Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 6E (Mathematical Biology)

The per-capita net growth rate is the birth rate minus the death rate,Multiplying by the population size gives the ordinary differential equationThis is the exponential density-dependent birth model.

At a positive equilibrium, division by $N$ givesSince $b>d$, this has the positive solution

Write the vector field as $F(N)=N(b{e}^{-aN}-d)$. Its derivative isAt $N=N^{\ast }$, the first two terms cancel and $b{e}^{-a{N}^{\ast }}=d$, soThe one-dimensional linear stability analysis therefore shows that $N^{\ast }$ is locally asymptotically stable.

The susceptible equation may be writtenThe factor $b{e}^{-aS}$ is a density-dependent per-capita birth rate. The law of mass action gives the infection incidence $\beta SI$, which transfers individuals out of the susceptible compartment.

Likewise,There is no birth term in the infective equation, so only susceptible individuals reproduce in this model. Infectives die at total per-capita rate $d+\delta$. These terms define the susceptible-infective model with exponential density-dependent birth.

The disease-free equilibrium is $(S,I)=(N^{\ast },0)$. Forthe Jacobian matrix there isBecause this matrix is triangular, its eigenvalue are its diagonal entries:The second eigenvalue is positive exactly whenUnder this condition the linear stability analysis gives an unstable disease-free equilibrium, agreeing with the epidemic invasion threshold.

At an endemic equilibrium, $I^{\ast }>0$. The equation $\dot{I}=0$ then givesSince also $S^{\ast }>0$, the equation $\dot{S}=0$ givesThis infective population is positive precisely whenwhich is exactly the assumed instability condition for the disease-free equilibrium. Thus the endemic equilibrium of the susceptible-infective model with exponential birth exists.

At this equilibrium the relations above reduce the Jacobian matrix toHenceBy the trace-determinant stability criterion, both eigenvalues have negative real part. The linearization stability theorem therefore proves that the endemic equilibrium is locally asymptotically stable.