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Math Help - Equilibrium solution to set of six differential equations

  1. #1
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    Equilibrium 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
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  2. #2
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    Re: Equilibrium solution to set of six differential equations

    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
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