Help with a differential equation problem. If I am given a population size, and an infected individual enters the population, I need to find the number of infected individuals at time t.
I am also given k.
How do I go about writing the equation?

Question

Help with a differential equation problem. If I am given a population size, and an infected individual enters the population, I need to find the number of infected individuals at time t.
I am also given k.
How do I go about writing the equation?

anonymous · Answer

More info?

anonymous · Answer

Is this for a DE class?

anonymous · Answer

Yes it is. I am quite lost (in coming up with the equation myself).
Population is 999. 1 infected individual. All susceptible to the disease. k=0.0005.
I need to find the number of infected invidivials after 15 days.

anonymous · Answer

I assume I use the model P=Poe^kt

amistre64 · Answer

dP/di  is the rate of change of the population with respect to the infection right?

i never was good at modeling with these either

amistre64 · Answer

if P' = kP

then we can determine

P' - kP = 0

amistre64 · Answer

in other words, if the rate of change in the population is dependant upon the size of the population at a given moment in time .... then kP is the size of the population appropriate to the rate of change in the population

anonymous · Answer

yes i started with dP/dt = kP
then i create an equation where dP/dt=0.0005P
and go on to separate? which gives me
P=Ae^-0.0005t
?

anonymous · Answer

When I plug in A=999 (initial population) and t=15 I get 1077
So i assume 1077-999 = 78 infected persons ???

anonymous · Answer

How does the infected person come into play in the equation?

anonymous · Answer

Initial infected was 1, not 999

irishboy123 · Answer

i think you ned to be more specific.  what for example do you think k actually means

you could go at this any number of ways.  eg you might look to see exponential growth in the number of infections.  thus the equation would be dI/dt = kI, but that gives nonsense answers

alternatively, you might say that the pop P will decay exponentially due to k (whatever it means):

dP/dt = -kt

in which case you get P = 999 e^(-kt) (clarify is the 1 infected person within the 999 or does she/he make it up to 1000)

this expo decay approach means about 6 or 7 people die in 16 days, which makes more sense

however that too depends upon what k means....

anonymous · Answer

I would suggest Kermack-McKendrick SIR model, quite funny if you get into it ;)