1. Modelling changes in HIV prevalence among women attending antenatal clinics in Uganda Brian Williams

3.  = birth rate N = S + I = rate at which new infections occur  = mortality  S I I  N S I /N  I S  The basic model

4. R 0 = 3.3

5.  = birth rate N = S + I = infection rate  I = Weibull mortality  S I I  N S I /N  I S Normal (Weibull 2) Exponential (Weibull 1)

7.  = birth rate N = population = e –  P  I = Weibull mort. ~ ~  S I I  N S I /N  I S  –P–P e Heterogeneity in sexual behaviour

9. ~  S I I  N S I /N  I S ~  = birth rate N = population = C(t)  I = mortality ~ ~ C(t)C(t) Including control

11. ~  S I I  N S I /N  I S *  = birth rate N = population = e  I = mortality ~ * –M–M –M–M e Mortality leads to behaviour change

13. Nairobi 6 yr Nunn P et al. Tuberculosis control in the era of HIV. Nat Rev Immunol. 2005 Oct;5(10):819-26.

14. TB incidence among gold miners in SA Corbett EL Stable incidence rates of tuberculosis (TB) among human immunodeficiency virus (HIV)-negative South African gold miners during a decade of epidemic HIV-associated TB. J Infect Dis. 2003;188: 1156-63.

15. SS+ Tuberculosis Prevalence Incidence Disease Duration (%) (%/yr) (yr) HIV+ 0.44 (0.02-1.05)2.87 (1.94-4.25)0.15 (0.05-0.48) HIV-0.55 (0.14–0.95)0.48 (0.27-0.84)1.15 (0.48-1.13) DDR = 0.13 (0.09–0.20) Gold miners in South Africa We define disease duration as prevalence divided by incidence

16. Repeat the model 4 times, once for each stage of HIV. Use time series of HIV prevalence to determine incidence. Incidence gives rate at which people enter first stage; overall (Weibull) survival determines rate at which people move to next stage. TB-HIV model Williams BG et al. The impact of HIV/AIDS on the control of tuberculosis in India. PNAS 2005 102: 9619-9624.

17. Impact of interventions on TB cases in Kenya TB incidence/100k/yr 800 600 400 200 0. Baseline ARV 80% TLTI (6 m) TLTI (life) ARV 100% TB detect. TB cure HIV incid Base line: CDR = 50% CR = 70% Interventions: 1% increase 1980 2000 2020 2040 Year Currie, C. et al. Cost, affordability and cost-effectiveness of strategies to control tuberculosis in countries with high HIV prevalence. BMC, 2005. 5: 130.

18. Percent HIV positive HIV negative Williams BG et al. HIV Infection, Antiretroviral Therapy, and CD4+ Cell Count Distributions in African Populations. J Infect Dis, 2006 194: 1450-8.

19. 1,000 2,000 10 20 Time to death (yrs) Initial CD4/  L Time to death (yrs) 1,000 2,000 10 20 Initial CD4/  L Model 1 CD4 decline independent of starting value Survival determined by pre- infection CD4 Model 2 Survival independent of starting value CD4 decline determine entirely by starting value and survival distribution 

20. Spatial Epidemiology of HIV Doubling time = 1 year Life expectancy = 10 years Number of partners = 4 Proportion of random partners chosen at random = 0 (left hand set) or 10% (right hand set) in the following slides. Note that in this model migrants have exactly the same sexual behaviour and individual risk as non-migrants.

27. 1.Can we combine spatial/network models with our more conventional continuous time models of HIV? 2.Can we get a better understanding of the host-viral interaction? 3.What are the population level implications of 2? 4.Do we have enough data to explore fully the joint dynamics of TB and HIV? Questions for all of us

28. Advice to young epidemiologists Never make a calculation until you know the answer. Make an estimate before every calculation, try a simple biological argument (R 0, generation time, selection, survival, control). Guess the answer to every puzzle. Courage: no one else needs to know what the guess is. Therefore, make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise. In either case, life as an epidemiologist, however long, is more fun. Plagiarised from E.F. Taylor and J.A. Wheeler Space-time Physics 1963