𝐼
= 𝛽𝐼 − 𝑣𝐼
The rate of change of people who have recovered with respect to time:
= 𝑣𝐼
What do these actually mean?
Well, just how gradient is rate of change on a graph, rate of change means the same here.
If 𝐼 is positive then the number of infected people is increasing, i.e. the disease is spreading. If 𝐼 is zero then there is no one is becoming infected, or people are being infected and recovering at the same rate (either way, there is no change). If 𝐼 is negative then the number of infected people decreases, or more people are recovering than become infected. Note that these equations also feature the constants 𝛽 and 𝜈 . 𝛽 represents the contact rate, i.e. probability of infection when exposed to ill person directly. 𝜈 is the recovery rate (time taken to recover- and for the purposes of a simple model, become immune).These values are given by the formulae:
𝐷 = 1/𝜈
0
= β/v
Where 𝐷 is the duration of infection.
How does this work in real life then?
A convenient online resource models measles for us (note format has been changed for ease of understanding).For measles we have an average infection of about a week. Working in days;
1 𝜈 1 7
7 =
𝜈 =
take R 0
= 15
𝛽 𝜈
0
=
15 = 𝛽 0.14 𝛽 = 2.14
Therefore our 3 equations for rates of change become:
= −2.14 𝐼
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