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Mathematical Biosciences & Engineering

2006 , Volume 3 , Issue 3

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From the guest editors
Abba Gumel and  James Watmough
2006, 3(3): i-ii doi: 10.3934/mbe.2006.3.3i +[Abstract](30) +[PDF](547.5KB)
In September of 2003, on the tail of the outbreak of Severe Acute Respiratory Syndrome (SARS) in the Greater Toronto Area, MITACS (, one of Canada's Networks of Centres of Excellence, teamed up with Health Canada in funding a workshop entitled the MITACS-PIMS-Health Canada meeting on SARS at the magnificent Banff International Research Station (BIRS) . A follow-up summer school and workshop were held the following year, at the same venue, under the auspices of the PIMS-MITACS-MSRI Special Program on Infectious Diseases, June 19 to July 02, 2004. Admission into the graduate summer school was quite competitive. Forty three students (10 from USA, 32 from Canada and 1 from China, as part of the MITACS exchange program with the Chinese Ministry of Higher Education) were selected.
Stochastic epidemic models with a backward bifurcation
Linda J. S. Allen and  P. van den Driessche
2006, 3(3): 445-458 doi: 10.3934/mbe.2006.3.445 +[Abstract](50) +[PDF](264.3KB)
Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region of bistability exists in a subset of parameter space. The dynamics of the stochastic epidemic models are investigated in this region of bistability, and compared with those of the deterministic model. In this region the probability distribution associated with the infective population exhibits bimodality with one mode at the disease-free equilibrium and the other at the larger endemic equilibrium. For population sizes $N\geq 1000$, the deterministic and stochastic models agree, but for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.
Modeling the potential impact of rectal microbicides to reduce HIV transmission in bathhouses
Romulus Breban , Ian McGowan , Chad Topaz , Elissa J. Schwartz , Peter Anton and  Sally Blower
2006, 3(3): 459-466 doi: 10.3934/mbe.2006.3.459 +[Abstract](21) +[PDF](162.7KB)
We evaluate the potential impact of rectal microbicides for reducing HIV transmission in bathhouses. A new mathematical model describing HIV transmission dynamics among men who have sex with men (MSM) in bathhouses is constructed and analyzed. The model incorporates key features affecting transmission, including sexual role behavior (insertive and receptive anal intercourse acts), biological transmissibility of HIV, frequency and efficacy of condom usage, and, most pertinently, frequency and efficacy of rectal microbicide usage. To evaluate the potential impact of rectal microbicide usage, we quantify the effect of rectal microbicides (ranging in efficacy from 10% to 90%) on reducing the number of HIV infections in the bathhouse. We conduct uncertainty analyses to assess the effect of variability in both biological and behavioral parameters. We find that even moderately effective rectal microbicides (if used in 10% to 50% of the sex acts) would substantially reduce transmission in bathhouses. For example, a 50% effective rectal microbicide (used in 50% of sex acts) would reduce the number of secondary infections by almost 13% at disease invasion. Our modeling analyses show that even moderately effective rectal microbicides could be very effective prevention tools for reducing transmission in bathhouses and also potentially limit the spread of HIV in the community.
The influence of infectious diseases on population genetics
Zhilan Feng and  Carlos Castillo-Chavez
2006, 3(3): 467-483 doi: 10.3934/mbe.2006.3.467 +[Abstract](36) +[PDF](434.1KB)
Malaria is the vector-transmitted disease that causes the highest morbidity and mortality in humans. Motivated by the known influence of sickle-cell anemia on the morbidity and mortality of malaria-infected humans, we study the effect of malaria on the genetic composition of a host (human) population where sickle-cell anemia is prevalent and malaria is endemic. The host subpopulations are therefore classified according to three genotypes, $A$$A$, $AS$, and $SS$. It is known that $A$$A$ malaria-infected individuals experience higher malaria-induced mortality than $AS$ or $SS$ individuals. However, individuals carrying the $S$ gene are known to experience a higher mortality rate in a malaria-free environment than those who lack such a gene. The tradeoffs between increased fitness for some types in the presence of disease (a population level process) and reduced fitness in a disease-free environment are explored in this manuscript. We start from the published results of an earlier model and proceed to remove some model restrictions in order to better understand the impact on the natural hosts' genetics in an environment where malaria is endemic.
An sveir model for assessing potential impact of an imperfect anti-SARS vaccine
Abba B. Gumel , C. Connell McCluskey and  James Watmough
2006, 3(3): 485-512 doi: 10.3934/mbe.2006.3.485 +[Abstract](42) +[PDF](289.5KB)
