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The SIR Model for Spread of Disease

Posted 3 years ago
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POSTED BY: Arnoud Buzing
32 Replies

Why, if the average recovery time is 14 days, does this imply that on average one third of the infected population recovers?

POSTED BY: Luc Hens

Sorry, that was a typo. I will see if I can fix the notebook text.

POSTED BY: Arnoud Buzing

I think I fixed it.

POSTED BY: Arnoud Buzing

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.

POSTED BY: Robert Nachbar

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?)

POSTED BY: Arnoud Buzing

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.

POSTED BY: Robert Nachbar

The sentence would be clearer if it would say: "... 1/14th of the infected population recovers every day:"

POSTED BY: Luc Hens

No, this is an instantaneous rate, not a daily rate.

POSTED BY: Robert Nachbar

It has to have units of recoveries/time, and by design the time units are days.

POSTED BY: Daniel Lichtblau

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.

POSTED BY: Robert Nachbar
Posted 3 months ago

Hi, I tried your method to estimate the total, with the command given below (" total= Integrate[solutionI[t_?NumericQ] , {t, 0, 100}] / i[0] ") but I don't get a number:

    b = 3.3925 // Rationalize; 
k = 5.9 // Rationalize;
 n = 157759;
 equationS = s'[t] == -b*s[t] i[t]/n; 
equationI = i'[t] == b*s[t] i[t]/n - k*i[t]; 
equationR = r'[t] == k i[t];

    solution =    NDSolve[{equationS, equationI, equationR, s[0] == 157756, 
        i[0] == 3, r[0] == 0}, {s, r, i}, {t, 0, 100}];

    solutionI[t_?NumericQ] := First[i /. solution][t];

  total=      Integrate[solutionI[t_?NumericQ] , {t, 0, 100}] / i[0] 
{max, arg} = NMaximize[{solutionI[t], 0 < t < 100}, t] 

p1 = Plot[i[t] /. solution, {t, 0, 100},   PlotRange -> All,   Epilog -> {Red, AbsolutePointSize[4],
        Point[{t /. arg, max}]}]

Is this model not suitable for your method?

Thanks

POSTED BY: Ser Man

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}]

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!

POSTED BY: Arnoud Buzing

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POSTED BY: Moderation Team
Posted 3 years ago

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?

POSTED BY: Updating Name

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.

POSTED BY: Arnoud Buzing
Posted 3 years ago

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?

POSTED BY: David G
Posted 2 years ago

Can this be replicated with discrete time?

POSTED BY: Y Getachew

I think so. But I have not tried it.

POSTED BY: Arnoud Buzing
Posted 2 years ago

II tried to replicate it in discrete time but RSolve/RSikvevakye couldn't solve it.

Attachments:
POSTED BY: Y Getachew

I've just finished making a stochastic solver for compartmental models, and will be making a post to Community later this week.

POSTED BY: Robert Nachbar

I would suggest either RecurrenceTable or an explicit loop for obtaining a numerical solution in discrete time steps. The system might not be amenable to exact analytic results such as would be provided by RSolve.

POSTED BY: Daniel Lichtblau
Posted 3 months ago

Hi Arnold, how do you calculate the total number of infected according to your model?

Thanks!

POSTED BY: Ser Man
Posted 3 months ago

Hi Arnoud! I was asked for help by a student with a similar problem. There was a similar model, but it differed in that the loss of immunity over time was added. Those. those who fell ill once could get sick again. Actually below is the code and graphics.

Params1 = {T0 -> 50.05, a -> 0.07, b -> 1.08, c -> 0.08};

Conditions1 = {T[0] == T0, II[0] == T0*0.1/100, R[0] == 0};

SIRm = {
   S'[t] == -a*S[t]*II[t] + c*R[t], 
   II'[t] == a*S[t]*II[t] - b*II[t], 
   R'[t] == b*II[t] - c*R[t], 
   T[t] == S[t] + II[t] + R[t]
   };

{Smsol, IImsol, Rmsol, Tmsol} = 
  First[{S, II, R, T} /. 
    NDSolve[Join[SIRm, Conditions1] /. Params1, {S, II, R, T}, {t, 0, 
      50}]];

Plot[{Smsol[t], IImsol[t], Rmsol[t], Tmsol[t]}, {t, 0, 50}, 
 PlotTheme -> "Scientific", ImageSize -> Large, 
 PlotLegends -> {"S(t)", "I(t)", "R(t)", "T(t)"}, 
 GridLines -> Automatic, PlotStyle -> Thick, 
 PlotLabel -> 
  "Changing S, I, R and T in time with the weakening of immunity"]

enter image description here

POSTED BY: Kirill Belov
Posted 3 months ago

Hi Kiril, this model is very interesting. It seems also to be more in agreement with the natural fluctuations of epidemics. But I wonder, how does one then calculate the total of infected, when a new "batch" of individuals is infected "continuously". ?

POSTED BY: Ser Man
Posted 3 months ago

I think it's like a regular flu epidemic - a certain number of people get sick consistently throughout the year. If, for example, we add the dependence of immunity on the season, that is, reduce and increase the rate of its loss depending on the season, then we will also see seasonal fluctuations in the number of infected people. And about your question - I think you just need to calculate the definite integral, but with the exact time. Then we get the number of cases per year. Oh yes, the result will have to be divided by the average time of illness

POSTED BY: Kirill Belov
Posted 3 months ago

But I meant how to calculate the number of infected over a specified period of time, say t=0 to t=10 with your model.

POSTED BY: Ser Man
Posted 3 months ago

Like this?

enter image description here

POSTED BY: Kirill Belov
Posted 3 months ago

Did you get an actual value of the total from your approach? I see only a plot, but no number, for say the period between day 0 and day 50.

POSTED BY: Ser Man
Posted 1 month ago

Great post! Does anyone know how to draw the curves of the infected individuals (I) for different values of b (say from 0.1 to 0.6) on the same graph? Here I am referring to the state variable I and parameter b in the original post. Thank you so much!

POSTED BY: Mad Hew
Posted 1 month ago

Please don't mind my noob question as i am trying to learn mathematica. I tried to do your work in my pc but the blue,Red,green lines are not showing. Can someone tell me what i have done wrong or what should i do?

enter image description here

POSTED BY: Paritosh Kumar
Posted 1 month ago

In your screenshot the values for the equationS, equationI and equationR variables are simply not set

POSTED BY: Kirill Belov
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