1. QR24, Fall 2005, Lecture 2
Modeling Epidemics

2. Outline Epidemics When disease becomes an epidemic
Concentrated v. generalized epidemics
Focus on high risk areas

3. Epidemics have dominated much of human history
Plague of Athens (430 BC) 75, ,000 dead
(likely typhoid fever)
Black Death ( ) million dead
1918 “Spanish” flu 75 million
HIV million
Ebola ( ) 11,000

4. Epidemiological modeling

5. How bad can an epidemic get?
Air/water born diseases – potentially everyone
Bloodstream infections (HIV) – people who are exposed to infected individuals

6. Modeling epidemics Susceptible – not yet infected
Infected – with the disease
Recovered/Removed – no longer susceptible (perhaps died)
Population (N)
For the mathematically inclined, N = S + I + R

7. Reminder Infected Population / = Prevalence Total population
Newly Infected Population / = Incidence
Number not yet infected

8. HIV Prevalence Trends Differently in Different Countries

9. For HIV, how do we know incidence and prevalence?
Random blood draws
More common recently
Women who visit antenatal clinics
These are extrapolated to the entire country (Spectrum model)

10. When will disease infect everyone?
Some notation
Share of people who are:
Infected: i = I/N
Susceptible: s = S/N
Recovered: r = R/N
If an infected person interacts with a susceptible person, the probability that the susceptible person becomes infected is β.
Reflects (1) average rate of contact; and (2) average transmission rate
A person who is infected recovers / dies with probability δ.

11. When will disease infect everyone?
Change in share = Inflow Outflow
of people infected:
Inflow
Infected people randomly meet susceptible people.
Inflow = β ∙ s ∙ i (β ∙ s is the transmission rate for
each infected person)
Outflow
Outflow = δ ∙ i (if you have had calculus:
di/dt = β ∙ s ∙ i - δ ∙ i)
Change in % infected = δ ∙ s ∙ i - γ ∙ i

12. When will disease infect everyone?
Epidemic is when change in % infected > 0
β ∙ s ∙ i - δ ∙ i > 0
β ∙ s / δ > 1
At the start of an epidemic, s ≈ 1. Thus, this becomes:
β / δ ≡ R0 > 1
R-naught

13. Where have you seen this before?
Contagion (2011)
Watch Kate Winslet
(In another scene, ‘she’ gets it wrong)

14. Let’s test it out

15. A Richer Model for HIV
There are different infected groups with different infection rates
i1, i2, i3, … (call these ij)
E.g., female sex workers, casual liaisons, wives
Different susceptible groups engage in different practices
s1, s2, s3, … (call these sk)
E.g., men who travel, men with multiple wives
Transmission probability is not the same for each group
E.g., depends on precautions taken
Death occurs after some time

16. A Richer Model for HIV
Change in infected = Σj βkj ∙ sk ∙ ij δk ∙ ik(t-l)
share for group k
high risk contacts latency
matter a lot period
The nature of interactions explains how widespread the infection becomes.

17. High risk aspects Multiple (concurrent) sexual partners
Including sexually experienced partners
Including partners with very high risk of AIDS
Infrequent use of condoms

18. Types of Epidemics

19. Concentrated v. Generalized Epidemic
Generalized epidemic: Prevalence > 1% in general population
Concentrated epidemic: Prevalence > 5% in any sub-population at a higher risk of infection (IVDUs, FSWs, MSM)
Low-level epidemics: Others

20. Concentrated epidemic
QR24, Lectures 6-8, Fall 2005
Concentrated epidemic
Population
MSM
SWs
IVDUs
Sex Workers (commercial & informal)
Men having Sex with Men
IV Drug Users
Women Men

21. Generalized epidemic Population 2-wives Women Men Older 1
QR24, Lectures 6-8, Fall 2005
Generalized epidemic
Population
2-sugar daddies
2-wives
Women Men
Older
Younger 1
SWs

22. Groups with Very High Prevalence: MSM

23. Groups with Very High Prevalence 2: IV Drug Users

24. Groups with Very High Prevalence 3: Sex Workers

25. Sex Work and HIV in sub-Saharan Africa
First African country hard hit – Uganda
Civil War and population mobility in 1970s/1980s
Currently hardest hit countries – Southern Africa
Significant mining
Men who travel for work
Want to see more?
The Trans-Africa Highway System

26. A Ugandan Truck Stop

27. HIV in the sex trade in Nairobi
Chance study of chancroid (an STD) since 1980, for which blood specimens had been frozen.
Prostitutes
1981: 4% HIV+; : 61% HIV+
>1,400 paid intercourses per year
Men who consulted with an STD clinic
1980: 0% HIV+; : 6% HIV+; : 15% HIV+

28. The clients of female sex workers

29. Share of Adult Women who are Sex Workers

30. Spread to other sexual partners
Wive(s)
Who has the power in the relationship?
Who controls the money?
Younger girls
Gift exchange
Sometimes
non-consensual

31. Non-Consensual Sex

32. A Close Call: Thailand
Bangkok has a very large sex industry

33. HIV Prevalence in Thailand

34. How to Reduce High Risk Interactions?
QR24, Fall 2005, Lecture 2
How to Reduce High Risk Interactions?

35. Are Sugar Daddies Unique to Sub-Saharan Africa?
American Sugar Daddies