The control of severe acute respiratory syndrome (SARS), a fatal contagious viral disease that spread to over 32 countries in 2003, was based on quarantine of latently infected individuals and isolation of individuals with clinical symptoms of SARS. Owing to the recent ongoing clinical trials of some candidate anti-SARS vaccines, this study aims to assess, via mathematical modelling, the potential impact of a SARS vaccine, assumed to be imperfect, in curtailing future outbreaks. A relatively simple deterministic model is designed for this purpose. It is shown, using Lyapunov function theory and the theory of compound matrices, that the dynamics of the model are determined by a certain threshold quantity known as the control reproduction number ($\R_{v}$). If $\R_{v}\le 1$, the disease will be eliminated from the community; whereas an epidemic occurs if $\R_{v}>1$. This study further shows that an imperfect SARS vaccine with infection-blocking efficacy is always beneficial in reducing disease spread within the community, although its overall impact increases with increasing efficacy and coverage. In particular, it is shown that the fraction of individuals vaccinated at steady-state and vaccine efficacy play equal roles in reducing disease burden, and the vaccine must have efficacy of at least 75% to lead to effective control of SARS (assuming $\R=4$). Numerical simulations are used to explore the severity of outbreaks when $\R_{v}>1$.
Global dynamics of a staged progression model for infectious diseases
Hongbin Guo and  Michael Yi Li
2006, 3(3): 513-525 doi: 10.3934/mbe.2006.3.513 +[Abstract](35) +[PDF](232.0KB)
We analyze a mathematical model for infectious diseases that progress through distinct stages within infected hosts. An example of such a disease is AIDS, which results from HIV infection. For a general $n$-stage stage-progression (SP) model with bilinear incidences, we prove that the global dynamics are completely determined by the basic reproduction number $R_0.$ If $R_0\le 1,$ then the disease-free equilibrium $P_0$ is globally asymptotically stable and the disease always dies out. If $R_0>1,$ $P_0$ is unstable, and a unique endemic equilibrium $P^*$ is globally asymptotically stable, and the disease persists at the endemic equilibrium. The basic reproduction numbers for the SP model with density dependent incidence forms are also discussed.
Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs
Robert G. McLeod , John F. Brewster , Abba B. Gumel and  Dean A. Slonowsky
2006, 3(3): 527-544 doi: 10.3934/mbe.2006.3.527 +[Abstract](47) +[PDF](285.4KB)
This paper presents a statistical study of a deterministic model for the transmission dynamics and control of severe acute respiratory syndrome (SARS). The effect of the model parameters on the dynamics of the disease is analyzed using sensitivity and uncertainty analyses. The response (or output) of interest is the control reproduction number, which is an epidemiological threshold governing the persistence or elimination of SARS in a given population. The compartmental model includes parameters associated with control measures such as quarantine and isolation of asymptomatic and symptomatic individuals. One feature of our analysis is the incorporation of time-dependent functions into the model to reflect the progressive refinement of these SARS control measures over time. Consequently, the model contains continuous time-varying inputs and outputs. In this setting, sensitivity and uncertainty analytical techniques are used in order to analyze the impact of the uncertainty in the parameter estimates on the results obtained and to determine which parameters have the largest impact on driving the disease dynamics.
Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000
Brandy Rapatski , Petra Klepac , Stephen Dueck , Maoxing Liu and  Leda Ivic Weiss
2006, 3(3): 545-556 doi: 10.3934/mbe.2006.3.545 +[Abstract](46) +[PDF](216.4KB)
The dynamics of HIV/AIDS epidemics in a specific region is de- termined not only by virology and virus transmission mechanisms, but also by region's socioeconomic aspects. In this paper we study the HIV transmission dynamics for Cuba. We modify the model of de Arazoza and Lounes [1] accord- ing to the background about the virology and the socioeconomic factors that affect the epidemiology of the Cuban HIV outbreak. The two main methods for detection of HIV/AIDS cases in Cuba are ''random'' testing and contact tracing. As the detection equipment is costly and depends on biotechnological advances, the testing rate can be changed by many external factors. Therefore, our model includes time-dependent testing rates. By comparing our model to the 1986-2000 Cuban HIV/AIDS data and the de Arazoza and Lounes model, we show that socioeconomic aspects are an important factor in determining the dynamics of the epidemic.
A note on epidemic models with infective immigrants and vaccination
Eunha Shim
2006, 3(3): 557-566 doi: 10.3934/mbe.2006.3.557 +[Abstract](57) +[PDF](304.8KB)
The roles of immigration and vaccination on disease dynamics are explored in a simple setting that considers the possibility of conferred immunity. We focus on SIR and SIS models with a vaccinated class. Conditions for the existence of multiple endemic steady states and a fold bifurcation are discussed.

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