Estimation of the final size of the coronavirus epidemic by the SIR model

Abstract

In the note, the SIR model is used for the estimation of the final size of the coronavirus epidemic. The current prediction is that the size of the epidemic will be about 85000 cases. The note complements the author’s note [1]

Full text

21.03.2020 21:57
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Estimation of the final size of the coronavirus epidemic by the SIR
model
Milan Batista
University of Ljubljana, Slovenia
milan.batista@fpp.uni-lj.si
(Feb 2020)
Abstract.
In the note, the SIR model is used for the estimation of the final size of the coronavirus
epidemic. The current prediction is that the size of the epidemic will be about 85 000
cases. The note complements the author’s note [1]
1. Introduction
In this note, we will try to estimate a final epidemic size by the SIR model [2, 3]. The
program implements the model is available at
https://www.mathworks.com/matlabcentral/fileexchange/74658-fitviruscovid19
2. The SIR model
The model equations are
dS IS
dt N


, (1)
dI IS I
dt N

, (2)
dR I
dt 
, (3)
where t is time,
 
St
is the number of susceptible persons at time t,

I It

is the
number of infected persons at time t,
 
Rt
is the number of recovered persons in time

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t.

is the contact rate, .and
1
is the average infectious period. From (1),(2),(3) we
obtain total population size N
const.N SIR  
(4)
The initial conditions are
 
0
0SS
,
 
0
0II
,
 
0
0RR
.
Eliminating I from (1) and (3) yield
 
00
expS S RR
N




 


. (5)
In the limit
t
the number of susceptible people left
S

is
 
00
exp
S S RR
N





 


, (6)
where
R
is the final number of recovered persons. Because the final number of
infected people is zero, we have, using (4),
NS R


. (7)
From this, and (6) the equation for
R
is

00
expR NS R R
N





  


. (8)
In order to use the model, we must estimate model parameters

,

and initial values
0
S
and
0
I
from available data (we set
0
0R
and
01
IC
).
Now the available data is a time series of the total number of cases C, i.e.,
CIR
. (9)
We can estimate the parameters and initial values by minimizing the difference between
the actual and predicted number of cases, i.e., by minimizing
 
2
0
ˆ, , min
tt
CC S
, (10)
where
 
12
,,,
tn
C CC C
are given number of cases in times
12
,, ,
n
tt t
and
 
12
ˆ ˆˆ ˆ
,,,
tn
C CC C
are corresponding estimates calculated by the model. For the

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practical calculation, we use MATLAB’s function fminsearch and for the
integration of the model equation MATLAB's function ode45.
3. Results
The results of the calculation are shown in Table 1 and on Figure 1. From data in
Table 1, we see that all data sets have high R2 (>0.98). Also, we can see that the final
number of recovered persons converge and the predicted values do not differ
substantially; however, the predicted total population involved differs substantially.
Here we note that from day 28, the collection of data changed. Until day 28, the
estimated epidemic size was about 52 000 infections; after that prediction change to
about 85 700 infections.
Table 1. Convergence study. After day 28, the method of data collection change.
Day N
S

R
, ,1nn
RR
 



2
R
12.feb.20
28
551513
473888
77625
1.429
2.897
2.689
1.077
0.988
13.feb.20
29
1300538
1189616
110922
1.048
4.026
3.854
1.045
0.988
14.feb.20
30
1434310
1318035
116275
0.925
4.157
3.988
1.042
0.990
15.feb.20
31
1203132
1095609
107523
0.932
3.905
3.730
1.047
0.991
16.feb.20
32
1002252
901990
100262
0.953
3.624
3.441
1.053
0.992
17.feb.20
33
864976
769448
95528
0.969
3.387
3.198
1.059
0.993
18.feb.20
34
774076
681530
92546
0.969
3.205
3.010
1.065
0.994
19.feb.20
35
683791
594070
89722
0.978
2.998
2.798
1.071
0.994
20.feb.20
36
619431
531645
87787
0.985
2.835
2.630
1.078
0.994
21.feb.20
37
574963
488460
86503
0.990
2.712
2.504
1.083
0.994
22.feb.20
38
544793
459126
85667
0.993
2.624
2.413
1.087
0.995
23.feb.20
39
522591
437513
85078
0.993
2.557
2.343
1.091
0.995
24.feb.20
40
506492
421823
84669
0.995
2.506
2.291
1.094
0.995

