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I am working on a problem involving mathematical modelling of a closed population using graph networks.

  • In this problem, people start of healthy - from there,
    • they can either $\tt recover$,
    • stay $\tt sick$
    • or $\tt die$.
    • If they get $\tt better$, they can fall $\tt sick$ again or still $\tt die$.

In the problem I am working on:

Let $G = (V, E)$ be an undirected graph where:

  • $V$ is the set of vertices (nodes)
  • with $|V| = n = 100$ (total number of nodes)
  • $E$ is the set of edges

In this graph, any person at time $t$, depending on the the health status of their neighbors (i.e. neighbors can infect a person), their state vector at time $t$ as:

$$ S(t) = \begin{bmatrix} S_{\text{HNI}}(t) \\ S_{\text{CI}}(t) \\ S_{\text{HPI}}(t) \\ S_{\text{D}}(t) \end{bmatrix} $$

Where

  • HNI = Healthy (Never Infected)
  • CI = Currently Infected
  • HPI = Healthy (Previously Infected)
  • D = Dead.

The transition matrix $P$ represents the probabilities of moving from one state to another (I am not sure if this description is fully accurate since the probability of infection depends on how many infected neighbors a person has):

$$ P = \begin{bmatrix} p_{\text{HNI,HNI}} & p_{\text{HNI,CI}} & p_{\text{HNI,HPI}} & p_{\text{HNI,D}} \\ p_{\text{CI,HNI}} & p_{\text{CI,CI}} & p_{\text{CI,HPI}} & p_{\text{CI,D}} \\ p_{\text{HPI,HNI}} & p_{\text{HPI,CI}} & p_{\text{HPI,HPI}} & p_{\text{HPI,D}} \\ p_{\text{D,HNI}} & p_{\text{D,CI}} & p_{\text{D,HPI}} & p_{\text{D,D}} \end{bmatrix} $$

$$ P = \begin{bmatrix} 0.69 & 0.3 & 0 & 0.01 \\ 0 & 0.87 & 0.1 & 0.03 \\ 0 & 0.3 & 0.68 & 0.02 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

I simulated this network using a computer (using R):

enter image description here

I understand this problem is similar to a SIR problem, but I was wondering if the following questions could be answered:

  • How long on average will it take for the whole population to be infected at least once ?
  • On the $50$th day of the simulation, what percent of the population will be currently infected ?
  • At some point, will the entire population be dead ?

These questions sound difficult and I am not sure if they have closed form solutions.

Thus, I tried to repeat the simulations to help me answer these kinds of questions:

enter image description here

I would repeat the simulation many times and take the average answer to each of these questions.

But I was just wondering - do these questions have analytical solutions or do they have to be numerically simulated?

  • PS $1$: I am aware that in some situations with rare events probabilities (e.g. rare disease transmission), I would need to run each simulation for many iterations to observe even one event - and then repeat this again and again, thus making simulation infeasible for such situations. In such situations, I think closed form solutions would be better.
  • PS $2$: Does anyone want the R code that I used in this problem ? I can provide it if someone wants.
Rodrigo de Azevedo
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stats_noob
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    You might look at the terminal probabilities. See Perron eigenvector. – A rural reader Oct 15 '24 at 01:43
  • thank you rural reader! I will check it out! – stats_noob Oct 15 '24 at 01:53
  • The transition matrix $P$ is for every vertice $v \in V$? Why does it not depend on the neighbors of $v$? – NN2 Oct 17 '24 at 12:37
  • thank you for your comment @NN2 : It does ... I just didn't know how to write it in that format... – stats_noob Oct 17 '24 at 12:52
  • The current transition matrix is then not correct. You could describe the relationship between $v$ and its neighbors (for example: if 50% of the neighbors is infected, then $v$ is infected,....). After that, perhaps someone (not sure it's me) can help you to write a more correct transition matrix $P$. – NN2 Oct 17 '24 at 13:09
  • I suppose that you have also what I will call the 'map' of all peoples (list of neighbors for every one). In your pictures, it seems that the map is always the same (normal), but you rotate the picture for some reason. Do you confirm. Normally, rows 1 and 3 of you matrix are highly dependant of number of neighbors that are infected. – Lourrran Oct 22 '24 at 18:32
  • Turn this question into a research paper @OP. – ThankYouForFlyingRyanair Feb 07 '25 at 13:42

2 Answers2

1

This is NOT an answer. Just some long comments in the spirit of "how to start working on these types of questions"... Note: I am an engineer, not a mathematician or population biologist. I have no background in modeling these type of problems and simply presenting a few things that came to my mind when I read the question (i.e. to list things that I would like to consider if I had the right mathematical skills to deal with them).

