# Modeling influenza at ECMTB/SMB 2016

This week, I am at the University of Nottingham for the joint meeting of the Society of Mathematical Biology and the European Conference on Mathematical and Theoretical Biology — ECMTB/SMB 2016. It is a huge meeting, with over 800 delegates in attendance, 308 half-hour mini-symposium talks, 264 twenty-minute contributed talks, 190 posters, 7 prize talks, 7 plenary talks, and 1 public lecture. With seventeen to eighteen sessions running in parallel, it is impossible to see more than a tiny fraction of the content. And impossible for me to give you a comprehensive account of the event. However, I did want to share some moments from this week. If you are at ECMTB and want to share some of your highlights for TheEGG then let me know, and we can have you guest post.

I did not come to Nottingham alone. Above is a photo of current/recent Moffitteers that made their way to the meeting this year.

On the train ride to Nottingham, I needed to hear some success stories of mathematical biology. One of the ones that Dan Nichol volunteered was the SIR-model for controlling the spread of infectious disease. This is a simple system of ODEs with three compartments corresponding to the infection status of individuals in the population: susceptible (S), infectious (I), recovered (R). It is given by the following equations

\begin{aligned} \dot{S} & = - \beta I S \\ \dot{I} & = \beta I S - \gamma I \\ \dot{R} & = \gamma I, \end{aligned}

where $\beta$ and $\gamma$ are usually taken to be constants dependent on the pathogen, and the total number of individuals $N = S + I + R$ is an invariant of the dynamics.

As the replicator dynamics are to evolutionary game theory, the SIR-model is to epidemiology. And it was where Julia Gog opened the conference with her plenary on the challenges of modeling infectious disease. In this post, I will briefly touch on her extensions of the SIR-model and how she used it to look at the 2009 swine flu outbreak in the US.

### Modeling cross-immunity

Gog praised the simplicity of the SIR-model. This simplicity allows it to be used as a component that can be extended or dropped into more complicated settings. The first half of her talk focused on such extension for cross-immunity. For infectious diseases like measles, mumps, and rubella, it is fine to have lifetime resistance. But for influenza — the focus of Gog’s work — this is not a reasonable approximation. The prevalent influenza viruses keep evolving and shifting. Although an immunity to one version does not (necessarily) provide immunity to another, a prior immunity does often provide some cross-immunity to similar mutants.

By viewing models of influenza as extensions of the simple SIR-model, Gog was able to classify them into two broad categories. The first categories updated the population variables to reflect the antigen’s evolution. Instead of having a fixed definition (like “susceptible to H1N1”), S, I, and R became a reference frame that tracked the common strain (like “susceptible to the currently prevalent influenza”). This allows for minimal modifications like adding a small flow from R to S, or introducing seasonality where the last season’s $\lim_{t \rightarrow \infty} R(t)$ is transformed — according to some reasonable function — into this season’s S(0). This allows us to maintain a low number of variables but limits the amount of strain-specific information that we can use to parameterize the model.

The second category creates new variables for each circulating strain. At this point, R becomes a bit meaningless, and it starts to make sense to instead create many S that are indexed by the subsets of all strains L. With $S_K$ meaning that the individual has had each of the strains in $K \subseteq L$ but is still susceptible to the strains in $L - S$ (thus, R would become $S_L$. Further, the strength of susceptibility to each $r \in L - S$ — i.e. the rate of flow from $S_K$ to $I_{r,K}$ — can depend on the prior exposure K. If we then assume that the time one is infected is negligible compared to the time one is susceptible — a reasonable assumption given that we don’t spend a large fraction of our life down with the flu — then we can remove the I compartments completely, and just have monotonic flows on the hypercube of $S_K$ for $\emptyset \subseteq K \subseteq L$.

To my one track mind, this looks like a fitness landscape without sign epistasis where the relative fitness of genotypes $K$ and $K + r$ is given by the flow between $S_K$ and $S_{K + r}$. I wonder to what extent this would allow the reuse of SIR-techniques for modeling the spread of microbial resistance, with each strain of influenza being replaced by a drug-resistance within a given type of microbes. Since resistance is often constly, I would expect that this would introduce more epistasis into the landscape.

In the second half of the talk, Gog discussed the integration of SIR-models with public health data. In particular, she reminded the audience to waste less time on proving existence and uniqueness results for their equations, and more time working with biologists to obtain datasets and taking the time to visualize and analyze those datasets. It seems to me that her advice is for mathematical biologists to be data scientists. A wisdom that I can agree with.

In her case, Gog gained access to a dataset of a substantial fraction of US medical insurance claims by week by zip code (for more, see Gog et al., 2014). She focused in on the claims related to influenza-like illness (ILI), cleaned it up by removing known seasonal base rates of non-flu ILI (like common colds or allergies), and then visualized her estimates of influenza by city across the US during the 2009 influenza pandemic (swine flu):

The striking feature of this data is the slow radial spread (taking around 3 months) from the first cases in the South-East US. Using this data to parametrize a spatialized SIR model, Gog could conclude that the spread was well explained by short-distance interaction. This stood in contrast to the common wisdom that pandemic spread by long-distance interactions (i.e. air travel), environmental factors, and population size. She also stressed the importance of the school term to the disease transmission. I guess that this should reassure the ECMTB/SMB delegates that our conference is not acting as a hub for influenza jumps.

I wonder if similar datasets could be gathered for microbial resistance. For instance, by tracking which antibiotics doctors write for a given patient and assuming that a new drug following quickly an old one means a resistance to the former. This would give an interesting extra scale to a fascinating evolutionary problem, and a fun machine learning puzzle to play with.

Gog, J.R., Ballesteros, S., Viboud, C., Simonsen, L., Bjornstad, O.N., Shaman, J., Chao, D.L., Khan, F., & Grenfell, B.T. (2014). Spatial Transmission of 2009 Pandemic Influenza in the US. PLoS Computational Biology, 10 (6) PMID: 24921923