I develop and analyze mathematical and computational models to examine biologically-motivated questions. In addition, the examination of biological problems often leads to rich quantitative challenges, which require the development of novel mathematics, models and quantitative tools.

My current and past research includes mathematical and computational models using diverse techniques with wide applications including infectious diseases, immunology, and ecology. A main focus of my work is understanding the pathogenesis and spread of infectious diseases, particularly those spread by the mosquito, such as malaria and dengue. I consider the interactions within a host organism, such as between an invading pathogen and the immune response, and how these within-host interactions impact transmission of disease throughout a population. 

Analysis of Systems of Differential Equations

Systems of differential equations are an ideal way to describe the average behavior of many biological systems. Assessing the long term dynamics, via stability theory, bifurcation theory, and asymptotics, still allows for description and understanding of dynamics when working with fields that are difficult to assess experimentally. I have analyzed solutions to a multi-scale partial differential equation (PDE) system of dengue dynamics (Nikin-Beers et. al., 2018), to the forced Kuramoto model (Childs and Strogatz, 2008), and to an ODE model of the activation of an immune response (Childs et. al., 2011).

Numerical Simulation of Systems of Differential Equations

Often the system of ODEs from biologically motivated problems becomes too unwieldy to solve exactly. In such cases, numerical solutions provide insight into dynamics of interest. I have performed extensive numerical work including the malaria parasite development within the mosquito (Childs and Prosper, 2017), the role of density effects on transgenic mosquitoes (Walker et. al., 2018), immune cell activation (Childs et. al., 2011), the role of interconnected populations in Ebola transmission dynamics (Blackwood and Childs, 2016), the impact of novel mosquito interventions in malaria dynamics (Shaw et. al., 2016), malaria parasite dynamcis in red blood cells (Archer et. al., 2018), and the role of sexual transmission in Zika dynamics (Maxian, Neufeld and Talis et. al., 2017). In such cases, I use numerical solvers to simulate solutions to the systems of equations in parameter regimes of interest. When parameters are uncertain, I performed extensive sensitivity analysis. For these analyses I routinely develop my own code, which I share with my publications.

Stochastically Interrupted Systems

Traditionally ODE systems are viewed as structurally static with a pre-defined number of variables, each of which describes a single entity. In biological systems, these entities, often representing individuals or species, are constantly evolving. When new variants of individuals appear, it is necessary to include new variables in the model; likewise, when variants go extinct, variables must be removed. Such a scenario arose naturally when tracking the coevolution of bacteria and viruses that attack them and in extensions ((Childs et. al., 2012; Held et. al. 2013; Childs and England et. al., 2014). In order to account for the addition and removal of bacterial and viral strains a novel implementation was required: the static ODE system was solved numerically for short time periods between the discrete events of birth and death of strains, which occurred with stochastically determined intervals.

Discrete Deterministic Systems

Although ODEs are important tools for the study of biological systems, at times the inherent continuous nature of change can be in contrast to the punctuated advance of life, such as the burst of viruses from a cell or the emergence of insects from larva. Such scenarios are better described by difference equations where events occur instantaneously, and in the simplest case at fixed intervals. I used discrete framework to examine two aspects of the malaria parasite life cycle. The first involved tracking within-host dynamics of interactions of the parasite with the human immune response (Childs and Buckee, 2015; Coleman et. al., 2014; Nilsson, Childs et. al., 2016). A discrete model was ideal as the parasite bursts from its home in human red blood cells every forty-eight hours. The second model examined the dynamics of the mosquito life cycle ( Childs, Cai, Kakani et. al., 2016; Paton et al., 2019). As mosquitos are generally most active for a few hours each evening, a discrete model accounts for the daily time lag between events such as feeding and mating.

Discrete Stochastic Systems

Stochastic models are useful when there is heterogeneity in the system of interest. By allowing for random variation in inputs through time, the likelihood of outcomes can be estimated. Sequence evolution is a natural place to use stochastic models, and I examined such evolution in the context of malaria parasite proliferation in the mosquito (Prosper and Childs, 2017) and in the generation and maturation of antibodies during an immune response (Childs et. al., 2015).

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