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Why, if the average recovery time is 14 days, does this imply that on average one third of the infected population recovers?
Sorry, that was a typo. I will see if I can fix the notebook text.
I think I fixed it.
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The statement is still incorrect. In the model with these parameter values, all the susceptible individuals get infected and all recover.
If the recovery rate is fast enough, R0 = b N / k = b / k will be less than 1 and the infection will die out before everyone gets infected. The exact value can be worked out from the equations in Brauer.
Sure, but why are the values I am using incorrect? I am just using example values (which anyone can vary to experiment with this model?)
I see the difficulty now.
I was interpreting the statement "Here we assume that the average recovery time is 14 days, which means that, on average, 1/14th of the infected population recovers” too broadly, that is, I was assuming you meant the total number of infected and not the current number of infected.
What you have is indeed correct. Sorry for the confusion.
The sentence would be clearer if it would say: “… 1/14th of the infected population recovers every day:”
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No, this is an instantaneous rate, not a daily rate.
It has to have units of recoveries/time, and by design the time units are days.
Yes, the units are day^-1, but the fraction of infected that recover in a given day is the integral
Integrate[k i[t], {t, t1, t2}] / i[t1]
where t2 - t1 = 1 and i[t] is the solution from NDSolve, is not necessarily exactly 1/k.
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What about if the Recovery equation changes to something like: equationR = r’[t] == (1 - w) k i[t], where w = a “death” parameter, such as w = 0.005?
Then, DSolve[equationR, {r[t]}, {t}] gives:
r[t] → C[1] +
Inactive[Integrate][(k i[K[1]] - k w i[K[1]]), {K[1], 1, t}]
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How do I calculate b from real data?
Here are the cases the begin of infecteds in São Paulo Brazilian State
data = {1,1,1,2,2,2,2,3,6,10,13,16,19,30,42,56,65,136,152,164,240,286,396,459,631,745,810};
Clear[a, k];
NonlinearModelFit[data, a Exp[k t], {a, k}, t]["BestFitParameters"]
{a → 2.35641, k → 0.219352}
It’s somehow related to k?
Yes, I think this basic set of differential equations can be extended in many ways to make it more useful and precise. It would be interesting to see what you can do with it!
I am not sure how to best calculate b and k from actual data. This model is very simple and the real world is quite complex (different spreading rates for different people and places, and time-dependent spreading rates due to changing policies). It may be possible to get a fit for a current situation, but this fit probably has limited predictive value.
This model does have value to understand how changing the parameters affects the spreading of infectious diseases.
I was wondering if you could change the WindowSize option to Automatic for this notebook. I downloaded it from WolframCloud and it displayed in Full Screen mode, making it impossible (AFAIK) to resize it. I was finally about to open options dialog and reset the WindowSize option. Would others have the same issue?
Can this be replicated with discrete time?
I think so. But I have not tried it.
II tried to replicate it in discrete time but RSolve/RSikvevakye couldn’t solve it.
SIR_Dicrete_time.nb (12.5 KB)