A time-since-infection model for populations with two pathogens

The pioneering work of Kermack and McKendrick (1927, 1932, 1933) is now most known for introducing the SIR model, which divides a population into discrete compartments for susceptible, infected and removed individuals. The SIR model is the archetype of widely used compartmental models for epidemics. It is sometimes forgotten, that Kermack and McKendrick introduced the SIR model as a special case of a more general framework. This general framework distinguishes individuals not only by whether they are susceptible, infected or removed, but additionally tracks the time passed since they got infected. Such time-since-infection models can mechanistically link within-host dynamics to the population level. This allows the models to account for more details of the disease dynamics, such as delays of infectiousness and symptoms during the onset of an infection. Details like this can be vital for interpreting epidemiological data. The time-since-infection framework was originally formulated for a host population with a single pathogen. However, the interactions of multiple pathogens within hosts and within a population can be crucial for understanding the severity and spread of diseases. Current models for multiple pathogens mostly rely on compartmental models. While such models are relatively easy to set up, they do not have the same mechanistic underpinning as time-since-infection models. To approach this gap of connecting within-host dynamics of multiple pathogens to the population level, we here extend the time-since-infection framework of Kermack and McKendrick for two pathogens. We derive formulas for the basic reproduction numbers in the system. Those numbers determine whether a pathogen can invade a population, potentially depending on whether the other pathogen is present or not. We then demonstrate use of the framework by setting up a simple within-host model that we connect to the population model. The example illustrates the context-specific information required for this type of model, and shows how the system can be simulated numerically. We verify that the formulas for the basic reproduction numbers correctly specify the invasibility conditions.