Fabio Sanchez

Drinking as an Epidemic—A Simple Mathematical Model with Recovery and Relapse

Publisher Summary This chapter discusses a simple mathematical model of drinking lapse. Problem drinking is modeled as an acquired state - the result of frequent or intense interactions among individuals in three drinking states within a specified drinking environment. The goal of the model is to identify mechanisms that facilitate or limit the conversion of a population of nondrinkers to one of drinkers within prespecified environments. The process of quantification helps to understand the role of social forces on the time evolution of drinking. The dynamics of the model support two distinct states. The nature of these distinct outcomes depends in general on the size of the initial proportion of drinkers, the overall average residence time in the drinking environment, and the intensity of the interactions between problem drinkers and the rest of the residents. Numerical simulations are used to illustrate the model results on drinking dynamics. It is found that the most general model can support two permanent prevalent states when treatment only has short-term effect and relapse rates are high.

The Role of Nonlinear Relapse on Contagion Amongst Drinking Communities

Relapse, the recurrence of a disorder following a symptomatic remission, is a frequent outcome in substance abuse disorders. Some of our prior results suggested that relapse, in the context of abusive drinking, is likely an “unbeatable” force as long as recovered individuals continue to interact in the environments that lead to and/or reinforce the persistence of abusive drinking behaviors. Our earlier results were obtained via a deterministic model that ignored differences between individuals, that is, in a rather simple “social” setting. In this paper, we address the role of relapse on drinking dynamics but use models that incorporate the role of “chance”, or a high degree of “social” heterogeneity, or both. Our focus is primarily on situations where relapse rates are high. We first use a Markov chain model to simulate the effect of relapse on drinking dynamics. These simulations reinforce the conclusions obtained before, with the usual caveats that arise when the outcomes of deterministic and stochastic models are compared. However, the simulation results generated from stochastic realizations of an “equivalent” drinking process in populations “living” in small world networks, parameterized via a disorder parameter p, show that there is no social structure within this family capable of reducing the impact of high relapse rates on drinking prevalence, even if we drastically limit the interactions between individuals (p ≈ 0). Social structure does not matter when it comes to reducing abusive drinking if treatment and education efforts are ineffective. These results support earlier mathematical work on the dynamics of eating disorders and on the spread of the use of illicit drugs. We conclude that the systematic removal of individuals from high risk environments, or the development of programs that limit access or reduce the residence times in such environments (or both approaches combined) may reduce the levels of alcohol abuse.

Vertical Transmission in a Two-Strain Model of Dengue Fever

Abstract The role of vertical transmission in vectors has rarely been addressed in the study of dengue dynamics and control, in part because it was not considered a critical population-level factor. In this paper, we apply the pioneering modeling ideas of Ross and MacDonald, motivated by the context of the 2000–2001 dengue outbreak in Peru, to assess the dynamics of multi-strain competition. An invading strain of dengue virus (DENV-2) from Asia rapidly circulated into Peru eventually displacing DENV-2 American. A host-dengue model that considers the competing dynamics of these two DENV-2 genotypes, the resident or the American type and the invasive more virulent Asian strain, is introduced and analyzed. The model incorporates vertical transmission by DENV-2 Asian a potentially advantageous trait. Conditions for competitive exclusion of dengue strains are established. The model is used to show that lower transmission rates of DENV-2 Asian are sufficient for displacing DENV-2 American in the presence of vertical transmission.