  1. You graph shows some connections among neighbors. It is not immediately clear to me what the basis for the graph connections is. One may define a ROI (radius of infection), if one were to uniformly distribute the population on a large symmetric shape, such that ignoring the periphery, for a large population, the number of neighbors for any individual is roughly the same. The radius could be just the nearest set of equidistant neighbors one is connected to, to simplify. I imagine starting with a healthy population in this model and randomly replace one (or $n$) healthy individuals with infected individuals and see how the disease spreads.

  2. I do not see a distinction between HNI and HPI wither in terms of them counting towards being sick or contirbuting towards further sickness in the population.

  3. Sick individuals should tend to get healthy unless reinfected within a specified time (if they become reinfected within the specified time, their sickness should accumulate). There should be a rate of propagation of sickness (assume radially equal in all directions). This may or may not depend on how sick the individual is. There has to be some sense of sickness threshold, beyond which an accumulated sickness results in death.

  4. Dead individuals can be considered not to propagate sickness anymore.

  5. Rate of recovering vs rate of propagation matters for spreading. This is where the proabilities of transmission to a neighbor will come into picture besides the transmission vs recovering rates.

Though I don't have the skills to analytically deal with these parameters, I can imagine it being possible depending upon the simplifications one does for each of these factors. I hope at least one of the points helps you in a better numeric modeling at least.

Srini
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But I was just wondering - do these questions have analytical solutions or do they have to be numerically simulated?

It is very unlikely for these types of models to have analytical closed form solutions.

However, you might be able to make approximations for your questions

  • On the $50$th day of the simulation, what percent of the population will be currently infected ?.

    • You could estimate a speed of spreading based on the incubation times of the disease and rate/probability of transmission. This is however tricky because inhomogeneity of the network and it's dependency on the state of evolution.

      Based on a toy model from here Detailed description/scripts of mathematical models for Coronavirus you see that there is some exponential behaviour that switches towards a power law behaviour.

      example

      It is not unimaginable that one could derive the coefficients in those power law and exponential law phases as well as the time of transition.

      The inhomogeneity makes it difficult however. If not every node is similarly connected, and if we also add variations in transmission probability per connection, then the transmission is not a straightforward computation as in a homogeneous SIR model.

      In this question a toy model is used to investigate the effect of a fraction of the population having the transmission rate reduced: Using different nodes in a networked Compartmental Model (SIR) for different regimes?. What you see is that as a function of the fraction of people that have their transmission rate limited, the amount of infections decreases, but this effect is only a simple linear relationship (where it is possibly easily approximated) in a restricted range.

      effects of different fractions after x days

  • How long on average will it take for the whole population to be infected at least once ?.

    • It might be that not the entire population gets infected (because after many people already having gotten the disease, the rate of transmission decreases). For example, with the homogenous SIR model, not everyone will be infected when time goes to infinity.

      In your model you seem to have a situation where people after being recovered, will still have a probability to become sick again, and you might have a certain endemic phase where the disease never dies out. Then you could estimate the average number of sick people and the probability of becoming sick. From that you can estimate the probability of getting sick and the probability for the entire population getting sick at least once.

  • At some point, will the entire population be dead ?.

    • If you wait long enough then almost certainly everyone dies.

      Your transition matrix is not very clear because it doesn't involve the network connections. But, if you have something like the minimal 0.01 probability for anyone to die, then the state 'death' will be an absorbing state where all cases eventually end up. Possibly, if you include the introduction of new people (births) then you can have a population that never dies out, or at least with some non-zero probability it will never die out.

Sextus Empiricus
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