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Beginner at modelling here.

To generate an equation for the spread of disease, my textbook gave me this:

  • x(t) denotes the number of people who have contracted the disease

  • y(t) denotes the number of people who have not yet been exposed

  • The number of interactions is jointly proportional to x(t) and y(t)

Therefore: dx/dt = kxy

Suppose a small community has a fixed population of 'n' people. If one infected person is introduced into this community, then: dx/dt = kx(n+1-x)

My question: why 'n+1'? If y=n+1-x, then wouldn't this mean the number of 'uninfected' people is greater since y(t) denotes the number of people who have not yet been exposed?

To introduce an infected person, wouldn't you change x to be x+1?

Thank you in advance!

noodles
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  • For every population of any size N, you are given that the number x(t) of infected individuals evolves according to dx/dt = kx(N-x). Before the introduction, N=n hence dx/dt = kx(n-x). After the introduction of a new individual, whether she is infected or not, N=n+1 hence dx/dt = kx(n+1-x). Nothing else matters. – Did Aug 07 '18 at 04:38

1 Answers1

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You start with a community of $n$ people. When you introduce on infected person the community now has $n+1$ people. If $x$ are infected, the number uninfected is $n+1-x$. At the moment the infected person is introduced we have $x=1$ infected person and $n=y=n+1-x$ uninfected people.

Ross Millikan
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