate of new infections is given by [$\dfrac{dI}{dt} = S \times c \times p \times v$]{style="color:blue"} where [$S$]{style="color:blue"} is the number of susceptible hosts, [$c$]{style="color:blue"} is the contact rate, and [$p$]{style="color:blue"} is the probability of contact with an infected host, and [$v$]{style="color:blue"} is the probability of transmission per contact.
------------------------------------------------------------------------
- [$p$]{style="color:blue"} is usually assumed to be the disease prevalence, [$\dfrac{I}{N}$]{style="color:blue"}.
- Hence, the rate of new infections, [$\dfrac{dI}{dt} = S \times c \times \dfrac{I}{N} \times v$]{style="color:blue;"}.
------------------------------------------------------------------------
- In frequency dependent transmission, the contact rate [$c$]{style="color:blue"} is also assumed to be constant, say [$c = \eta$]{style="color:blue"} irrespective of population density, [$\dfrac{N}{A}$]{style="color:blue"}, where [$N$]{style="color:blue"} is the population size and [$A$]{style="color:blue"} is the area occupied by the population.
- Hence, [$\dfrac{dI}{dt} = S \times \eta \times v \times \dfrac{I}{N}$]{style="color:blue"}
- Therefore, [$\dfrac{dI}{dt} = \beta S \times \dfrac{I}{N}$]{style="color:blue"}, where [$\beta = \eta \times v$]{style="color:blue"}, and [$\lambda = \beta \times \dfrac{I}{N}$]{style="color:blue"}.
------------------------------------------------------------------------
- [*Frequency-dependent/mass action transmission*]{style="color:tomato;"} is often used to model sexually-transmitted diseases and diseases with heterogeneity in contact rates.
- Sexual transmission in this case does not depend on how many infected individuals are in the population.
------------------------------------------------------------------------
#### Density dependent transmission
- The rate of contact between individuals depends on the population density, [$\dfrac{N}{A}$]{style="color:blue"}.
- Transmission also depends on [$p$]{style="color:blue"} - the probability that a given contact is with an infected individual, often taken to be [$\dfrac{I}{N}$]{style="color:blue"}.
------------------------------------------------------------------------
- The density-dependent transmission is therefore given as [$\lambda = \beta \times \dfrac{I}{A}$]{style="color:blue"}.
- Here, because transmission increases with the density of infected individuals, it is called density-dependent transmission.
::: {.callout-note collapse="true" icon="false"}
- Notice that [$\lambda = \beta \times \dfrac{I}{N} \times \dfrac{N}{A}$]{style="color:blue"} and the $N's$ cancel out.
- [$A$]{style="color:blue"} is often ignored.
:::
------------------------------------------------------------------------
- [*Density dependent transmission*]{style="color:tomato;"} can be used to model airborne and directly transmitted diseases, for example, measles.
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Density-dependent vs frequency dependent transmission
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 @begon2002clarification. Most of the clarifications provided here are based on this paper.
:::
- We will only use the [density-dependent]{style="color:tomato;"} formulation in this course.
------------------------------------------------------------------------
------------------------------------------------------------------------
#### Process 2: Recovery
{width="60%" fig-align="center"}
[Recovery]{style="color:tomato;"}, governed by the recovery rate, $\gamma$ (rate at which infected individuals recover and become immune).
------------------------------------------------------------------------
##### Some notes on the recovery process
- If the duration of infection is [$\dfrac{1}{\gamma}$]{style="color:blue"}, then the rate at which infected individuals recover is [$\gamma$]{style="color:blue"}.
- The average infectious period is often [estimated from epidemiological data]{style="color:tomato;"}.
::: {.callout-note collapse="true" icon="false"}
You will learn about parameter estimation in the model fitting and calibration lectures.
:::
------------------------------------------------------------------------
------------------------------------------------------------------------
### Putting it all together
{width="80%"}
------------------------------------------------------------------------
### Formulating the model equations
[Continuous time compartmental]{style="color:tomato;"} models are formulated using [differential equations]{style="color:tomato;"} that describe the change in the number of individuals in each compartment over time.
------------------------------------------------------------------------
The SIR model can be formulated as:
::::: columns
::: {.column width="50%"}
{fig-align="center"}
:::
::: {.column width="50%"}
\begin{align*}
\frac{dS}{dt} & = \dot{S} = \color{orange}{-\beta S I} \\
\frac{dI}{dt} & = \dot{I} = \color{orange}{\beta S I} - \color{blue}{\gamma I} \\
\frac{dR}{dt} & = \dot{R} = \color{blue}{\gamma I}
\end{align*}
:::
:::::
where [$\beta$]{style="color:blue;"} is the transmission rate, and [$\gamma$]{style="color:blue;"} is the recovery rate.
::: notes
- We may or may not use the dot notation to denote the rate of change.
- The terms are just **inflows** and **outflows**.
- **Pause** for questions and clarifications.
:::
------------------------------------------------------------------------
The initial conditions are given by:
\begin{equation*}
\begin{aligned}
S(0) & = N - 1,\\
I(0) & = 1, \text{and} \\
R(0) & = 0.
\end{aligned}
\end{equation*}
where $N$ is the total population size.
------------------------------------------------------------------------
::: {.callout-note collapse="true" icon="false"}
- We represent the compartments as [population sizes]{style="color:tomato;"}.:
- Some modellers often use [proportions]{style="color:tomato;"} instead of population sizes as a way to remove the dimensions from the equations.
:::
------------------------------------------------------------------------
### Model assumptions
- The population is closed: no births, deaths, or migration.
- Implicitly: the epidemic occurs much faster than the time scale of births, deaths, or migration.
- Individuals are infectious immediately after infection and remain infectious until they recover.
