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A Model for the Study of HIV/AIDS Brandy L. Rapatski Juan Tolosa Richard Stockton College of NJ Classic Model – Compartment Model S Susceptible I Infected R Recovered Removed/Recovered Class •Immunity (can’t infect or be infected) •Death •Isolation Assumptions An average infective makes contact sufficient to transmit infection with bN others per unit time. (N = total population). Probability a random contact by an infective is with a susceptible is S/N The number of new infections in unit time is bN (S/N)I = bSI There is no entry into or departure from the population except possibly through death from the disease. S dS bSI dt dI bSI I dt dR I dt I R Qualitative Approach dS bSI dt dI ( bS ) I dt dS 0 S(t) decreases for all time dt dI 0 if S , I(t) decreases to 0. dt b dI 0 if S , I(t) increases to maximum then decreases. dt b bS (0) threshold value Basic Reproduction Number bS (0) R0 The number of secondary infections caused by a single infective introduced into a wholly susceptible population over the course of the infection of this single infective. R0 1, epidemic dies out R0 1, there is an epidemic Progression of Infection by Stage We investigate how infectivity varies with stage of infection. 1/6 Year First Stage 7 Years Second Stage 3 Years Third Stage including AIDS Before modern medical intervention AIDS is roughly the last year of the third stage Death Progression of Infection by Stage an alternative set of durations Infectivity varies by stage of the disease 1/6 Year First Stage 7 Years Second Stage 2 Years AIDS & Death Third Stage* Before modern medical intervention *Here last year of AIDS is omitted due to sexual inactivity Untreated Viral Loads [------------------2nd Stage-------------------] [1st Stage] [-----3rd Stage----] Anderson, R. M. “The spread of HIV and sexual mixing patterns,” pp. 71-86 in Mann, J. and D. Tarantola (eds.), AIDS in the World II: Global Dimensions, Social Roots, and Responses. The Global AIDS Policy Coalition. (New York: Oxford University Press, 1996). Modes of HIV Transmission Sexual Contact IV Drug Use Vertical Transmission (Mother to Child) Blood Transfusion Modeling SF Gay Population: the Data HIV exploded in San Francisco between the years 1978 and 1984. San Francisco obtained high quality data on 6875 gay men: Infection rates and number of sexual contacts. Blood samples were taken and frozen during the years HIV was quietly breaking out – as part of a hepatitis vaccine study. San Francisco San Francisco Transmission Dynamics Analysis of 1978-1984 Six Activity Levels (from survey data) Infectiousness depends on stage (3 stages) Bathhouse Assumption Men vary in how often they visit the bathhouses but once inside choose partners at random. Model Compartments Individuals can be in any of four stages (including susceptible) and in any of six activity groups. So the model keeps track of fraction of people in each of the 4 x 6 subpopulations. Simple Transmission Dynamics dS bIS new case rate dt S fraction of susceptibl es, between 0 and 1 I fraction infected 1 S b effective contact rate Multiple Group Transmission Dynamics: from Group j to group i dSi bijIjSi dt rate of new group i cases caused by group j Si fraction of susceptibl es, between 0 and 1 Ij fraction infected 1 Sj bij effective contact rate, which equals numbers of contacts times infectious ness Go to Six-Group Models Six-Group Models 80 70 60 50 model 40 data 30 20 10 0 0 1 2 3 4 5 6 7 8 Current Work Test and Treat US MSM HIV/AIDS population Go to Test and Treat Model Juan’s Stuff Contact Information [email protected] [email protected] Current MSM Epidemic in US MSM accounted for 71% of all HIV infections among male adults and adolescents in 2005 even though only about 5% to 7% of male adults and adolescents in the United States identify themselves as MSM Through its National HIV Behavioral Surveillance system, CDC found that 25% of the MSM surveyed in 5 large cities were infected with HIV and 48% of those infected were unaware of their infections. Young black MSM in this study were more likely to be unaware of their infection – approximately 9 of 10 young black MSM compared to 6 of 10 young white men. Of the men who tested positive, most (74%) had previously tested negative for HIV infection and 59% believed that they were at low or very low risk.