Question

1. Using the three difference equations listed of the SIR model, create a 3 column population model in Excel which shows the population from time step 1 to time step 300. Let your starting populations be: S[t] = 99, I[t] = 1, R[t] = 0. Let a = 0.001, and let γ = 0.05.

a. Create a line graph showing the populations of S[t], I[t], and R[t] through time.

b. Holding all other variables constant, try increasing and decreasing the infection coefficient a. How does changing the value of a affect the behavior of the model?

c. Holding all other variables constant, vary the infection coefficient γ (by increasing and decreasing the value). How does changing the value of γ affect the behavior of the model? Why?

Answer 1

1. SIR model using Excel sheet

N= total number of individuals. Which will be constant.
T= Time step
S = Susceptible
I= Infected
R= Recovered

T	S	I	R	N
1	99	1	0	100
2	98.95	1.049	0.001	100
3	98.89758	1.100375	0.002049	100
4	98.84261	1.154236	0.003149	100
5	98.78499	1.210702	0.004304	100
6	98.72459	1.269895	0.005514	100
7	98.66127	1.331943	0.006784	100
8	98.5949	1.39698	0.008116	100
9	98.52534	1.465146	0.009513	100
10	98.45243	1.536587	0.010978	100
11	98.37603	1.611455	0.012515	100
12	98.29596	1.689909	0.014126	100
13	98.21207	1.772113	0.015816	100
14	98.12417	1.858242	0.017588	100
15	98.03208	1.948474	0.019447	100
16	97.93561	2.042996	0.021395	100
17	97.83456	2.142005	0.023438	100
18	97.72872	2.245702	0.02558	100
19	97.61787	2.3543	0.027826	100
20	97.5018	2.468017	0.03018	100
21	97.38027	2.587082	0.032648	100
22	97.25303	2.711733	0.035235	100
23	97.11984	2.842215	0.037947	100
24	96.98043	2.978785	0.040789	100
25	96.83452	3.121707	0.043768	100
26	96.68185	3.271257	0.04689	100
27	96.52212	3.427718	0.050161	100
28	96.35502	3.591387	0.053589	100
29	96.18025	3.762567	0.05718	100
30	95.99748	3.941575	0.060943	100
31	95.80638	4.128735	0.064884	100
32	95.6066	4.324384	0.069013	100
33	95.3978	4.528867	0.073337	100
34	95.17959	4.742542	0.077866	100
35	94.95162	4.965776	0.082609	100
36	94.71348	5.198946	0.087574	100
37	94.46479	5.442439	0.092773	100
38	94.20513	5.696653	0.098216	100
39	93.93409	5.961993	0.103912	100
40	93.65125	6.238877	0.109874	100
41	93.35616	6.527728	0.116113	100
42	93.04838	6.82898	0.122641	100
43	92.72746	7.143073	0.12947	100
44	92.39293	7.470455	0.136613	100
45	92.04434	7.811579	0.144084	100
46	91.6812	8.166904	0.151895	100
47	91.30304	8.536895	0.160062	100
48	90.90938	8.922017	0.168599	100
49	90.49974	9.322739	0.177521	100
50	90.07363	9.73953	0.186844	100
51	89.63056	10.17286	0.196583	100
52	89.17005	10.62319	0.206756	100
53	88.69163	11.09099	0.217379	100
54	88.19483	11.5767	0.22847	100
55	87.67917	12.08078	0.240047	100
56	87.1442	12.60367	0.252128	100
57	86.58949	13.14578	0.264731	100
58	86.01459	13.70753	0.277877	100
59	85.41912	14.2893	0.291585	100
60	84.80266	14.89146	0.305874	100
61	84.16487	15.51437	0.320765	100
62	83.50539	16.15833	0.33628	100
63	82.82392	16.82364	0.352438	100
64	82.12018	17.51056	0.369262	100
65	81.39393	18.21929	0.386772	100
66	80.64498	18.95003	0.404992	100
67	79.87314	19.70291	0.423942	100
