• Infectious diseases operate on different temporal and spatial scales.
• Building models that connect scales can allow one to answer new questions.

• Static: A within-host model is analyzed/simulated. Results are being fed into a between-host model, which is subsequently being run.
• Dynamic: A within-host model is being simulated inside a between-host model. Requires an ABM for the between-host model, each agent has its own infection model running.

It is easiest to discuss multi-scale models in the context of an example. Let’s consider spread of an acute viral infection (e.g. influenza) at the within-host and the population level.

At the within host level, we can start with the basic virus model.

\[ \begin{aligned} \dot{U} & = n - d_UU - bUV \\ \dot{I} & = bUV - d_I I \\ \dot{V} & = pI - d_V V - gb UV \\ \end{aligned} \]

• At the population level, we’ll look at the standard SIR model, with compartments being susceptible, infected and infectious, and recovered.
• To avoid confusion, we give all the parameters on the population level model Greek letters.

\[ \begin{aligned} \dot S & = \nu - \beta SI - \mu S \\ \dot I & = \beta S I - \gamma I - \mu I \\ \dot R & = \gamma I - \mu R \\ \end{aligned} \]

• Assume transmission rate is linked to virus load, e.g. \(\beta = kV\), with \(k\) some parameter.

\[ \begin{aligned} \dot S & = \nu - {\bf kV} SI - \mu S \\ \dot I & = {\bf kV} S I - \gamma I - \mu I \\ \dot R & = \gamma I - \mu R \\ \end{aligned} \]

Now the between-host model is connected to the within-host model through the variable \(V\).

• For a chronic infection model, we can compute \(V\) at steady state as function of model parameters. \[V = \frac{n(p-d_Ig)}{d_Id_V}-\frac{d_U}{b}\]
• Changes in the within-host parameters now impact the between-host dynamics.
• A similar model could be made that computes total virus load for an acute infection, and assumes this to be proportional to \(\beta\) (Handel et al. 2013).

• We could also assume that the duration of the infectious period, \(1/\gamma\) is determined by the time \(V\) in the within host model drops below a certain level.
• To investigate this:
  • Set within-host model parameters. Run model. Determine time at which \(V<1\) from time-series.
  • Use that time as \(1/\gamma\) in the between-host model.
• This approach could be done static (compartmental), or dynamic (ABM).

• We could now answer questions such as: Does increased virus infection (parameter \(b\)) lead to more spread on the population level? If we assume link through \(\beta\) and/or \(\gamma\).
• For a chronic infection, we can see it from the equation: \[V = \frac{n(p-d_Ig)}{d_Id_V}-\frac{d_U}{b}\]
• For an acute infection, we would need to run simulations.

• So far, we assumed that the lower scale (within-host) affects the higher scale (between-host).
• One could also consider the population level dynamics to impact the within-host level. E.g. if we had a new (flu) strain spreading on the population level which can partially avoid pre-existing immunity, it might impact the within-host dynamics.
• It gets complicated. One either needs to break down the pieces and look at them individually, or put them all in one large simulation.

Does low-temperature environmental persistence versus high-temperature within-host persistence pose a potential trade-off for avian influenza (Handel et al. 2013, 2014)?

Connect a within-host model and a population level model. Explore how different decay rates at different temperatures affect overall virus fitness.

How does drug resistance emergence within an HIV infected individual impact the population level dynamics (Saenz and Bonhoeffer 2013)?

• Virus infection within-host model with drug sensitive and resistant strain and drug treatment.
• The epi model parameters for infection duration and transmission rate are linked to virus load.

Infected cases for different levels of treatment (color) and without (left) and with (right) within-host drug resistance.

A fully dynamic multi-scale model for influenza (Lukens et al. 2014).

These review papers can provide a good further introduction to the topic: (Childs et al. 2019; Garira 2017; Mideo, Alizon, and Day 2008; Murillo, Murillo, and Perelson 2013; Handel and Rohani 2015)