You know you're on the right track when you discover that other people are thinking the same way you are.
Here is a model of contagious epidemics.
What's unique about it is, it uses the method in algebraic topology that I've grown very fond of, which is approximating surfaces with discrete maps. (Sometimes it's called "triangulation", if we're in 3 dimensions with a 2 dimensional surface).
Normally, surfaces ("manifolds") are forced to be continuous, and that way you can use the methods of calculus (derivatives etc) to analyze them. This is the approach taken by most of modern physics, for instance Einstein's spacetime manifold is "locally Euclidean" even in a curved universe.
But there's another way, and it makes the math a lot easier. In math language it has a fancy name ("homology"), but really it's very simple. The benefit is you can use ordinary linear algebra instead of worrying about tangent planes and fiber bundles and such.
This paper is a great application, and it illustrates the benefits of the discrete approach. In this paper we're looking at a contagious epidemic, like Covid. We're trying to describe the dynamics of the epidemic.
Usually this is done by averaging over the population, to make the math easier. (The physicists call this an "ensemble"). But that's not really the way a disease looks. For instance Covid had hot spots, and super spreaders. Mom and pop sitting at home don't catch Covid because they never go out. You have retirement communities like Palm Desert with very low rates of COVID because people stay inside watching TV all the time.
So in this paper, they treat the landscape as a "simplicial complex", which is the discrete version of a continuous surface. Instead of averaging the population over wide geographic areas, they take the opposite approach. They use "nodes", which could be any size. Nodes in this case, are geographic locations, could be a house, could be a grocery store, could be a shopping mall or even an entire city. Could even be a single person walking around town. We don't really care how big the nodes are. Conceptually though, they represent pockets where the disease has consistent characteristics. For instance if you go see a movie at a theater, or perhaps attend an indoor Trump rally, the disease has a consistent behavior within the geographic boundary. Everyone who attends gets exposed in the same way.
So you make a graph, the vertices are the nodes and the edges are the probabilities of contagion from one node to the next. You have probabilities within nodes, and between nodes. The graph might look something like this:
The black dots are the nodes (vertices), and the blue lines are the probabilities of contagion (edges). When you have a bunch of nodes near each other you can fill in the triangle (in green) to show connectedness - this might apply for instance, in a shopping mall. The stores each have their own clientel, but most people who go shopping visit more than one store, so the contagion rates are more closely related than a bunch of neighboring houses where people stay inside.
Now instead of averaging over the population, we consider that every blue line is a random walk. Covid when it's airborne can go in any direction, we don't really know where it will end up. But if it lands in the shopping mall, all the theater goers are likely to be infected.
So that's what's going on in this paper.
It's worth a read. To solve the system they use the same methods Wall Street uses, only instead of stock tickers they're dealing with social networks.
Here is a model of contagious epidemics.
What's unique about it is, it uses the method in algebraic topology that I've grown very fond of, which is approximating surfaces with discrete maps. (Sometimes it's called "triangulation", if we're in 3 dimensions with a 2 dimensional surface).
Normally, surfaces ("manifolds") are forced to be continuous, and that way you can use the methods of calculus (derivatives etc) to analyze them. This is the approach taken by most of modern physics, for instance Einstein's spacetime manifold is "locally Euclidean" even in a curved universe.
But there's another way, and it makes the math a lot easier. In math language it has a fancy name ("homology"), but really it's very simple. The benefit is you can use ordinary linear algebra instead of worrying about tangent planes and fiber bundles and such.
This paper is a great application, and it illustrates the benefits of the discrete approach. In this paper we're looking at a contagious epidemic, like Covid. We're trying to describe the dynamics of the epidemic.
Usually this is done by averaging over the population, to make the math easier. (The physicists call this an "ensemble"). But that's not really the way a disease looks. For instance Covid had hot spots, and super spreaders. Mom and pop sitting at home don't catch Covid because they never go out. You have retirement communities like Palm Desert with very low rates of COVID because people stay inside watching TV all the time.
So in this paper, they treat the landscape as a "simplicial complex", which is the discrete version of a continuous surface. Instead of averaging the population over wide geographic areas, they take the opposite approach. They use "nodes", which could be any size. Nodes in this case, are geographic locations, could be a house, could be a grocery store, could be a shopping mall or even an entire city. Could even be a single person walking around town. We don't really care how big the nodes are. Conceptually though, they represent pockets where the disease has consistent characteristics. For instance if you go see a movie at a theater, or perhaps attend an indoor Trump rally, the disease has a consistent behavior within the geographic boundary. Everyone who attends gets exposed in the same way.
So you make a graph, the vertices are the nodes and the edges are the probabilities of contagion from one node to the next. You have probabilities within nodes, and between nodes. The graph might look something like this:
The black dots are the nodes (vertices), and the blue lines are the probabilities of contagion (edges). When you have a bunch of nodes near each other you can fill in the triangle (in green) to show connectedness - this might apply for instance, in a shopping mall. The stores each have their own clientel, but most people who go shopping visit more than one store, so the contagion rates are more closely related than a bunch of neighboring houses where people stay inside.
Now instead of averaging over the population, we consider that every blue line is a random walk. Covid when it's airborne can go in any direction, we don't really know where it will end up. But if it lands in the shopping mall, all the theater goers are likely to be infected.
So that's what's going on in this paper.
It's worth a read. To solve the system they use the same methods Wall Street uses, only instead of stock tickers they're dealing with social networks.
