Mathematical models to study the biology of pathogens and the infectious diseases they cause

Abstract

Mathematical models have many applications in infectious diseases: epidemiologists use them to forecast outbreaks and design containment strategies; systems biologists use them to study complex processes sustaining pathogens, from the metabolic networks empowering microbial cells to ecological networks in the microbiome that protects its host. Here, we (1) review important models relevant to infectious diseases, (2) draw parallels among models ranging widely in scale. We end by discussing a minimal set of information for a model to promote its use by others and to enable predictions that help us better fight pathogens and the diseases they cause.

Keywords: Computer modeling; Infection control in health technology; Microbiology.

Graphical abstract

Figure 1

Mathematical models in infectious diseases study the interaction between elements defined at a range of scales, from the molecules inside a pathogen’s cell all the way to the people infected by pathogen as the disease spreads across a population

Figure 2

The SIR model is a simple mathematical model of epidemics (A) In its classic form, the SIR model considers the individuals in three states: susceptible, infectious, and recovered. (B) Individuals transition between the states and those state transitions model the processes of infection and recovery. Their dynamics assume mass-action kinetics. (C) Despite its simplicity, the model can be used to study real-world scenarios such as outbreaks, and compare the outcome of interventions to slow an epidemic such as social distancing. A variation of the model that includes an additional state (“immunized”) and a new process of immunization that converts individuals to a new state could be used to predict how a vaccination campaign can contribute to stopping the spread. (D) The classical assumption of mass action assumes that encounters are random in the population. (E) Expansions of the SIR model include spatial structure and other refinements to increase the realism of the model and the accuracy of the predictions.

Figure 3

Genome-scale models provide a way to study how a pathogen can thrive in different environments, for example by using the nutrients available in the various organs of its human host (A) A model describes the intracellular metabolism of the pathogen as a network of metabolites connected by biochemical reactions. (B) Each reaction has a stoichiometry and a rate. (C) Flux balance analysis (FBA) assumes that the pathogen is in balanced growth, which means that the concentration of each intracellular metabolite is in a steady state and its time derivative equals 0. Balanced growth is a simplifying assumption that turns a system of differential equations into a system of algebraic linear equations. The system is then solved using FBA by adding experimental constraints and an objective function, such as the biomass equation. (D) A high-quality genome-scale network can describe a reference strain with a well annotated genome, and be used to predict its growth phenotypes. (E) The model of the reference strain may also be customized to model other closely related strains. A set of network models can be used to compare the strains' abilities to use different nutrients.

Figure 4

Lotka-Volterra equations model ecosystem dynamics in a multispecies system like the gut microbiome (A) The dynamics of predator and prey species in animal ecosystems inspired the Lotka-Volterra model, a system of two coupled ordinary differential equations that assumes that the law of mass action applies. The right-hand side of each differential equation has terms that represent the gains (sources) or losses (sinks) of each species. (B) The Lotka-Volterra system can be generalized to any number of species (n). (C) Network models inferred from timeseries data can be used to study the relationship between network structure and function in the gut microbiome. One application is in finding gut bacteria such as Clostridium scindens that inhibit the growth of pathogens like Clostridioides difficile.