Calculating the cumulative incidence is easy for a simple SIR model - the decline in susceptibility shows us how many people have become infected . First, notice that for every incident case, the value of
S
goes down by 1. So to calculate the total incidence in a given time interval, we just need to calculate the amount by which susceptiblity has fallen in that same time period . This is the approach in the following example .
Given parameters
γ=0.2;​​β=0.4;​​tMax=100;​​n=1000;
Define the force of infection (λ)
λ[t_]:=βi[t]/n;
Define the system of differential equations
eqns={s'[t]==-λ[t]s[t],i'[t]==λ[t]s[t]-γi[t],r'[t]==γi[t],s[0]==1000,i[0]==1,r[0]==0};
Solve the system
sol=NDSolve[eqns,{s,i,r},{t,0,tMax}][[1]];
In[]:=
case0=s[0]/.sol
Out[]=
1000.
In[]:=
case101=s[100]/.sol
Out[]=
202.615
In[]:=
Difference=case0-case101
Out[]=
797.385
From this,we can see that a total of 797 cases occurred during the epidemic . This number is the cumulative incidence .

A flexible approach

For the simple SIR model, the previous approach is sufficient, but if we want to calculate incidence in more complex models, the following code will help us
Cumulative incidence function
inc[tmax_]:=NIntegrate[λ[t]s[t]/.sol,{t,0,tmax}];
In[]:=
cump=Plot[inc[u],{u,0,100},AxesLabel->{"Time","Incidence"},PlotLegends->{"cumulative incidence"},PlotStyle->{Black,Dashed}]
Out[]=
cumulative incidence
In[]:=
p=Plot[Evaluate[{i[t]}/.sol],{t,0,tMax},PlotLegends->{"Infected"},AxesLabel->{"Time","Population"},PlotStyle->{Red}]
Out[]=
Infected
In[]:=
Show[p,cump,PlotRange->All]
Out[]=
Infected
cumulative incidence
The plot shows us that at the peak of the epidemic on day 34, 154 people were infected . By that time, a total of 506 people had been infected . By the end of the epidemic, around 75 days after the start of the simulation, nearly 800 people had been infected in total .
Create a table with S, I, R, and inc values
timeValues=Range[0,tMax,1];​​table=Table[{t,s[t]/.sol,i[t]/.sol,r[t]/.sol,inc[t]},{t,timeValues}];
Display the combined table with appropriate headers
Grid[Prepend[table,{"Time","Susceptible","Infected","Recovered","Cumulative Incidence"}],Dividers->All]
Out[]=
Time
Susceptible
Infected
Recovered
Cumulative Incidence
0
1000.
1.
0.
0.
1
999.557
1.2213
0.221396
0.442694
2
999.017
1.49128
0.491761
0.983038
3
998.358
1.8205
0.821852
1.64235
4
997.553
2.22176
1.22476
2.44652
5
996.573
2.71049
1.71638
3.42688
6
995.379
3.30531
2.31602
4.62133
7
993.924
4.02852
3.04705
6.07558
8
992.155
4.90682
3.93775
7.84458
9
990.006
5.97195
5.02222
9.99417
10
987.397
7.26139
6.34148
12.6029
11
984.236
8.81908
7.94464
15.7637
12
980.414
10.696
9.89037
19.5864
13
975.801
12.9506
12.2482
24.1988
14
970.251
15.6487
15.1002
29.7489
15
963.595
18.8629
18.5421
36.405
16
955.644
22.6711
22.6849
44.356
17
946.191
27.1537
27.6555
53.8091
18
935.015
32.3889
33.5966
64.9854
19
921.888
38.4463
40.6659
78.1122
20
906.588
45.3783
49.0335
93.4118
21
888.914
53.2085
58.8772
111.086
22
868.705
61.9191
70.3757
131.295
23
845.863
71.438
83.6989
154.137
24
820.377
81.6274
98.9957
179.623
25
792.343
92.2767
116.381
207.657
26
761.979
103.103
135.918
238.021
27
729.627
113.762
157.611
270.373
28
695.745
123.869
181.386
304.256
29
660.875
133.03
207.095
339.125
30
625.615
140.875
234.51
374.385
31
590.566
147.097
263.337
409.434
32
556.297
151.476
293.226
443.703
33
523.304
153.9
323.797
476.696
34
491.984
154.361
354.655
508.017
35
462.627
152.956
385.417
537.373
36
435.418
149.858
415.724
564.582
37
410.44
145.297
445.262
589.56
38
