or coincide with your expectations
.
1.
Describe, as clearly and concisely as you can, the
epidemic model you came up with in class on Thursday.
(It is ok to change it if you've had second thoughts,
and ok to differ from your group-mates if you want to.)
2. I have written this Maple worksheet, mws version, HTML version, inspired by what you developed in class. It generates an animated histogram, showing the fractions of the population that are susceptible (green, 1st column), infected (red), recovered (turquoise), and dead (grey, last column). I think it can be modified fairly readily to represent any of the models you came up with.
Take it and modify it so that it represents your own model. For all the parameters, choose what you think are reasonable values for whatever disease and public health policy you are modelling.
3. Suppose that initially the entire population is susceptible
except for those who are initially infected.
Investigate how the total number of deaths for the entire epidemic depends
on the size of the fraction of the population that is initially infected.
Discuss whether the results are surprising or coincide with your expectations.
4. How would a quarantine policy be expressed in your Maple-implemented model? Investigate quantitatively the effect of some feasible quarantine policy on the total number of deaths, and summarize your results.
5. If you have noticed anything else that is interesting or informative coming out of your simulations, describe it clearly and concisely here.
IF YOU ARE UNSURE ABOUT HOW TO GET STARTED, TALK TO ME EARLY IN PERSON OR VIA EMAIL.
1. I went with the model whose implemention in Maple
I had supplied to you in the HW assignment. This is a discrete-state
discrete-time deterministic model. Here is the state diagram:
S is the susceptible state, I1 thru Ini are infected states,
R represents recovery (and immunity), and D denotes dead.
In the numerical experiments, I used 7 infected states.
From each infected state In, a specific fraction krn recover,
a fraction kdn die from the disease, and the remainder
move on to the next infected state, In+1 (or recovery in the case
of the last infected state).
Populations of each state are expressed for convenience as fractions of the total population.
Each infected state has an infectiousness coefficient, kin. New infections (i.e. transitions from state S to state I1) are supposed to occur in the amount number in S times sum over n of kinIn, based on the "mass-action" idea of chemistry.
2. My Maple implementation, including the experiments I did, are here: mws, HTML.
3. In my investigation of how the total mortality for the epidemic depends on the number initially infected, I set all the infectiousness coefficients to the same value, 0.4, and all the mortality fractions to 0.2 except for the first infected state which I gave zero mortality (considering it to be a "latently infected" pre-symptomatic state. All the recovery fractions were set to 0.1, except for 0.8 for the last infected state, which is viewed as an almost-recovered state.
I then ran the model 12 times starting from initial condition with 12 different numbers of initially infected (population of I1). All those not in I1 were put in the susceptible state S. (Thus there was initially no-one in any other infected state, nor in R nor D.) I observed that the epidemic took longer to get off the ground the smaller the number of initially infected. Therefore, to make sure I captured the entire epidemic in all of the 12 cases, I chose to run for 100 time steps which I determined was long enough for the epidemic to play out even with the smallest initial infection number I was going to use. My initial numbers were 2-k for k running from 0 to 11.
A plot for the total mortality for these 12 cases is shown below, and the results were somewhat surprising to me.
Except for very large initial infections, the total number of deaths is almost independent of the size of the initial infection. About 32% of the population died, no matter how small the initial infection. The size of the initial infection affected only the time it took for the epidemic to reach its maximum intensity.
4. Quarantine means isolating those who are identified as having the disease so they cannot easily spread it to others. I represented this by reducing the infectiousness coefficients for states I4 through I7 by a factor q, which I varied through the values 1 (no quarantine), 0.5, 0.25, 0.125, and 0 (perfect quarantine). I left the infectiousness coefficients for states I1 through I3 unchanged to model the assumption that the disease would only be recognized at the end of stage 3, at which point the person would be put in quarantine.
As in answering Question 3, I determined the total mortality for the epidemic as a function of number of initially infected individuals, now repeating that whole set of runs for each of the 5 values of the quarantine "leakage coefficient" coefficient, q. The results are shown below. (I note however, that for small q, the epidemic starts so slowly that I believe I may have not taken enough time steps to capture the entire course of the epidemic, so the true mortalities may be higher than shown for the small-q runs.)
If we focus on, say a 1% initial infection (-2 on the base-10 log scale), we see that a 50% effective quarantine saves a significant number of lives, dropping the mortality from about 32% to about 20%. That's a reduction in the number of deaths by over 1/3. Reducing the quarantine leakage to 0.25 further reduces the mortality to about 15% of the total population. The gains achieved by further improvements in the quarantine are progressively smaller. And even a "perfect" quarantine (q=0) cannot prevent losing 7% of the total population to the disease. (This is because the pre-detection infected people infect others.)
5. Looking at the quarantine plot, I realize that it at some point it becomes more effective to put effort into detecting the disease earlier, than to clamp down tighter on those in quarantine. I suppose this is obvious, but the model provides a way of quantifying the issue.