# Introduction¶

Since the emergence of the COVID-19 pandemic in early 2020, epidemiological modelling has seen a considerable boost in interest. Researchers, journalists, and laymans alike have since been working on a large number of different models that are supposed to either predict case numbers on a short time scale or to discuss the implications of various containment/mitigation scenarios for longer time frames.

The complexity of such models often mitigates simple replication and/or adaption to local circumstances. Typically, such modifications are quickly thought up, but their influence on system properties such as the epidemic threshold are less clear, hence new evaluations have to be performed and new numerical solutions have to be implemented for every model iteration.

epipack aims at solving this issue by providing a simple, process-based framework to quickly prototype any compartmental epidemiological model and to investigate their implications based on analytical, numerical, stochastical, and agent-based or network-based simulations. The research process is further facilitated by a one-size-fits-all visualization framework and interactive analysis routines that give immediate visual feedback regarding a system's inner workings.

In this documentation, we will define general Markovian compartmental models and demonstrate how specific models can be constructed and studied using epipack.

## Constant Rates¶

Simple epidemiological models are usually based on the assumption that two individuals, an infected (I) and a susceptible (S), can interact in a way that transmits the infection from the infected to the susceptible individual. In a no-memory (Markovian) picture, such a process can be defined by means of chemical reactions, such as

$S + I \stackrel{\alpha}{\longrightarrow} I + I$

which formalizes what we said above: a contact between an S particle (individual) and an I particle (individual) leads to the decay of the S particle to an I particle. This reaction takes place with rate $$\alpha > 0$$.

Additionally, individuals are assumed to spontaneously recover, i.e. an I particle decays to an R particle with rate $$\beta$$, such that

$I \stackrel{\beta}{\longrightarrow} R.$

S, I, and R are usually referred to as compartments which individuals can be part of, which means that S quantifies the number (or fraction, depending on the definition) of susceptibles in the population, etc.

Sometimes, models also assume that previously non-existent susceptible individuals are born with constant rate $$\gamma$$ as

$\varnothing \stackrel{\gamma}{\longrightarrow} S.$

These reaction equations can be formalized in a system of ODEs

$\begin{split}\frac{d}{dt}S &= -\alpha SI/N + \gamma\\ \frac{d}{dt}I &= \alpha SI/N - \beta I\\ \frac{d}{dt}R &= \beta I\\ N &= S + I + R.\end{split}$

Note that this is a system of constantly growing population size which is an usual situation but for the sake of having a non-trivial example we will keep it like that.

The ODEs above could now be implemented in an ODE solver and numerically integrated or investigated analytically with either pen and paper or computer algebra systems.

While reaction equations such as the ones above are often used, specific situations may need careful adjustment of the basic reactions, e.g. by introducing compartments for asymptomatic infectious or compartments for quarantined individuals. Such adaptions would then have to be translated into new ODE systems which takes time to set up and debug. In principle, the following reaction equations describe all possible epidemiological models

$\begin{split}Y_i + Y_j &\stackrel{\alpha_{ijk\ell}}{\longrightarrow} Y_k + Y_\ell\\ Y_i &\stackrel{\beta_{ij}}{\longrightarrow} Y_j\end{split}$

where $$Y_i$$ is any of $$C$$ compartments and $$Y_i = \varnothing$$ is a valid choice in any of the reaction equations. These reaction equations are still somewhat restrictive, considering that in odd cases, asymmetric reactions are mathematically allowed. For constant rates, a generalized deterministic epidemiological model can therefore be defined based on a system of second-order coupled ordinary differential equations (ODEs)

$\frac{d}{dt}Y_i = \sum_{j,k} \alpha_{ijk} Y_jY_k/N + \sum_j \beta_{ij} Y_j + \gamma_i$

where the population of size N is assumed to be sorted into C compartments $$Y_1, Y_2, ..., Y_C$$ such that

$N = \sum_{i=1}^C Y_i.$

epipack allows one to set up systems like these explicitly. For constant rates, one may use epipack.numeric_epi_models.EpiModel, epipack.symbolic_epi_models.SymbolicEpiModel, epipack.numeric_matrix_epi_models.MatrixEpiModel, or epipack.symbolic_matrix_epi_models.SymbolicMatrixEpiModel.

While EpiModel and SymbolicEpiModel are event-based implementations that allow for stochastic mean-field simulations, too, they can be slow to set up and to run for increasingly complex systems. Hence, if you're dealing with constant-rate systems of a large number of compartments/couplings, you may fall back to MatrixEpiModel or SymbolicMatrixEpiModel which are defined based on sparse matrix implementations and therefore faster to both set up and for numeric integrations. Yet, mean-field stochastic simulations only work with the first two base models.

## Functional Rates¶

In general, we do not have to assume that rates are constant. They can depend both on the current system state as well as on time explicitly.

The generalized Markovian system therefore reads

$\frac{d}{dt}Y_i = \sum_{j,k} \alpha_{ijk}(t,Y_1,Y_2,...) Y_jY_k/N + \sum_{j} \beta_{ij}(t,Y_1,Y_2,...\}) Y_j + \gamma_i(t,Y_1,Y_2,...).$

Such systems can be set up and analyzed analytically, numerically, or based on mean-field stochastic simulations with epipack.numeric_epi_models.EpiModel, epipack.symbolic_epi_models.SymbolicEpiModel.