Deterministic Compartmental Models, Application: Modeling the Interaction of HIV and Malaria

Transcription

1 Deterministic Compartmental Models, Application: Modeling the Interaction of HIV and Malaria Mathematical Modeling of Infectious Diseases: Tools of the Trade Laith J. Abu-Raddad Fred Hutchinson Cancer Research Center Seattle, Washington, U.S.A

2 The big question Why is the HIV epidemic in sub- Saharan Africa has been so massive?

3 HIV global distribution

4 Where is the high risk sexual behavior? (Gregson et al. 2004) Manicaland Study: Baseline Survey (women) % Number reporting X partners % 40% 20% Percentage HIV positive N umber of lifetime sexual partners 0% Courtesy of S. Gregson.

5 Geographic overlap between HIV and malaria

6 What is mathematical modeling? Mathematics A quantified rigorous and logical analytical thinking.

7 Mathematical modeling is a tautology Empirical data Mathematical Modeling Implications

8 What does malaria do for an HIV infected person? (Kublin et al. 2005) Logarithmic increase in virus concentration Pre-malaria Malaria Post-malaria

9 HIV-1 Viral load and transmission probability per sex act p = p Log 10 ( 2 1 ) vl vl (Quinn et al. 2000) Transmission probability increase (relative scale) Logarithmic increase in viral concentration

10 Are HIV infected persons more likely to acquire malaria? (Patnaik et al. 2005) Percentage increase in risk of malaria infection 120% 100% 80% 60% 40% 20% 0% Time from HIV infection (years)

11 HIV and malaria: A vicious cycle? HIV Malaria

12 HIV infectious spread

13 From sex act to sexual partnership β is the transmission probability per sex act Prob partnership = 1 (1- β) nτ n (frequency of sex acts) ~ 10 times per month τ (duration of partnership) ~ minutes to years

14 Sexual partnership formation: An entangled web

15 A reduction from individual dynamics to population dynamics?? Interacting Populations Interacting Individuals

16 A reduction from atoms to planes??

17 Renormalization group and effective field theory: An elegant mathematical strategy Small microscopic scale Large macroscopic scale g( x x x x x...) g( y y y ) 1, 2, 3, 4, 5, 1, 2, 3 Very large number of degrees of freedom Small number of degrees of freedom Relevant and irrelevant observables Only Relevant observables

18 Kenneth Wilson: 1982 Nobel Prize in Physics

19 Simplifying the complex: Representation of network properties using summary measures Interacting Populations Interacting Individuals

20 How often do people change their sexual partners? Men 80% 70% 60% 50% 40% 30% 20% 10% 0% Number of sex partners in last 12 months

21 How often do people effectively change their sexual partners in each risk group? Effective rate of partner change = ρ + baseline ρ social desirability bias + Summary measures ρ ρ sexual variability concurrency + Theoretical reduction

22 Mapping the risk of exposure to HIV in the population General Population Core Group Bridging Population

23 How do risk groups mix? Harborview STD Clinic Seattle Garnett et al Proportion Partner's number of partners random Index woman's number of partners

24 Malaria infectious spread

25 Mosquito and malaria transmission dynamics

26 Structural similarity across diseases Two-host systems: Sex acts (bites) Men are the vectors (mosquitoes) Women are the hosts (humans)

27 The HIV-malaria interaction model Host (human): Vector (mosquito): ds() i V H i = μn0() i + νi0() i μs() i qλm S() i ΛHIVS() i dvs dt dt di0() i V H I0 () i = qλm S() i μi0() i νi0() i ΛHIV I0() i dvi dt dt dy1 () i S() i V H =Λ HIV S() i + νi1() i μy1() i ω1y1() i g1qλm Y1() i dt di1() i I0 () i V H =Λ HIV I0() i + g1qλm Y1() i μi1() i νi1() i ω1i1() i dt dy2 () i V H = ω1y1() i + νi2a() i + νi2b() i μy2() i ω2y2() i g2qλm Y2() i dt di2a () i V H = fhiv + g2qλ M Y2() i + fhiv + ω1i1() i μi2a () i νi2a () i ω2i2a () i dt di2b() i V H = (1 fhiv + ) g2qλ M Y2() i + (1 fhiv + ) ω1i1() i μi2b() i νi2b() i ω2i2b() i dt dy3 () i V H = ω2y2() i + νi3() i μy3() i ω3y3() i g3qλm Y3() i dt di3() i V H = gq 3 Λ M Y3() i + ω2i2a() i + ω2i2b() i μi3() i νi3() i dω3i3() i dt = μ V μ V Λ V H V V V S M S H V μτ V =Λ ( t τ) V ( t τ) e μ V M S V I

28 A synergy between HIV and malaria! (Abu-Raddad et al. 2006) Prevalence (%) Time (years) HIV prevalence (no interaction) HIV prevalence (interaction) M alaria prevalence (no interaction) M alaria prevalence (interaction) Antenatal clinic survey Population-based survey

29 Impact of malaria on HIV incidence 10 9 Fraction of new HIV infections due to malaria 8 Percentage (%) Time (years)

30 Impact of HIV on adult malaria incidence 14 Fraction of new adult malaria infections due to HIV Percentage (%) Time (years)

31 More on the results of Kisumu Cumulative number of cases attributed to interaction up to 2005 (adult population of 200,000): 8,500 excess HIV infections 980,000 excess malaria episodes

32 Impact of HIV on shifting malaria endemicity threshold Prevalence (%) Macdonald's stability index No interaction Interaction Excess prevalence

33 A bigger question? Co-infections that increase HIV virus load a leading cause of the HIV epidemic in Africa?