Mathematical Approaches to Infectious Disease Prediction and Control

Mathematics has long been an important tool for understanding and controlling the spread of infectious diseases. Here, we begin with an overview of compartmental models, the traditional approach to modeling infectious disease dynamics, and then introduce contact network epidemi- ology, a relatively new approach that applies bond percolation on random graphs to model the spread of infectious disease through heterogeneous populations. As we illustrate, these methods can be used to address public health challenges and have recently been coupled with powerful computational methods to optimize epidemic control strategies

Evolution of Genetic Potential

Organisms employ a multitude of strategies to cope with the dynamical environments in which they live. Homeostasis and physiological plasticity buffer changes within the lifetime of an organism, while stochastic developmental programs and hypermutability track changes on longer timescales. An alternative long-term mechanism is “genetic potential”—a heightened sensitivity to the effects of mutation that facilitates rapid evolution to novel states. Using a transparent mathematical model, we illustrate the concept of genetic potential and show that as environmental variability decreases, the evolving population reaches three distinct steady state conditions: (1) organismal flexibility, (2) genetic potential, and (3) genetic robustness. As a specific example of this concept we examine fluctuating selection for hydrophobicity in a single amino acid. We see the same three stages, suggesting that environmental fluctuations can produce allele distributions that are distinct not only from those found under constant conditions, but also from the transient allele distributions that arise under isolated selective sweeps

Optimal H1N1 vaccination strategies based on self-interest versus group interest

Background\ud Influenza vaccination is vital for reducing H1N1 infection-mediated morbidity and mortality. To reduce transmission and achieve herd immunity during the initial 2009-2010 pandemic season, the US Centers for Disease Control and Prevention (CDC) recommended that initial priority for H1N1 vaccines be given to individuals under age 25, as these individuals are more likely to spread influenza than older adults. However, due to significant delay in vaccine delivery for the H1N1 influenza pandemic, a large fraction of population was exposed to the H1N1 virus and thereby obtained immunity prior to the wide availability of vaccines. This exposure affects the spread of the disease and needs to be considered when prioritizing vaccine distribution.\ud \ud Methods\ud To determine optimal H1N1 vaccine distributions based on individual self-interest versus population interest, we constructed a game theoretical age-structured model of influenza transmission and considered the impact of delayed vaccination.\ud \ud Results\ud Our results indicate that if individuals decide to vaccinate according to self-interest, the resulting optimal vaccination strategy would prioritize adults of age 25 to 49 followed by either preschool-age children before the pandemic peak or older adults (age 50-64) at the pandemic peak. In contrast, the vaccine allocation strategy that is optimal for the population as a whole would prioritize individuals of ages 5 to 64 to curb a growing pandemic regardless of the timing of the vaccination program.\ud \ud Conclusions\ud Our results indicate that for a delayed vaccine distribution, the priorities that are optimal at a population level do not align with those that are optimal according to individual self-interest. Moreover, the discordance between the optimal vaccine distributions based on individual self-interest and those based on population interest is even more pronounced when vaccine availability is delayed. To determine optimal vaccine allocation for pandemic influenza, public health agencies need to consider both the changes in infection risks among age groups and expected patterns of adherence

Quasispecies Made Simple

Quasispecies are clouds of genotypes that appear in a population at mutation–selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. The literature on these important evolutionary processes is, however, quite challenging. Here we use simple models to link mutation–selection balance theory to the most novel property of quasispecies: the error threshold—a mutation rate below which populations equilibrate in a traditional mutation–selection balance and above which the population experiences an error catastrophe, that is, the loss of the favored genotype through frequent deleterious mutations. These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe

From Bad to Good: Fitness Reversals and the Ascent of Deleterious Mutations

Deleterious mutations are considered a major impediment to adaptation, and there are straightforward expectations for the rate at which they accumulate as a function of population size and mutation rate. In a simulation model of an evolving population of asexually replicating RNA molecules, initially deleterious mutations accumulated at rates nearly equal to that of initially beneficial mutations, without impeding evolutionary progress. As the mutation rate was increased within a moderate range, deleterious mutation accumulation and mean fitness improvement both increased. The fixation rates were higher than predicted by many population-genetic models. This seemingly paradoxical result was resolved in part by the observation that, during the time to fixation, the selection coefficient (s) of initially deleterious mutations reversed to confer a selective advantage. Significantly, more than half of the fixations of initially deleterious mutations involved fitness reversals. These fitness reversals had a substantial effect on the total fitness of the genome and thus contributed to its success in the population. Despite the relative importance of fitness reversals, however, the probabilities of fixation for both initially beneficial and initially deleterious mutations were exceedingly small (on the order of 10(−5) of all mutations)