Universidad avanzadaApplied MathematicsPlanificado
Mathematical biology
Mathematical models in biology: population growth and the logistic equation, predator–prey and competition models, epidemic models such as SIR, stability analysis by linearisation, and the limits of models.
math.applied.mathematical-biology
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 37
Aprenderás a
- Analyse the logistic equation and discrete-time population models, including the onset of chaos in the logistic map.
- Find the equilibria of predator–prey and SIR models and classify their stability using the Jacobian.
- Interpret the basic reproduction number R₀ and the assumptions behind it.
Definición
- The logistic equation is dN/dt = rN(1 − N/K).
- The SIR model is S′ = −βSI/N, I′ = βSI/N − γI, R′ = γI.
- In the simple SIR model, R₀ = β/γ is the expected number of new infections caused by one infected individual in a fully susceptible population.
Teorema
- In the SIR model the number of infected individuals initially grows if and only if R₀ S(0)/N > 1.
- An equilibrium at which every eigenvalue of the Jacobian has negative real part is locally asymptotically stable.
- The Lotka–Volterra predator–prey model has a conserved quantity, so its orbits are closed curves.
Errores frecuentes
- Treating model predictions as forecasts independent of their assumptions.
- Thinking R₀ is a fixed biological constant; it depends on behaviour and environment.
Descubrimientos opcionales
Itinerarios de aprendizaje
Lecturas complementarias
- J. D. Murray, Mathematical Biology (1989)
- Leah Edelstein-Keshet, Mathematical Models in Biology (1988)
- Steven H. Strogatz, Nonlinear Dynamics and Chaos (1994)