Deterministic models

Steps in developing a model

1. Formulate objectives
2. Conceptual diagram
3. Dynamic equations
4. Computer code

A classic model for an acute immunizing infection that classified individuals according to infection status

• Susceptible
• Infections
• Recovered/Immune

These quantities will be our state variables

First studied extensively by Kermack & McKendrick (1927) A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society of London. Series A 115 (772): 700–721

Transmission systems

• Sexual transmission
• Environmental transmission
• Vector-borne transmission

Structure

• Age
• Sex
• Location

Assign each state variable a unique variable (typically a Roman letter)

• Susceptible: (\( S \) for proportion, \( X \) for number)
• Infectious: (\( I \) for proportion, \( Y \) for number)
• Recovered: (\( R \) for proportion, \( Z \) for number)

Assign parameters a unique variable (typically a Greek letter)

• Contact rate: \( \kappa \)
• Pathogen infectivity: \( \nu \)
• Transmissibility (contact rate times infectivity): \( \beta = \kappa \times \nu \)
• Recovery rate: \( \gamma \)

Note: Notation varies among authors; one should not assume that a symbol used in two different papers means the same in each.

Sum the positive and negative flows that affect each state variable

\[ \begin{aligned} \frac{dX}{dt} &= - \beta XY \\ \frac{dY}{dt} &= \beta XY - \gamma Y \\ \frac{dZ}{dt} &= \gamma Y \end{aligned} \]

Note: These equations can also be derived using difference quotients and the fundamental theorem of calculus. I find it more straightforward to think in terms of flows.

Note: Sometimes “overdot notation” is used to signifiy time derivatives, i.e. \( \dot{X} = \frac{dX}{dt} \).

It is common to think of the rate at which susceptibles become infected as caused by a force acting on each susceptible individually.

This force of infection, which we might denote by \( \lambda \), may be identifed by factoring the \( X \) out of our equation for \( \frac{dX}{dt} \), i.e.

\[ \begin{aligned} \frac{dX}{dt} &= - \beta XY \\ &= - (\beta Y) \times X \\ &= - \lambda \times X \\ \end{aligned} \]

The force of infection may take many different forms depending on the conditions under which transmission occurs. A common distinction is between density dependent transmission (\( \lambda = \beta Y \)) and frequency dependent transmission (\( \lambda = \beta \frac{Y}{N} \), where \( N = X+Y+Z \) is the total population size).

If the population size does not change (i.e. \( N \) is a constant) then these two models differ only by the units of \( \beta \). However, if the population size does change (for instance due to disease-induced mortality) then these two forces of infection result in very different dynamical outcomes.

• Can we make general statements about epidemics without resorting to extensive numerical integration?
• Under what conditions will an infectious disease invade a system

Assume an infectious individual arises in a Wholly Susceptible and otherwise Disease Free population with frequency-dependent transmission

Initial conditions: \( X(0)=N-1 \approx N \), \( Y(0)=1 \), and \( Z(0)=0 \)

Invasion occurs only if \( dY/dt > 0 \)

\[ \begin{align} \frac{dY}{dt} = \beta X Y/N - \gamma Y &> 0 \\ Y(\beta X/N - \gamma) &>0 \\ X/N &> \gamma/\beta \end{align} \]

Since \( X \approx N \), this is satisfied when \( 1>\gamma / \beta \), which is equivalent to \( \beta/\gamma > 1 \)

Kermack & McKendrick (1927)

• For the \( SIR \) model, the ratio \( \beta/\gamma \) gives the number of secondary cases that will be infected before the index case recovers
• This quantity is universally referred to as \( R_0 \) and called the Basic Reproduction Number or Basic Reproductive Ratio
• We can extend the concept of \( R_0 \) to other models with the following definition
• The basic reproduction number is the number of secondary cases generated a single typical infected case in an entirely suspectible population
• Hence, \( R_0 \) depends on both properties of the pathogen and properties of the population into which it is introduced

• As before, pathogen invasion occurs if \( dY/dt > 0 \)
• Assuming nearly everyone initially susceptible \( X \approx N \), then \( \beta NY - \gamma Y > 0 \implies Y(\beta N- \gamma) > 0 \)
• Now, we define \( R_0 = \beta N / \gamma \)
• Conclusion: For density dependent transmission there is a threshold population density required for invasion

We assume infectious individuals die at rate \( \alpha \)

\[ \frac{dY}{dt} = ... -\gamma Y - \alpha Y \]

• If the population size doesn't change substantially then frequency- and density-dependent transmission are equivalent (\( \beta_{FD} = N \times \beta_{DD} \))
• Otherwise
  • Frequency dependent transmission is typically assumed to be more appropriate for large populations with heterogeneous mixing, sexually transmitted diseases, and vector-borne pathogens
  • Density dependent transmission is assumed to represent wildlife and livestock diseases, especially those with smaller population sizes
  • Empirical evidence suggests that many disease systems are intermediate

When we talk about “solving” a model we mean that we want to know what the state of the system will be at a future time given the system equations and some initial conditions, requiring integration of the equations. These are also known as boundary value problems.

Systems of nonlinear differential equations (like the \( SIR \) model) are typically not analytically tractable, although clever mathematicians have found numerous approximations or special cases, like the final size relation.

\[ 1-R(\infty)-e^{-R(\infty)R_0} \]

Thus, we use numerical algorithms such as the 4th order Runge-Kutta algorithm to take advantage of Euler's approximation to obtain an approximate solution by solving a sequence of tractable linear approximations at smaller and smaller step sizes until a specified tolerance is achieved.

In R, numerical integration of ordinary differential equations is readily performed using the package deSolve. Particularly, the function ode is useful since it automatically selects the optimal solving algorithm based on numerical performance.

Modelers must be cautious, however, and alert to seeming anomalous results resulting from numerical instabilities.

Presentations and exercises draw significantly from materials developed with Pej Rohani, Ben Bolker, Matt Ferrari, Aaron King, and Dave Smith used during the 2009-2011 Ecology and Evolution of Infectious Diseases workshops and the 2009-2019 Summer Institutes in Statistics and Modeling of Infectious Diseases.

Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/bync/3.0/. Please share and remix noncommercially, mentioning its origin.