Thread: Equilibirum solution to set of six differential equations

Dear All,

I'm a biologist and recently started mathematical modelling. I'm trying to find the equilibrium solutions to a set of six differential equations that represent a six-compartment model of infectious disease transmission. Although I can set this model up and run it, I'm interested in how I can solve for the unknown equilibirum prevalences in the model in terms of the known parameters (as this would be a useful way to describe aspects of the system). These are the equations:

dA/dt = m + o * C - A * (p + q + n)

dB/dt = p * A - B * (s + q * r + n)

dC/dt = s * B - C * (q + o + n)

dD/dt = q * A + o * F - D * (p * t + u + n)

dE/dt = p * t * D + q * r * B - E * (s * v + u + n)

dF/dt = s * v * E + q * C - F * (o + u + n)

And I'd like to solve for equilibrium when:
dA/dt = dB/dt = dC/dt = dD/dt = dE/dt = dF/dt = 0

i.e. solve for (A*, B*, C*, D*, E*, F*) in terms of m, o, p, q, n, s, r, t, v and u.

Any help or advice on this would be very much appreciated.

Thanks,
Z

Dear All,

Apologies, I've just realised that two of the above terms multiply out, so that the equations above are sadly more complex than I thought, and should have been posted as:

dA/dt = m + o * C - A * ((p * x * (B + D) / (A + B + C + D + E + F)) + (q * x * (D + E +F) / (A + B + C + D + E + F)) + n)

dB/dt = p * A - B * (s + (q * x * (D + E +F) / (A + B + C + D + E + F)) * r + n)

dC/dt = s * B - C * ((q * x * (D + E +F) / (A + B + C + D + E + F)) + o + n)

dD/dt = (q * x * (D + E +F) / (A + B + C + D + E + F)) * A + o * F - D * ((p * x * (B + D) / (A + B + C + D + E + F)) * t + u + n)

dE/dt = (p * x * (B + D) / (A + B + C + D + E + F)) * t * D + (q * x * (D + E +F) / (A + B + C + D + E + F)) * r * B - E * (s * v + u + n)

dF/dt = s * v * E + (q * x * (D + E +F) / (A + B + C + D + E + F)) * C - F * (o + u + n)

Sorry for any inconvenience, and any help much appreciated.

Thanks,
Z