A simple simulation model

• We’ll start with a very simple model, a population of entities (pathogens/immune cells/humans/animals) that grow or die.
• We’ll implement the model as a discrete time equation, given by:

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \]

• \(P_t\) are the number of pathogens in the population at current time \(t\), \(dt\) is some time step and \(P_{t+dt}\) is the number of pathogens in the future after that time step has been taken.
• The processes/mechanisms modeled are growth at rate \(g\) and death at rate \(d_P\).

A simple simulation model

• Assume we have a population of individuals (e.g., pathogens) that grow at a rate of \(g=12\) and die at rate of \(d_P=2\) (per some time unit, e.g. days or weeks or years).
• If we started with 100 individuals (pathogens) at time t=0 and took time steps of \(dt=1\), how many individual would we have after 1,2,3… time units?

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \]

A simple simulation model - variant 1

Original:

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \] Alternative:

\[ P_{t+dt} = P_t + dt ( g - d_P P_t ) \] What’s the difference? Is this a good (biologically potentially reasonable) model?

A simple simulation model - variant 2

Original:

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \] Alternative:

\[ P_{t+dt} = P_t + dt ( g P_t - d_P) \]

What’s the difference? Is this a good (biologically potentially reasonable) model?

Discrete time models

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \]

• The model above is updated in discrete time steps (to be chosen by the modeler).
• Good for systems where there is a “natural”” time step. E.g. some animals always give birth in spring or some bacteria divide at specific times.
• Used in complex individual based models for computational reasons.
• For compartmental models where we track the total populations (instead of individuals), continuous-time models are more common. They are usually formulated as ordinary differential equations (ODE).
• If the time-step becomes small, a discrete-time model approaches a continuous-time model.

Continuous time models

Discrete:

\[ P_{t+dt} = P_t + dt ( g P_t - d_P P_t ) \] Re-write:

\[ \frac{P_{t+dt} - P_t}{dt} = g P_t - d_P P_t \]

Continuous (dt -> 0): \[ \frac{dP}{dt} = gP - d_P P \]

• If we simulate a continuous time model, the computer uses a smart discrete time-step approximation.

Some notation

The following are 3 equivalent ways of writing the differential equation:

\[ \begin{aligned} \frac{dP(t)}{dt} &= gP(t) - d_P P(t) \\ \frac{dP}{dt} &= gP - d_P P \\ \dot{P} &= gP - d_P P \\ \end{aligned} \] We will use the ‘dot notation’.

Some terminology

\[ \dot{P} = gP - d_P P \]

• The left side is the change in time of the indicated variable.
• Each term on the right side represents a (often simplified/abstracted) biological process/mechanism.
• Any positive term on the right side is an inflow and leads to an increase of the indicated variable.
• Any negative term on the right side is an outflow and leads to a decrease of the indicated variable.

Extending the model

\[ \dot{P} = gP - d_P P \]

For different values of the parameters g and \(d_P\), what broad types of dynamics/outcomes can we get from this model?

Extending the model

\[ \dot{P} = gP - d_P P \]

How can we extend the model to get growth that levels off as we reach some high level of \(P\)?

Model with saturating growth

\[ \dot{P} = gP(1-\frac{P}{P_{max}}) - d_P P \]

• We changed the birth process from exponential/unlimited growth to saturating growth. \(P_{max}\) is the level of \(P\) at which the growth term is zero.
• If \(P > P_{max}\), the growth term is negative.
• The population settles down at a level where the growth balances the decay, i.e. when \(gP(1-\frac{P}{P_{max}}) = d_P P\).

Adding a second variable

• A single variable model is ‘boring’.
• The interesting stuff happens if we have multiple compartments/variables that interact.
• Let’s introduce a second variable.
• Let’s assume that P is a population of some bacteria (but could also be some animal), which gets attacked and consumed by some predator, e.g. the immune system or another animal. We’ll pick the letter H for the predator (any label is fine).

Adding a second variable

\[ \begin{aligned} \dot{P} & = gP(1-\frac{P}{P_{max}}) - d_P P \ \pm \ ?\\ \dot{H} & = ? \end{aligned} \]

• The predator attacks/eats the prey. What process could we add to the P-equation to describe this?

Adding a second variable

\[ \begin{aligned} \dot{P} & = gP(1-\frac{P}{P_{max}}) - d_P P - kPH\\ \dot{H} & = ? \end{aligned} \]

• The more P there is, the more the predator will grow, e.g. by eating P or by receiving growth signals.
• What term could we write down for the growth dynamics of H?
• Finally, H individuals have some life-span after which they die. How can we model this?

Predator-prey model

The model we just built is a version of the well-studied predator-prey model from ecology. \[ \begin{aligned} \dot{P} & = g_P P(1-\frac{P}{P_{max}}) - d_P P - kPH\\ \dot{H} & = g_H P H - d_H H \end{aligned} \]

The discrete-time version of the model is: \[ \begin{aligned} P_{t+dt} & = P_t + dt(g_P P_t(1-\frac{P_t}{P_{max}}) - d_P P_t - kP_tH_t)\\ H_{t+dt} & = H_t + dt( g_H P_t H_t - d_H H_t) \end{aligned} \]

Bacteria and immune response model

The names of the variables and parameters are arbitrary. If we think of bacteria and the immune response, we might name them B and I instead.

\[ \begin{aligned} \dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - kBI\\ \dot{I} & = r BI - d_I I \end{aligned} \] \[ \begin{aligned} B_{t+dt} & = B_t + dt(g B_t(1-\frac{B_t}{B_{max}}) - d_B B_t - k B_t I_t)\\ I_{t+dt} & = I_t + dt( r B_t I_t - d_I I_t) \end{aligned} \]

Graphical model representation

• It is important to go back and forth between words, diagrams, equations.
• Diagrams specify a model somewhat, but not completely. The diagram below could be implemented as ODEs or discrete time or stochastic models.

Model exploration

Homework

• If you haven’t already, install DSAIRM and take a look at the Basic Virus Model and Basic Bacteria Model apps.
• Start looking at and going through the “What to do” tasks. You can try yourself first (better for learning), or look at the solutions right away (only suggested if you feel completely lost).
• Take an especially close look at the solution for task 6 in the Basic Bacteria app, we will use a similar approach tomorrow.
• Links to solution files are in the schedule of the class website.