1. How complicated do we want to make this?
Modelling HIV:
How complicated do we want to make this?
Brian Williams and John Hargrove
South African Centre for Epidemiological Modelling and Analysis 1

2. The basic SI model
Working with proportions, susceptible people are infected at a per capita rate b ; Infected people die at a per capita rate m.

3. We can fit the rise or the peak but not both at the same time.
Exponential rise give R0 = b /m = 4.7. Logistic increase to a steady state prevalence of (R0 –1)/R0 = 0.8.

4. Heterogeneity in risk: 1
If those most at risk are infected first then transmission will decline as prevalence rises. This is how they do it at Imperial College.
Imperial College: A proportion P of the population are at risk; (1 – P) are at no risk

5. Heterogeneity: 1
We can now fit the peak prevalence (but not the decline). Model suggest that only 16% of people are at risk of HIV.
N.B. If the mortality is 10% per year, it is always 10% of the prevalence. The scales differ by a factor of 10 so the prevalence and incidence curves lie exactly on top of each other.

6. Heterogeneity in risk: 2
SACEMA: Risk of infection declines with prevalence.
But I prefer to do it like this.
n = 1: Exponential
n = 2: Gaussian
n = 3: Step-function

7. All fits are equally good but the implied incidence and the steady state prevalence are quite different. we will use n = 2.
n = 1
n = 2
n = 4
Linear
The problem is that the data do not allow us to distinguish between them. If we had good measures of the incidence we would know which is best.

8. Control Let risk decline logistically with time ac rc tc
It is clear that transmission rates fell before ART was introduced. This could have been less sex, fewer partners, condoms or something else. We simply put in a control variable to fit the data but this tells us what level of change must have occurred and we need then to find an explanation. rc ac tc

9. Control
A reduction of 85% in transmission seems a lot!
We now get a good fit to all the data. Transmission fell by 85% between 1992 and 2002

10. Control only
And even worse: if we ignore heterogeneity in risk then we can take up the slack by increasing the level of ‘control’. Again, the data are not sufficient to tell us which is most likely.
But we get an equally good fit if we drop the heterogeneity and increase the ‘control’. Transmission fell by 90% between 1987 and 2003.

11. Heterogeneity v. control
Heterogeneity and control
Transmission: b * or b †
Control only
Without heterogeneity control has to start 3 years earlier and has to fall further 90% v. 85%.

12. Weibull survival Survival on HIV follows a Weibull
survival curve with a median survival of
10 years and a shape parameter of 2.25.
Close to a survival for a G-function with a shape parameter of 4. We use four successive compartments for those with HIV.
One thing of which we are sure is the survival is Weibull, not exponential, and this introduces a very important delay between the rise and fall of incidence and the rise and fall of mortality.

13. Weibull survival
Mortality now rises about 8 years after the rise in incidence and about 3 years after the rise in prevalence. Weibull survival introduces an important delay.

14. Weibull survival and control
Heterogeneity and control
Transmission: b †
Weibull survival
It also reduces the level of control that we need in order to fit the data.
Allowing for the Weibull survival control has to start one year later and has to fall less (76% v. 85%).

15. Population growth
Population growth is easy to include.
Separate the birth rate from the death rate and allow people to die from natural causes in each stage. (Birth rate = 3.6% p.a.; death rate = 1.2% p.a.)

16. Population growth
But makes a very small difference.
Allowing the population to grow (2.4% p.a.) increases incidence and reduces mortality but only slightly.

17. Weibull survival and control
Population growth
Transmission: b †
Weibull survival
Allowing for the population growth has little effect on the level of control needed to fit the data.

18. Including treatment
People may start treatment and go onto ART (A) from any of the infected classes. Let the rate of starting treatment increase logistically with time.
We must, of course, include treatment coverage even though the estimates of coverage are weak.
Rate of starting
treatment p.a.

19. Treatment
And we might then want to refit the prevalence.
Assuming the same treatment rate in all classes. Treatment (pink) changes the proportion of people not on ART (green) has some effect on incidence (red) and a bigger effect on mortality (brown). People would have to be tested every two years (on average).

