Fix age and risk group mixing

[darcyrao]

Revise mixing parameters and implementation:

• [ ] Confirm that it is implemented correctly - my interpretation of the methods papers Nick and Monisha cited is that the delta matrix should be an identity matrix if it is implemented as stated with the epsilon parameter to set the degree of assortativity.
• [ ] Try to align it better with empirical data from Ott et al.
• [ ] Make sure that the risk group distribution, risk mixing parameters, and age mixing parameters are consistent. If we set the risk group distribution by age for females to equal that of males, I don't think the implied number of partnerships between different groups will be consistent with the mixing parameters. One option would be to derive the risk distribution for females given the risk distribution for males and the parameters for age and risk group mixing.

[darcyrao]

For the September WHO results deadline: Keep current approach for age and risk mixing and derive the risk group distribution for females from the mixing matrices and the risk group distribution for males.

Eventually:

• Try different approaches for specifying mixing matrices
• Check model output to see what the resulting mixing pattern is and how it changes over time with changes in population structure
• Consider changing the mixing matrix at some point in time to re-adjust to the shifts in population structure.

Because HPV is so prevalent, it is not as important to represent mixing patterns as precisely as for an bacterial infection.

[darcyrao]

Related to this, consider the implications for mixing if we adjust sexual activity to be lower for persons in late stages of HIV infection and those with cervical cancer.

We currently reduce the probability of HIV transmission for persons in late stages of HIV infection as a proxy for reduced sexual behavior, and we plan to do the same for HPV transmission probability in late stages of HIV and for both HIV and HPV transmission probabilities in cervical cancer states (see issue #58), but think about whether we could reduce behavior to accomplish this. Need to think about how it would influence mixing matrices and the algorithm to balance the number of partnerships in the matrices.

[carajbro]

I made a structure-related change to mixing in the model. I think it will have a small effect on calibration and outcomes, but is still good to make.

We assume that mixing by age is partially random and partially assortative. We assume that in a mixing pattern purely assortative by age, females are most likely to form partnerships with males of the next youngest age group. This off-diagonal assortative mixing pattern results in 10-14 year-old males and 75-79 year-old females having less than 100% of their partnerships, but the assumption is that these age groups have relatively few partnerships and contribute marginally to overall infection transmission.

However, as I'm updating the appendix I noticed some modifications were made to the assortative mixing matrix by age that I think are a little funky. These modifications remove the gender-age bias in persons <=19, and make it so 20-24 year-old males instead of 10-14 year-old males have less than 100% of their partnerships. I think 20-24 year olds are more influential to transmission in the model so we should remove these modifications.

Doing so affects HPV/HIV prevalence in this age group marginally. I guess this means the modifications don't make that big a difference, but I think it would still be better to remove them. And it also tells us we shouldn't be too worried about <100% of partnerships in the youngest and oldest age groups.

Here's a visual. Top picture is how I think it should be, bottom picture is with the modifications. The 0.30 or 0.70 numbers represent the proportion of partnerships from that age group.

The updated code is: if fivYrAgeGrpsOn %% Assign deltaR and deltaA (nature of assortative mixing by age and gender; Kronecker delta) deltaR = eye(3 , 3); deltaAF = eye(16) . 0.3 + diag(ones(15 , 1) . 0.7 , 1); deltaAM = eye(16) . 0.3 + diag(ones(15 , 1) . 0.7 , -1); deltaAF(1 , 1) = 0.0; deltaAF(1 , 2) = 0.0; deltaAF(2 , 2) = 0.0; deltaAF(2 , 3) = 0.0; deltaAM(1 , 1) = 0.0; deltaAM(2 , 1) = 0.0; deltaAM(2 , 2) = 0.0; deltaAM(3 , 2) = 0.0;

[carajbro]

Relevant source for age discrepancies in sexual mixing by gender in KZN:

(de Oliveira, 2017, Lancet HIV) Transmission networks and risk of HIV infection in KwaZulu-Natal, South Africa: a community-wide phylogenetic study