68	79.07833	20.47803	0.443645	100
69	78.26047	21.27541	0.464123	100
70	77.41955	22.09506	0.485398	100
71	76.55561	22.93689	0.507493	100
72	75.66877	23.8008	0.53043	100
73	74.75919	24.68658	0.554231	100
74	73.82709	25.59399	0.578917	100
75	72.87278	26.52271	0.604511	100
76	71.89663	27.47234	0.631034	100
77	70.89907	28.44243	0.658506	100
78	69.88061	29.43244	0.686949	100
79	68.84185	30.44177	0.716381	100
80	67.78343	31.46975	0.746823	100
81	66.70609	32.51562	0.778293	100
82	65.61064	33.57855	0.810808	100
83	64.49796	34.65765	0.844387	100
84	63.369	35.75196	0.879045	100
85	62.22477	36.86043	0.914797	100
86	61.06637	37.98197	0.951657	100
87	59.89495	39.11541	0.989639	100
88	58.71171	40.25954	1.028754	100
89	57.51792	41.41307	1.069014	100
90	56.31489	42.57468	1.110427	100
91	55.10399	43.74301	1.153002	100
92	53.8866	44.91665	1.196745	100
93	52.66418	46.09416	1.241661	100
94	51.43816	47.27408	1.287756	100
95	50.21003	48.45494	1.33503	100
96	48.98129	49.63523	1.383485	100
97	47.75341	50.81347	1.43312	100
98	46.52789	51.98817	1.483933	100
99	45.30623	53.15785	1.535921	100
100	44.08987	54.32105	1.589079	100
101	42.88027	55.47633	1.6434	100
102	41.67884	56.62228	1.698877	100
103	40.48694	57.75756	1.755499	100
104	39.30592	58.88082	1.813256	100
105	38.13705	59.99081	1.872137	100
106	36.98156	61.08631	1.932128	100
107	35.84062	62.16617	1.993214	100
108	34.71533	63.22929	2.055381	100
109	33.60673	64.27466	2.11861	100
110	32.51579	65.30133	2.182885	100
111	31.4434	66.30841	2.248186	100
112	30.39039	67.29512	2.314494	100
113	29.3575	68.26071	2.381789	100
114	28.3454	69.20455	2.45005	100
115	27.35467	70.12607	2.519255	100
116	26.38585	71.02477	2.589381	100
117	25.43936	71.90024	2.660406	100
118	24.51557	72.75212	2.732306	100
119	23.61478	73.58016	2.805058	100
120	22.73722	74.38414	2.878638	100
121	21.88303	75.16395	2.953022	100
122	21.05232	75.9195	3.028186	100
123	20.24511	76.65079	3.104106	100
124	19.46137	77.35788	3.180756	100
125	18.70102	78.04087	3.258114	100
126	17.96392	78.69992	3.336155	100
127	17.2499	79.33524	3.414855	100
128	16.55873	79.94708	3.49419	100
129	15.89013	80.53573	3.574137	100
130	15.24381	81.10152	3.654673	100
131	14.61942	81.64481	3.735775	100
132	14.01659	82.16599	3.817419	100
133	13.43493	82.66549	3.899585	100
134	12.87402	83.14373	3.982251	100
135	12.33341	83.60119	4.065395	100
136	11.81266	84.03834	4.148996	100
137	11.31129	84.45568	4.233034	100
138	10.82881	84.8537	4.31749	100
139	10.36474	85.23292	4.402344	100
140	9.918571	85.59385	4.487576	100
141	9.489799	85.93703	4.57317	100
142	9.077917	86.26298	4.659107	100
143	8.682418	86.57221	4.74537	100
144	8.302794	86.86526	4.831943	100
145	7.938539	87.14265	4.918808	100
146	7.589153	87.4049	5.00595	100
147	7.254138	87.65251	5.093355	100
148	6.933005	87.88599	5.181008	100
149	6.625271	88.10584	5.268894	100
150	6.33046	88.31254	5.357	100
151	6.048107	88.50658	5.445312	100
152	5.777755	88.68843	5.533819	100
153	5.518957	88.85854	5.622507	100
154	5.271277	89.01736	5.711366	100
155	5.034289	89.16533	5.800383	100
156	4.80758	89.30287	5.889548	100
157	4.590747	89.4304	5.978851	100
