Jorge X. Velasco-Hernandez

A simple vaccination model with multiple endemic states.

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a diseases reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.

Extinction Thresholds and Metapopulation Persistence in Dynamic Landscapes

Models of metapopulations have focused on the effects of extinction and colonization rate upon metapopulation persistence and dynamics, assuming static landscapes wherein patches are neither created nor go extinct. However, for species living in ephemeral (patchy) habitats, landscapes are highly dynamic rather than static. In this article, we develop a lattice metapopulation model, of the patch occupancy type, based on interacting particle systems that incorporate explicitly both metapopulation and patch dynamics. Under this scenario, we study the effects of different regimes of patch dynamics upon metapopulation persistence. We analyze the lattice behavior by numerical simulations and a mean field approximation (MF). We show that metapopulation persistence and extinction are strongly influenced by the rate at which the landscape changes, in addition to the amount of habitat destroyed. We derive MF analytical expressions for extinction thresholds related to landscape properties such as habitat suitability and patch average lifetime. Using numerical simulations, we also show how these thresholds are quantitatively overestimated by the MF equations, although the qualitative behavior of the spatial model is well explained by the MF when the array of habitat patches is dynamic or static but connected in space and time. The implications for conservation are also discussed.

Climate, environmental and socio-economic change: weighing up the balance in vector-borne disease transmission

Arguably one of the most important effects of climate change is the potential impact on human health. While this is likely to take many forms, the implications for future transmission of vector-borne diseases (VBDs), given their ongoing contribution to global disease burden, are both extremely important and highly uncertain. In part, this is owing not only to data limitations and methodological challenges when integrating climate-driven VBD models and climate change projections, but also, perhaps most crucially, to the multitude of epidemiological, ecological and socio-economic factors that drive VBD transmission, and this complexity has generated considerable debate over the past 10–15 years. In this review, we seek to elucidate current knowledge around this topic, identify key themes and uncertainties, evaluate ongoing challenges and open research questions and, crucially, offer some solutions for the field. Although many of these challenges are ubiquitous across multiple VBDs, more specific issues also arise in different vector–pathogen systems.

Modelling the Effect of Treatment and Behavioral Change in HIV Transmission Dynamics

In this paper we analyze a model for the HIV-infection transmission in a male homosexual population. In the model we consider two types of infected individuals. Those that are infected but do not know their serological status and/or are not under any sort of clinical /therapeutical treatment, and those who are. The two groups of infectives differ in their incubation time, contact rate with susceptible individuals, and probability of disease transmission. The aim of this article is to study the roles played by detection and changes in sexual behavior in the incidence and prevalence of HIV. The analytical results show that there exists a unique endemic equilibrium which is globally asymptotically stable under a range of parameter values whenever a detection /treatment rate and an indirect measure of the level of infection risk are sufficiently large. However, any level of detection/ treatment rate coupled with a decrease of the transmission probability lowers the incidence rate and prevalence level in the population. In general, only significant reductions in the transmission probability (achieved through, for example, the adoption of safe sexual practices) can contain effectively the spread of the disease.

FEEDBACK CONTROL OF THE CHEMOTHERAPY OF HIV

Using a model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce a feedback control strategy of chemotherapy in an early treatment setting, where the control represents the percentage of effect chemotherapy has on the viral production. We seek to regulate the viral count by manipulating the percentage of effect chemotherapy has on the viral production. We show via numerical simulations that the proposed feedback control strategy can handle strong uncertainties in the HIV dynamics induced by imperfect modeling and sampled/delayed cell measurements.

Modeling Contact Structures in Biology

The transmission of diseases, genetic characteristics, or cultural traits is influenced by many factors including the contact/social structure of the interacting subpopulation, that is, the social environment. Classical demography (see MacKendrick, 1926; Lotka, 1922; and Leslie, 1945) ignores social dynamics and usually concentrates on the birth and death processes of female populations under the assumption that they have reached a stable age distribution. They usually ignore the specific mating/contact structure of the population. The incorporation of mating structures or marriage functions, as they are commonly referred to in human demography, was pioneered by Kendall (1949) and Keyfitz (1949). However, despite the fact that their work was extended by Parlett (1972), Predrickson (1971), McFarland (1972), and Pollard (1973) two decades ago, their impact on demography, epidemiology, and population biology has been minimal.