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Figure 1. Actual and predicted number of cases by the SIR model and logistic model
(data up to 25 Feb 2020)
Having a series of final predictions, we can estimate the series limit by Shanks
transformation [4, 5]
2
,1 ,1 ,
,1 ,1 ,
2
nn n
nn n
RR R
RRR R
  

  


. (11)
For data from Table 1, the current prediction is 84085 cases (Table 2).
Table 2. Iterated Shanks transformation.
day
R
R(
R
) R(R(
R
)) R(R(R(
R

)))
R(R(R(R(
R

))))
32
100262
33
95528 87470
34
92546
39247
62344
35
89722 83575
83974
84181
36
87787 83971
84179
84084
84085
37
86503
84107
84003
84499
38
85667 83673
83731
39 85078 83740
40 84669
4. Conclusion

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If a method of data collection will not change again, and if the situation will remain
stable, then by the SIR model, the predicted size of the epidemic is about 84 100 cases.
This prediction is comparable with the current prediction 84 000 infections, by the
empirical logistic model [6] (see Fig 2).
Figure 2. Comparison of models predictions (data up to 25.Feb 2020)
Appendix. MATLAB script
%SIR model - Dynamic of epidemic
%
% Reference:
% https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology
%
% Variables
% S - # of susceptible
% I - # of infected
% R - # recovered
%
% Parameters:
% beta -- 1/beta = Tc - typical time between contacts
% gamma -- 1/gamma = Tr - typical time until recovery
% N -- = S + I + R = const
%
% Derived parameters:
% RN = beta/gamma -- basic reproduction number (ratio)
%
% History:
% 23.Feb.2020 MB Created
global beta gamma N % parameters
global S0 I0 R0 % initial values (normalized S0=1)

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global C % number of infected
global init % initial guess
close all
% get data
[CC,date0] = getData();
fprintf('%12s %3s %10s %10s %10s %7s %7s %7s %7s\n',...
'Date','Day','N','Sinf','Rinf','beta','gamma','R0','R2')
init = false;
for ii = 28:length(CC)
% get data
C = CC(1:ii);
% estimate model parameters
b = parest();
% final number
Rinf = Rmax();
Sinf = S0*exp(-beta/gamma/N*(Rinf - R0));
% calculate R2
tspan = 0:length(C)-1; % final time
ic = [S0 I0 R0]'; % initial conditions
opts = []; % no options set
[~,z] = ode45( @SIR, tspan, ic, opts);
z = z(:,2)+z(:,3);
zbar = sum(C)/length(C);
SStot = sum((C - zbar).^2);
SSres = sum((C - z').^2);
R2 = 1 - SSres/SStot;
% print results
fprintf('%12s %3d %10d %10d %10d %7.3f %7.3f %7.3f %7.3f\n',...
datestr(date0+ceil(length(C)-
1)),ceil(length(C)),round(N,0),...
round(Sinf,0),round(Rinf,0),beta,gamma,beta/gamma,R2)
% fprintf('Estimated parameters\n')
% fprintf(' End date
%s\n',datestr(date0+ceil(length(C)-1)));
% fprintf(' Day number %d\n',ceil(length(C)));
% fprintf(' Population size %d\n',round(N,0));
% fprintf(' Initial infected %d\n',round(z(1,2),0));
% fprintf(' Remain susceptible %d\n',round(Sinf,0));
% fprintf(' Total recovered %d\n',round(Rinf,0));
% fprintf(' Contact rate %g\n',beta);
% fprintf(' recovery rate %g\n',gamma);
% fprintf(' Recovery time %g\n',1/gamma);
% fprintf(' Reproduction number %g\n',beta/gamma);
% fprintf(' R2 %g\n',R2);
end
% set parameters
tspan = 0:2*length(C); % final time
ic = [S0 I0 R0]'; % initial conditions
opts = []; % no options set
% simulate
[t,z] = ode45( @SIR, tspan, ic, opts);