20. Treatment
Assuming that only those in Stage 4 are treated. This has a bigger impact on mortality but a smaller impact on transmission. People would have to be tested every 5 weeks (on average).

21. Projections
Assuming the same treatment rate in all classes will drive the epidemic down slowly

22. Projections
Keeping the same number of people on ART but only treating those in Stage 4 will have a bigger impact on mortality in the short term but will not bring the epidemic under control in the long term.

23. Uncertainty
The good news is that with a model as simple as this, it is very easy to get estimates of the uncertainty.
With a simple compartmental model we can estimate confidence limits on the parameters and the fitted curves. (This is an MCMC estimate.)

24. Demography: The model
We have, of course, omitted almost all the demography….
As people age they can remain uninfectious, be infected, progress with HIV, start treatment, remain on treatment, or fail treatment. They can die of other causes, AIDS, or die on ART.

25. The computational problem
Without demography
9 states (Susceptible 1; Infected 4; ART 4)
16 possible state transitions (20 with treatment failure)
500 time steps (50 years at intervals of 0.1 year)
104 calculations of state transitions
With demography
Each state can have 1,000 ages (100 years at intervals of 0.1 year)
107 calculations of state transitions
In both cases about 6 variable parameters
Many more fixed parameters with demography
But if we include even the simplest aspects of the demography we increase the computational burden by about 1,000 times.

26. The computational problem
Without demography
Solver in Excel converges after about 20 iterations taking about 5 seconds.
MCMC estimates of uncertainty require about 1,000 iterations taking about 5 minutes.
With demography
This is going to take about 5x104 seconds = 14 hours to optimizing the fit and about 5x104 minutes = 35 days to estimate the uncertainty.
Is it worth it?
Does the added information warrant the increase in computational complexity?

27. Demography: Fixed parameters
Assume that the current age-distribution is a stable age-distribution.
2. From the overall population growth rate calculate the age-specific mortality for HIV-negative people.
3. Check that this gives the current stable age-distribution and population growth rate.
4. Use the age mortality as a function of age for those not on ART. This declines linearly with age and the survival distribution is Weibull with a shape parameter of 2.25 at all ages.
5. Estimate the (relative) age-specific incidence of infection.
6. Run and fit the demographic equivalent of the model (without ART).
7. Note that this is still a one-sex model so we are averaging over a loop of transmission male female male ignoring age-matching of partners and other aspects of the network.
This is some of what we need to do but leaves out the most important thing which is the underlying network structure….

28. Demography
And, given the data, it doesn’t seem to add much!
The demography only changes the fits and predictions by a small amount. Incidence declines more quickly, mortality is a little lower (next slide). But both within the uncertainty limits.

29. Population growth Repeat of Slide 14.
As can be seen by making a comparison with the earlier fits….
Repeat of Slide 14.

30. As complicated as this (but no more….)
States
Susceptible; infected (x4); ART (x4)
Transitions
Births, deaths, incidence, progression, treatment, failure
Variable parameters
Initial value, transmission, heterogeneity,
control (x3), treatment (x12)*.
Fixed parameters
Population growth rate, survival on ART, failure rate.
So this seems to be the most appropriate but informative level of detail.
* Timing, rate of roll out and asymptotic rate x 4 stages. Collapse values where appropriate.

31. Outputs Prevalence Incidence Starting ART Mortality
In each of 4 states off treatment and on treatment with implications for rates of opportunistic infections.
Incidence
Overall incidence of HIV
Starting ART
Rate a which people start ART with implications for testing rates.
Mortality
Death rates off ART.
Which will give us all the outputs we need….

32. Interventions
The model indicates the extent to which transmission appears to have fallen pre-ART. Time trends in the levels of different interventions should be used to see if they can reasonably explain the pre-ART decline.
Possible interventions include: fewer partners, less age-discordancy, condoms, PreP, PEP, VMMC.
And allow us to make cost benefit calcualtions.