A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease

This work presents a new model for the linking of within- and between-host dynamics. We use this as a conceptual model for the dynamics of Toxoplasma gondii, in which the parasites life cycle includes interactions with the environment. We postulate the infection process to depend on the size of the infective inoculum that susceptible hosts may acquire by interacting with a contaminated environment. Because the dynamical processes associated with the within- and between-host occur on different time scales, the model behaviors can be analyzed by using a singular perturbation argument, which allows us to decouple the full model by separating the fast- and slow-systems. We define new reproductive numbers for the within-host and between host dynamics for both the isolated systems and the coupled system. Particularly, the reproduction number for the between-host (slow) system dependent on the parameters associated with the within-host (fast) system in a very natural way. We show that these reproduction numbers determine the stability of the infection-free and the endemic equilibrium points. Our model may present a so-called backward bifurcation.

Community treatment of HIV-1: initial stage and asymptotic dynamics

Treatment with antiviral drugs (zidovudine and ddI) has been reported to delay progression to AIDS, and may even possibly lower the infectiousness of the infectives. However, its effect on the community level is still uncertain. The latter is important since a successful community treatment program must meet both public health and individual health goals. Our study will focus on the effect of a community-wide treatment program initiated at the early stages of the disease as well as the long-term effect of the program. Using a simple mathematical model, we demonstrate that a community-wide treatment program could be instrumental in decreasing HIV incidence rate and eradicating the disease in the future if certain conditions on the parameters are met. On the other hand, when the above mentioned conditions on the parameters are not satisfied, we show that even if the treatment does improve survival in AIDS patients and decrease the rate at which HIV infection spreads in the community, it is still possible for the treatment program to have an adverse effect on the spread of AIDS in the population in the long run. Hence, a public health policy maker must exercise caution in order to design an effective treatment program for HIV/AIDS.

Facies Recognition Using Multifractal Hurst Analysis: Applications to Well-Log Data

Well-log (radioactivity, density and resistivity) analysis constitutes a standard approach for inferring lithology from wells. However, due to inherent complexity of the signals (such as highly heterogeneous deep-water sedimentary sequences) lithology recognition is not straightforward. We used a rescaled range analysis, calibrated with cores, to recognize lithological patterns from signal recorded along wireline logs. The detected intervals coincide with zones of visual electro-facies associations proposed by geologist well-log interpreters. In addition, we propose a rescaled range multifractal analysis to identify ranges of well-log signal complexities, which could be related to sedimentary process variations at specific stratigraphic order cycles.

Multistability in an open recruitment food web model

A general model for three species food web found in marine systems is analyzed. Following single-species approaches to model the dynamics of marine organisms, we relaxed the assumption of closed population structure and examine the behavior of webs in which some or all the component populations have completely open dynamics, that is, there is no connection between the arrival of new individuals and local reproductive output. Recent reviews have shown that coexistence among species with self-recruitment with those with completely open-recruitment is the norm in marine habitats. As part of a study of tri-trophic food web models with combinations of self-versus open-recruitment, we describe here the stability properties of a food web with omnivory where the basal and the intermediate predator species have open populations and the top predator reproduces locally. This system can have a maximum of four critical points and at least one corresponding to the omnivore-free equilibrium point. In this case our system reduces to a two level food web without omnivory. This equilibrium point has stability properties that depend on the capacity of invasion of the omnivore species. If the omnivore succeeds in invading the community then a three level food web can be established but with more complex stability properties. When the equilibrium point without the top predator is unstable, then there may exist three more critical points, at least one of which is asymptotically stable. The exact number of critical points that may exist depends on the food web parameter space. We speculate that the restrictive conditions for three-species stability could explain the scarcity of this particular combination of dispersal abilities in natural communities.