------------------------------------------------------------------------
- Mixing is *homogeneous*, i.e., [individuals mix randomly]{style="color:tomato;"}:
- Individuals have an equal probability of coming into contact with any other individual in the population.
- Transition rates are constant and do not change over time.
- Individuals acquire "lifelong" immunity after recovery.
::: notes
The $R$ compartment is an absorbing state in this model.
:::
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Discussion
- What diseases do you think the SIR model is appropriate for?
:::
------------------------------------------------------------------------
- The SIR model is appropriate for diseases that confer immunity after recovery. For example, measles and chicken pox.
- Popularised by Kermack and McKendrick in 1927 [@kermack1927contribution]
- A must-read paper for budding infectious disease modellers.
------------------------------------------------------------------------
### What questions can we answer with the SIR model?
- The SIR model can be used to understand the dynamics of an epidemic:
- How long will the epidemic last?
- How many individuals will be infected (final epidemic size)?
- When will the epidemic reach its peak?
------------------------------------------------------------------------
## Solving the SIR model
- Compartmental models cannot be solved analytically.
- We often perform two types of analyses to understand the [long term dynamics]{style="color:tomato;"} of the model:
- Qualitative: threshold phenomena and analysis of equilibria (disease-free and endemic).\
- Numerical simulations.
::: callout-note
- For this introductory course, we will use focus on simulation.
:::
- But first, let's do some qualitative analysis of the SIR.
------------------------------------------------------------------------
### Threshold phenomena
- Here, we study the conditions under which an epidemic will [grow or die out]{style="color:tomato;"} using the model equations.
------------------------------------------------------------------------
- Consider the case where $I(0) = 1$ individual is introduced into a population of size $N$ at time $t = 0$.
- That means in a completely susceptible population, we have $S(0) = N - 1$ susceptible individuals.
------------------------------------------------------------------------
- At time 0, the disease will not spread if the rate of change of infections is negative, that is [$\dfrac{dI}{dt} < 0$]{style="color:blue;"}.
- Recall from the SIR model that [$\dfrac{dI}{dt} = \beta S I - \gamma I$]{style="color:blue;"}.
- Let's solve this equation at $t = 0$ by setting $I = 1$, assuming [$\dfrac{dI}{dt} < 0$]{style="color:blue;"}.
------------------------------------------------------------------------
At $t = 0$, we have
\begin{equation*}
\frac{dI}{dt} = \beta S I - \gamma I < 0 \end{equation*}
Factor out $I$, and we get
\begin{equation*}
I (\beta S - \gamma) < 0
\end{equation*}
------------------------------------------------------------------------
- Since at $t=0$$, I > 0$, we have [$S < \dfrac{\gamma}{\beta}$]{style="color:blue;"}.
- [$\dfrac{\gamma}{\beta}$]{style="color:blue;"} is the relative removal rate.
------------------------------------------------------------------------
- **Interpretation**:
- At $t = 0$, $S$ must be less than $\dfrac{\gamma}{\beta}$ for the epidemic to die out.
- If the rate of removal/recovery is greater than the transmission rate, the epidemic will die out.
- Any infection that cannot transmit to more than one host is going to die out.
------------------------------------------------------------------------
For the SIR model, the quantity [$\dfrac{\beta}{\gamma}$]{style="color:blue"} is called the [reproduction number, $R0$]{style="color:tomato"} (pronounced "R naught" or "R zero").
------------------------------------------------------------------------
#### The basic reproduction number, R0
- The basic reproduction number, [$R0$]{style="color:blue"}, is the average number of secondary infections generated by a [single primary infection]{style="color:tomato"} in a [completely susceptible]{style="color:tomato"} population.
- The basic reproduction number is a key quantity in infectious disease epidemiology.
------------------------------------------------------------------------
- It is often represented as a single number or a range of high-low values.
- For example, the $R0$ for measles is popularly known to be $12$ - $18$ [@guerra2017measlesR0].
------------------------------------------------------------------------
- $R0$ is often used to express the threshold phenomena in infectious disease epidemiology:
- If $R0 > 1$, the epidemic will grow.
- If $R0 < 1$, the epidemic will decline.
- A pathogen's $R0$ value is determined by biological characteristics of the pathogen and the host's behaviour.
------------------------------------------------------------------------
- Conceptually, $R0$ is given by
$$
R0 \propto \dfrac{\text{Infection}}{\text{Contact}} \times \dfrac{\text{Contact}}{\text{Time}} \times \dfrac{\text{Time}}{\text{Infection}}
$$
- $R0$ is unitless and dimensionless.
------------------------------------------------------------------------
### Numerical simulations
- Numerical simulations can be performed with any programming language.
- This course focuses on the R programming language because:
- It is a popular language for data analysis and statistical computing.
- It has a rich ecosystem of packages for solving differential equations.
- It is free and open-source.
------------------------------------------------------------------------
- In R, we can use the `{deSolve}` package to solve the differential equations.
- To solve the model in R with `{deSolve}`, we will always need to define at least three things:
- The model equations.
- The initial parameter values.
- The initial conditions (population sizes).
------------------------------------------------------------------------
#### Practicals
- Let's do a code walk through in R using the script `sir.Rmd`.
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Discussion
- What happens if we increase or decrease the value of $R0$?
- What happens if we increase or decrease the value of the infectious period?
- How can we flatten the curve?
:::
::: notes
- We can flatten the curve by reducing the value of $R0$.
- In reality, we can reduce $R0$ by implementing control measures such as social distancing, wearing masks, and vaccination.
- These control measures reduce the number of contacts between individuals or the probability of infection, which in turn reduces the transmission rate.
:::
------------------------------------------------------------------------