387.699
139.538
473.763
612.301
39
367.133
132.851
501.016
632.867
40
348.639
125.502
526.86
651.361
41
332.081
117.731
551.188
667.919
42
317.31
109.752
573.938
682.69
43
304.168
101.745
595.087
695.832
44
292.501
93.8552
614.644
707.499
45
282.158
86.1977
632.645
717.842
46
272.999
78.8571
649.144
727.001
47
264.894
71.8933
664.213
735.106
48
257.726
65.3447
677.929
742.274
49
251.388
59.2327
690.38
748.612
50
245.784
53.5644
701.652
754.217
51
240.829
48.3365
711.835
759.171
52
236.447
43.5376
721.015
763.553
53
232.572
39.1505
729.277
767.428
54
229.144
35.1541
736.702
770.856
55
226.111
31.525
743.364
773.889
56
223.427
28.2382
749.334
776.573
57
221.051
25.2686
754.68
778.949
58
218.948
22.5911
759.461
781.052
59
217.085
20.1812
763.734
782.915
60
215.434
18.0158
767.55
784.566
61
213.972
16.0728
770.955
786.028
62
212.676
14.3314
773.993
787.324
63
211.528
12.7724
776.7
788.472
64
210.509
11.3781
779.113
789.491
65
209.607
10.1321
781.261
790.393
66
208.806
9.01949
783.174
791.194
67
208.096
8.02664
784.877
791.904
68
207.467
7.14116
786.392
792.533
69
206.908
6.35187
787.74
793.092
70
206.413
5.64862
788.939
793.587
71
205.973
5.02229
790.004
794.027
72
205.583
4.46468
790.952
794.417
73
205.237
3.96838
791.794
794.763
74
204.93
3.5268
792.543
795.07
75
204.658
3.13399
793.208
795.342
76
204.416
2.78465
793.8
795.584
77
204.201
2.47402
794.325
795.799
78
204.011
2.19786
794.791
795.989
79
203.842
1.95239
795.206
796.158
80
203.692
1.73423
795.574
796.308
81
203.558
1.54035
795.901
796.442
82
203.44
1.36808
796.192
796.56
83
203.335
1.21502
796.45
796.665
84
203.242
1.07905
796.679
796.758
85
203.159
0.958254
796.882
796.841
86
203.086
0.850958
797.063
796.914
87
203.021
0.755654
797.224
796.979
88
202.963
0.671006
797.366
797.037
89
202.912
0.59583
797.493
797.089
90
202.866
0.529064
797.605
797.134
91
202.825
0.469772
797.705
797.175
92
202.79
0.417119
797.793
797.211
93
202.758
0.370362
797.872
797.242
94
202.729
0.328842
797.942
797.271
95
202.704
0.291974
798.004
797.296
96
202.682
0.259236
798.059
797.318
97
202.662
0.230168
798.108
797.338
98
202.644
0.204358
798.151
797.356
99
202.629
0.18144
798.19
797.371
100
202.615
0.161092
798.224
797.385
◼
  • From the cumulative incidence at each timestep, we can easily calculate how many infections occurred between any 2 timesteps, as follows:
    • We calculate the difference between the cumulative incidence at each given timestep minus the cumulative incidence at the preceding timestep . Adding a 0 as the first value, the incident_infections vector shows you how many infections have occurred in the intervals between each timestep .
    Calculate the difference in cumulative incidence
    Create tables for S, I, R, Cumulative Incidence, and Incident Infections
    Display the combined table with appropriate headers
    We might also want to know how many new cases occurred during a specific time, such as after the peak of the epidemic, e . g . between day 40 and 75.
    Difference of cumulative incidence at day 75 and cumulative incidence at day 40 :
    Calculate cumulative incidence at day 40 and 75
    Calculate the difference of cumulative incidence at day 75 and day 40
    Display the result
    Even after the peak of the epidemic was already reached, 144 new infections still occurred .

    CITE THIS NOTEBOOK

    Calculating incidence in a deterministic SIR model​
    by Athanasios Paraskevopoulos​
    Wolfram Community, STAFF PICKS, September 6, 2023
    ​https://community.wolfram.com/groups/-/m/t/3007127