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% plot results
figure
hold on
plot(t,z,'LineWidth',2)
legend('Susceptible','Infected','Recovered',...
'Location','best','FontSize',12)
xlabel('Day (after 16 jan 2020)')
ylabel('Cases')
grid on
hold off
shg
% plot comparsion
figure
hold on
plot(t,(z(:,2)+z(:,3))','k','LineWidth',2)
scatter(1:length(C),C,50,'filled')
legend('Predicted','Actual',...
'Location','best','FontSize',12)
xlabel('Day (after 16 jan 2020)')
ylabel('Cases')
grid on
hold off
shg
% save to global
ta = t;
Ca = z(:,2)+z(:,3);
function dzdt = SIR(~,z)
%SIR model
% x(1) = S, x(2) = I, x(3) = R
global beta gamma N
S = z(1);
I = z(2);
R = z(3);
N = S + I + R;
dzdt = [ -beta*I*S/N; beta*I*S/N - gamma*I; gamma*I];
end
function b = iniguess()
%INIGUESS Obtain initial guess
global beta gamma
global S0 I0 R0 % initial values
global C
global init
if ~init
beta = 1/0.00267103;
gamma = 1/0.00267232;
S0 = 1e8;
I0 = C(1);
R0 = 0;
init = true;
end
b(1) = beta;
b(2) = gamma;
b(3) = S0;
end
function b = parest

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%PAREST Parameter estimation
global beta gamma N
global S0 I0 R0 % initial values
maxiter = 20000;
maxfun = 20000;
b0 = iniguess();
options = optimset('Display','off','MaxIter',maxiter,...
'MaxFunEvals',maxfun);
[b, fmin,flag] = fminsearch(@fun,b0,options);
% disp('Exit condition:')
% disp(flag)
% disp('Smallest value of the error:');
% disp(fmin)
fun(b); % obtain x ped
beta = b(1);
gamma = b(2);
S0 = b(3);
N = S0 + I0 + R0;
end
function f = fun( b)
%FUN Optimization function
global beta gamma S0 I0 R0
global C Ca
% set parameters
beta = b(1);
gamma = b(2);
S0 = b(3);
tend = length(C);
tspan = 0:tend-1; % time interval
ic = [S0 I0 R0]'; % initial conditions
% solve ODE
try
[tsol,zsol] = ode45(@SIR,tspan,ic);
catch
f = NaN;
return
end
% check if calculation time equals sample time
if length(tsol) ~= length(tspan)
f = NaN;
return
end
Ca = (zsol(:,2)+zsol(:,3))'; % calculated number of cases
f = norm(C - Ca);
end
function r = Rmax()
%FSOLVE Calculate number of recoverd individuals after t=inf
global N S0 beta gamma R0
RN = beta/gamma;
r = fzero(@f,[0,S0]);
%-----------------------
function z = f(x)

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z = x - (N - S0*exp(-RN*(x - R0)/N));
end
end
function [C,date0] = getData()
%GETDATA Coronavirus data
% data from 16 Jan to 21 Jan https://i.redd.it/f4nukz4ou9d41.png
% data from from 22 Jan 2020 to 13 Feb 2020 are from
% https://www.worldometers.info/coronavirus/
date0=datenum('2020/01/16'); % start date
C = [
45
62
121
198
291
440
580
845
1317
2015
2800
4581
6058
7813
9821
11948
14551
17389
20628
24553
28276
31439
34876
37552
40553
43099
44919
60326
64437
67100
69197
71329
73332
75198
75700
76676
77673
78651
79400
80088
%----------- add new data here
]';
end
References

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[1] M. Batista, Estimation of the final size of the coronavirus epidemic by the logistic
model, medRxiv (2020) 2020.02.16.20023606.
[2] H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review 42(4)
(2000) 599-653.
[3] I. Nesteruk, Statistics based predictions of coronavirus 2019-nCoV spreading in
mainland China, medRxiv (2020) 2020.02.12.20021931.
[4] D. Shanks, Non-linear Transformations of Divergent and Slowly Convergent
Sequences, Journal of Mathematics and Physics 34(1-4) (1955) 1-42.
[5] C.M. Bender, S.A. Orszag, Advanced mathematical methods for scientists and
engineers I asymptotic methods and perturbation theory, Springer, New York, 1999.
[6] M. Batista, Estimation of the final size of the coronavirus epidemic by the logistic
model, 2020 DOI: 10.13140/RG.2.2.36053.37603.