Can this be solved? Logistic problem

[esloveday]

If the spread of a virus follows a logistic model find the percentage of citizens infected after 3 days if initially 2% had the infection, after 1 hour 10% were infected and after 5 hours 35% were infected.

my working:

dx/dt=kx
lnx=kt+c
x=e[sup:2vb458oy]kt+c[/sup:2vb458oy]=e[sup:2vb458oy]kt[/sup:2vb458oy]*e[sup:2vb458oy]c[/sup:2vb458oy] where e[sup:2vb458oy]c[/sup:2vb458oy] is the initial amount infected = 0.02n where n is the population size
so x=e[sup:2vb458oy]kt[/sup:2vb458oy]*0.02n

at t=1, x=0.01n --> k=ln5

but at t=5, x=0.35n --> k=(ln17.5)/5

is there a flaw in the question? Would the entire population be infected in approx 2.4 hours given after 1 hour 10% are infected?

 

[tkhunny]

The flaw is in your formulation. You hvae simple exponential, not a logistic.

Try this dx/dt = x(1-x) and give it another go.

Or maybe this version \(\displaystyle \frac{dx}{dt}\;=\;rx\left(1-\frac{x}{k}\right)\)

 

[esloveday]

dx/dt=rx(1-x)

lnx-ln(1-x)=rt+c

when t=0, x=0.02 --> c=ln0.25

so x=e[sup:youv1ssx]rt[/sup:youv1ssx]/(e[sup:youv1ssx]rt[/sup:youv1ssx]+4)

now when t=1, x=0.1 --> r=-0.81093
and when t=5, x=0.35 --> r=0.153451 ???

what would r and k be in dx/dt=rx(1-x/k)??

 

[tkhunny]

Those are just hints. You have three pieces of information. You will need a model with three parameters.

 

[DrMike]

what would r and k be in dx/dt=rx(1-x/k)??

If you solve this DE, you'll have three unknowns - r, k and the constant of integration.

You're given 3 values of x at 3 values of t, these will give you three equations. You should be able to solve these equations to find r, k and c.