Research Publication (math, not economics): Using Graph Theory To Try To Stop the Spread of HIV/AIDS in Africa

 

03-26-2026

8:37 p.m.

You have a digraph.

The edges are "ability to spread" of the HIV.

Each node in the digraph has a weight that is "percentage infected."

Each node in the digraph can be "zoomed in on" to yield more and more detailed nodes within each node.

The challenge is, assign two types of resources--edge blockers / inhibitors and node treaters--to try to decrease the spread of the disease.  The edge blockers are the main one...treating an incurable disease doesn't really get rid of it, you could think of the node treaters as "hospitalize and quarantine" government people.

8:41 p.m.

Note, one very efficient edge blocker would be, post fliers and tell citizens in certain areas that are captured by both nodes (??--I forgot what I meant when I wrote this yesterday, 03-27) about the risk.  Publish, in some form, the node weights of the two areas, and, the two digraph edges between the two nodes (going in different directions).

More tomorrow!

03-27

9:07 a.m.

Define each node as a geometric square or rectangle on a particular geographic location in the continent of Africa.

Each node has:

 - a population

 - a number of people within that node with AIDS/HIV

 - edges to other adjacent nodes--have the rectangle/square dimensions all be the same--that describe the flow of AIDS from one node to the other, each node will have exactly 8 edges coming from it and 8 edges pointing to it, from the 8 adjacent nodes (we consider diagonal nodes adjacent, too).

It might be kind of hard to calculate the edge weights...but I'm claiming it's possible, with some study and attention to the task.  Sufficiently talented scientists ought to be able to figure it out and get the data.

9:15 a.m.

Each edge means, "after one time step (let's say one month), the source/from node infected population contributes exactly EDGE_WEIGHT percent of that total number of infected people to the target/to node."

Additionally, we have for each node a "death percentage," which calculates how much the infected population decreases.  We don't want deaths to happen, but they do decrease the infectedness of the population at each node/rectangle.  (03-27:  Note, quarantine also decreases the infection possibilities...but that quarantining will be measured and accounted for in the edge probabilities, not the nodes...although you could have an alternate version of this, where you treat both nodes and edges, and you treat nodes by quarantining people who have AIDS to the point where they can't impact anyone else.)

We're going to assume that aside from non-profit intervention to modify the edges, the edges stay the same.

Aside from the edges, there are 3 numbers associated with each node:

1 - total population

2 - infected population

3 - infected population decrease rate per time step (death or leaving the area)

So the idea is, at each time step, we expect the infected population at each node to decrease based on the death rate, and, we expect the edge weights to stay the same, but to govern an increase in the node.  There will be a margin of error, i.e., we expect the increase and decrease in the infected population to change by an amount that is within a certain bounds, based on the decrease rate per time step and the edge values and nearby nodes' infected population amounts.

Also, when the edges and adjacent nodes contribute to the infected population rate, what they do is add an integer to the value of infected people.  The value leads to a certain number of people who had not been infected before becoming infected.

Really, we can jettison the "total population" number.  We don't need that.  So it's really two things we're looking at:

1 - infected population

2 - infected population decrease rate per time step

And then we have the edges and the 8 adjacent rectangles.

The challenge becomes:  Identify the rectangle edges that are the most valuable to "intervene" at.  In other words, calculate how much it costs to do an "education and warning prevention plan" at a particular edge.  Basically, you dispatch a team of non-profit philanthropic workers to an area--the border of two adjacent rectangles, or a shared corner of two diagonally adjacent rectangles--and you get them to do an intensive campaign of warning people in that area about unsafe sex, AIDS, and the consequences of not adhering to the safety guidelines presented.  The philanthropic team should also somehow take action to try to prevent rapes in the geographic area that they are assigned to; perhaps coordination with local government and law enforcement could help to deter rape in these selected areas.  (Of course, you'd want to prevent rape everywhere--and prevent the spread of HIV/AIDS everywhere--but resources are limited.)  It might also be a good idea if the assigned charity workers brought food with them, which they could distribute to locals to help make their message about abstinence and safety with regard to sex more palatable.  The charity could also offer to provide medical help that would come with a sort of quarantine for people infected with HIV/AIDS.

The math problem is, identify the best rectangle edges to swoop in on and decrease the edge values.  We would need to calculate the percentage of improvement that the non-profit groups can cause empirically, i.e., we'd need to test out how effective the teams are, and then identify what percentage improvement we can achieve for each edge.  E.g., we might say, the teams can reduce the edge value by 65%, e.g., if the rectangle value for Node A is 1000 people, and the rectangle value for Node B is 600 people, and the edge from A to B is 50%, then we would expect the team to be able to reduce the edge from A to B down to .5 - .5*.65 = .5*.35 = 0.175 = 17.5%.  Remember that we assume that there is a margin of error involved.  If we reduced the edge by this much using the team of non-profit workers, we might get an increase of 175 people, i.e., Node B winds up at approximately 775 people infected instead of 1100 people infected at the next time step.

This is my strategy for improving AIDS in Africa.  If it is discovered, after doing an analysis like this, that there wouldn't be a feasible way to achieve success with this technique, then the answer would be, try to accumulate more resources before beginning the project.

Mathematically, I don't know how to optimize success for this idea algorithmically.  Surely, the problem is in NP, so quantum computers would likely be able to help at solving the problem.  Even without QC, though, there is a pretty good technique for getting "pretty good" results from this problem:  Just select the best edge first, i.e., pretend you have only one team that you can deploy to an edge, and then select the best edge.  Next, modify that graph in accordance with that assignment.  Now, assuming you have some finite number of teams you can deploy--let's say 100--continue this process and repeat finding the best edge you can go to again and again.  In some cases, you might send multiple teams to the same edge.  After you've calculated how much benefit you can cause using however many teams you have, you can get a grip on how much you can actually do.  In some cases, you might decide to wait before you have more resources before beginning a costly campaign to limit the spread of AIDS.

9:34 a.m.

I re-read these notes and reviewed them before publication to my blog, which I'm planning to do a little bit later today.

Like I said above, we could do a different version of this with teams sent to both nodes, to quarantine, and edges, to mainly educate people and warn them about the spread of AIDS.  There's quarantine-side intervention, and education-side intervention.  The main approach I explained above was mainly education-side intervention.

And yes, it is possible to simulate the evolution of the digraph over a number of time steps.  In general, I would see no reason not to do all of the interventions at the first available time step...but in theory, if there were some good reason to wait to do an intervention, you could do that, too.  I would imagine it could be mathematically proved that early intervention is better than later intervention in every case...but I don't have a proof of that right now.

OK, I'm ready to publish this.

Quick update (9:44 a.m.):

You probably do need to have a "natural decrease" percentage value for each node.  It could be based on "natural" quarantining, including self-quarantining, and death.  The idea is, the model needs to have some way to have the number of infected people decrease.  So you can't just have edges, because the edges always add a positive percentage to each target node; you have to have a natural decrease per time step value for each node.

Another update (9:51 a.m.):

Also, I support that quarantining would be done by governments, not foreign philanthropists.  The governments, even if they are imperfect/corrupt, should be allowed to control the behavior of infected citizens, and outsiders who care about the region should not get involved with ordering people quarantined...that would be highly inappropriate, for outsiders to swoop in and start quarantining people in Africa.