158	4.383397	89.54832	6.068282	100
159	4.185152	89.65702	6.15783	100
160	3.995642	89.75687	6.247487	100
161	3.814513	89.84824	6.337244	100
162	3.641418	89.93149	6.427092	100
163	3.476025	90.00695	6.517024	100
164	3.318012	90.07496	6.607031	100
165	3.167068	90.13583	6.697106	100
166	3.022893	90.18987	6.787241	100
167	2.885199	90.23737	6.877431	100
168	2.753707	90.27862	6.967669	100
169	2.628151	90.3139	7.057947	100
170	2.508273	90.34347	7.148261	100
171	2.393826	90.36757	7.238605	100
172	2.284571	90.38646	7.328972	100
173	2.180281	90.40036	7.419359	100
174	2.080736	90.4095	7.509759	100
175	1.985727	90.4141	7.600169	100
176	1.895052	90.41437	7.690583	100
177	1.808516	90.41049	7.780997	100
178	1.725936	90.40266	7.871407	100
179	1.647133	90.39106	7.96181	100
180	1.571938	90.37586	8.052201	100
181	1.500188	90.35723	8.142577	100
182	1.431727	90.33534	8.232934	100
183	1.366406	90.31032	8.32327	100
184	1.304083	90.28234	8.41358	100
185	1.24462	90.25152	8.503862	100
186	1.187889	90.218	8.594114	100
187	1.133763	90.18191	8.684332	100
188	1.082124	90.14336	8.774514	100
189	1.032858	90.10248	8.864657	100
190	0.985857	90.05938	8.95476	100
191	0.941015	90.01417	9.044819	100
192	0.898235	89.96693	9.134833	100
193	0.857421	89.91778	9.2248	100
194	0.818483	89.8668	9.314718	100
195	0.781335	89.81408	9.404585	100
196	0.745893	89.75971	9.494399	100
197	0.712079	89.70376	9.584158	100
198	0.679818	89.64632	9.673862	100
199	0.649039	89.58745	9.763509	100
200	0.619672	89.52723	9.853096	100
201	0.591653	89.46572	9.942623	100
202	0.56492	89.40299	10.03209	100
203	0.539412	89.3391	10.12149	100
204	0.515073	89.2741	10.21083	100
205	0.49185	89.20805	10.30011	100
206	0.46969	89.141	10.38931	100
207	0.448544	89.073	10.47845	100
208	0.428365	89.00411	10.56753	100
209	0.40911	88.93436	10.65653	100
210	0.390734	88.8638	10.74547	100
211	0.373198	88.79247	10.83433	100
212	0.356462	88.72042	10.92312	100
213	0.340489	88.64767	11.01184	100
214	0.325245	88.57427	11.10049	100
215	0.310695	88.50024	11.18906	100
216	0.296808	88.42563	11.27756	100
217	0.283553	88.35046	11.36599	100
218	0.2709	88.27476	11.45434	100
219	0.258823	88.19856	11.54262	100
220	0.247294	88.12189	11.63081	100
221	0.236288	88.04478	11.71894	100
222	0.22578	87.96724	11.80698	100
223	0.21575	87.8893	11.89495	100
224	0.206173	87.81099	11.98284	100
225	0.197029	87.73232	12.07065	100
226	0.188299	87.65332	12.15838	100
227	0.179963	87.574	12.24603	100
228	0.172003	87.49439	12.33361	100
229	0.164403	87.4145	12.4211	100
230	0.157145	87.33434	12.50852	100
231	0.150213	87.25394	12.59585	100
232	0.143594	87.1733	12.6831	100
233	0.137272	87.09245	12.77028	100
234	0.131234	87.0114	12.85737	100
235	0.125467	86.93015	12.94438	100
236	0.119958	86.84873	13.03131	100
237	0.114696	86.76714	13.11816	100
238	0.10967	86.6854	13.20493	100
239	0.104869	86.60352	13.29161	100
240	0.100282	86.5215	13.37822	100
241	0.0959	86.43936	13.46474	100
242	0.091713	86.35711	13.55118	100
243	0.087713	86.27475	13.63753	100
244	0.083891	86.1923	13.72381	100
245	0.080239	86.10976	13.81	100
246	0.07675	86.02714	13.89611	100
247	0.073415	85.94445	13.98214	100
