Stage structured hybrid model

Stochasticity and spatial distribution of the pathogen play a very critical role in determining the outcome of an infection. 1 in 10⁶ B-cells are specific to a particular pathogen. The serendipitous encounter of such a rare cognate B-cell with its fated antigen can determine host mortality. Mosquito vectors inject an average of 10⁵ PFU of WNV into an animal however there is a lot of variation around this mean. If a mosquito injects into a vein, the pathogen can spread systemically instead of being localized in tissue, leading to faster progression disease progression but possibly faster recognition by immune system cells. If a mosquito only injects into tissue, the pathogen will remain localized in a small volume of tissue and will probably be able to evade immune recognition while proliferating.

Such stochastic and spatial aspects of pathogenesis likely play a role in other diseases also. For example, macaques experimentally inoculated with HIV became infected with a very low probability in a dose dependent manner suggesting the role of initial stochastic events in shaping the trajectory of pathogenesis.

Current efforts at investigating the effect of stochasticity and space in modeling of host immune response and pathogens uses agent based models (ABMs). An ABM represents each entity or agent (each cell or virion in our case) explicitly, and a computer program encodes each rule or behavior for interacting with other agents. The agents move about in space and interact with other agents in their neighborhood according to the encoded rules. ABMs emphasize local interactions based on first principles, and these interactions give rise to the complex high-level phenomena of interest.

Due to the level of detail at which individual components are represented, ABMs can be computationally expensive and sometimes intractable. Population level approaches like ordinary differential equations (ODEs) are computationally tractable (paper) and can scale up to simulate host pathogen dynamics in large organisms (paper). However they make simplifying assumptions. For example they subsume individuals into a homogeneous compartment. They also assume that populations are homogeneously mixed. For example, the implicit assumption is that at initialization, a population of injected virions and normal cells would be “well-mixed” i.e. each virion has the opportunity to come in contact with every normal cell. This is unlikely to be satisfied during the initial stage of infection, when inoculated virions localize at the site of infection. Such spatial effects assume more importance during the onset of infection, when the number of virions is low, and we need an ABM to address this.

We proposed an approach that aims to strike a balance between the detail of representation of an ABM and the computational tractability of an ODE model. We call this a stage-structured hybrid model (paper). It uses a detailed and spatially explicit, but computationally intensive ABM in the initial stage of infection, and a coarse-grained but computationally tractable ODE model in the latter stages of infection (when the assumptions of homogeneous mixture of population are likely to satisfied and spatial effects can be ignored). Read more about it in the published paper here.

Such an approach might hold promise in modeling of other pathogens where the initial stochasticity of the pathogen and host response dictates the trajectory of pathogenesis. A general schematic of the approach is shown below