To vaccinate, or not to vaccinate

Dimitri Breda, Giulia Gava

In recent models of SIR epidemics the number of vaccinated newborns depends on the information available to their parents with respect to the concerned disease, [1]. This is formulated as

\[ M(t)=\int_{0}^{+\infty}g(S(t-\sigma),I(t-\sigma))K(\sigma)d\sigma,\]

for \(S\) and \(I\) the number of susceptible and infected individuals and

\[ K(\sigma)={\rm Erl}_{n,a}(\sigma):=\frac{a^{n}\sigma^{n-1}}{(n-1)!}e^{-a\sigma}\]

the Erlangian distribution with shape parameter \(n\in\mathbb{N}\) and rate parameter \(a>0\).

The interesting cases are that of fading memory, corresponding to the exponential non-delayed case \(n=0\) and, on the opposite side, that of fixed memory \(\tau>0\), corresponding to the limit as \(n\rightarrow\infty\) when \(a=n/\tau\), for which

\[ K(\sigma)=\delta(\sigma-\tau),\]

Dirac’s delta distribution at \(\tau\).

It is possible to prove that fading memory leads to at most one instability island due to Hopf bifurcations, while fixed memory leads to infinitely many such islands.

Moreover, the number of Hopf bifurcations increases as \(O(n)\). This can be proved by using the linear chain trick [2] and by analyzing the relevant characteristic equations via Rouché’s Theorem.

Left: Convergence of the Erlangian distribution to Dirac’s delta. Right: Real part of the rightmost eigenvalues as \(n\rightarrow\infty\).

  • [1] A. d’Onofrio, P. Manfredi and E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoret. Population Biol., 71 (2007), pp. 301–317, DOI: 10.1016/j.tpb.2007.01.001.
  • [2] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge Stud. in Math. Biol. 8, Cambridge Univeristy Press, Cambridge, 1989.