248	0.070228	85.86169	14.06808	100
249	0.067183	85.77887	14.15395	100
250	0.064272	85.696	14.23972	100
251	0.061491	85.61309	14.32542	100
252	0.058832	85.53014	14.41103	100
253	0.056291	85.44715	14.49656	100
254	0.053861	85.36413	14.58201	100
255	0.051539	85.28109	14.66737	100
256	0.049319	85.19803	14.75266	100
257	0.047197	85.11495	14.83785	100
258	0.045168	85.03186	14.92297	100
259	0.043228	84.94877	15.008	100
260	0.041374	84.86568	15.09295	100
261	0.0396	84.78258	15.17781	100
262	0.037905	84.6995	15.2626	100
263	0.036283	84.61642	15.3473	100
264	0.034733	84.53335	15.43191	100
265	0.03325	84.4503	15.51645	100
266	0.031832	84.36727	15.6009	100
267	0.030475	84.28426	15.68526	100
268	0.029178	84.20127	15.76955	100
269	0.027937	84.11831	15.85375	100
270	0.02675	84.03538	15.93787	100
271	0.025615	83.95248	16.0219	100
272	0.024529	83.86962	16.10586	100
273	0.02349	83.78678	16.18973	100
274	0.022496	83.70399	16.27351	100
275	0.021545	83.62124	16.35722	100
276	0.020635	83.53853	16.44084	100
277	0.019764	83.45586	16.52438	100
278	0.018931	83.37324	16.60783	100
279	0.018134	83.29066	16.69121	100
280	0.017371	83.20813	16.7745	100
281	0.016641	83.12565	16.8577	100
282	0.015943	83.04323	16.94083	100
283	0.015274	82.96085	17.02387	100
284	0.014634	82.87853	17.10683	100
285	0.014022	82.79627	17.18971	100
286	0.013435	82.71406	17.27251	100
287	0.012874	82.6319	17.35522	100
288	0.012337	82.54981	17.43785	100
289	0.011822	82.46777	17.5204	100
290	0.01133	82.3858	17.60287	100
291	0.010858	82.30388	17.68526	100
292	0.010407	82.22203	17.76756	100
293	0.009975	82.14024	17.84978	100
294	0.009561	82.05851	17.93192	100
295	0.009165	81.97685	18.01398	100
296	0.008785	81.89526	18.09596	100
297	0.008422	81.81372	18.17785	100
298	0.008074	81.73226	18.25967	100
299	0.007741	81.65086	18.3414	100
300	0.007422	81.56953	18.42305	100

2. Holding the other variable constant,
If you increase the infection coefficient γ, then the population move from susceptible to infected with a high rate that means Susceptible and infected line intersect each other on less time duration than the given condition.
If you decrease the infection coefficient γ, then the population move from susceptible to infected with a low rate that means Susceptible and infected line intersect each other on extended time duration than the given condition.

3. Holding the other variable constant,
If you increase the infection coefficient 'a', then the population moves from infected to recovery with a high rate which means more people in recovery than the given condition. It will help to attend the equilibrium stage faster.
If you decrease the infection coefficient a, then the population moves from infected to recovery with a low rate which means fewer people in recovery than the given condition. It will slow down to attend the equilibrium stage.
This happens because of infection coefficient 'a' is responsible for the movement of the population form the infected population to the recovery population. If it increases then the movement of population increase and if it decreases then the movement of population decrease.