Reproductive Numbers

An Introduction to Thresholds

Author
Affiliation

David L Smith

University of Washington

The basic reproductive number, \(R_0\) is a useful way of understanding dynamical systems describing malaria.

See Related for links to closely related vignettes.

Generically, suppose we have a dynamical system describing malaria transmission dynamics. We want to develop a metric that serves as a threshold criterion for closed systems, and that can be used to benchmark models. This is an outline:

Simple Systems

In autonomous, 1-patch systems with no control:

\[R_0 = V D\]

Spatial Dynamics

  • \([V]\) - VC Matrix

  • \([D]\) - HTC Matrix

  • Next generation spatial matrices:

    • \([R_x] = [V]\cdot[D]\) - from human to human

    • \([R_z] = [D]\cdot[V]\) - from mosquito to mosquito

Thresholds:

  • \([R_x] \cdot \tilde X = \lambda \tilde X\) where \(\lambda\) is a scalar, then \(\tilde X\) is an eigenvector

  • \([R_z] \cdot \tilde Z = \lambda \tilde Z\) where \(\lambda\) is a scalar and \(\tilde Z\) is an eigenvector.

The largest \(\lambda\) is called the dominant eigenvector or the spectral average, and \(\tilde X\) or \(\tilde Z\) are the associated eigenvectors.

Connectivity

We can rewrite:

  • \([D] = D [K_D]\), where \(D\) is a scalar value and \([K_D]\) describes parasite dispersal among patches by humans

  • \([V] = V [K_V]\), where \(V\) is a scalar value and \([K_V]\) describes parasite dispersal among patches by mosquitoes

Now connectivity from any starting set of bites is

  • \([C_x] = [V]\cdot[D]\cdot \left< X \right>\)

  • \([C_z] = [D]\cdot[V]\cdot \left< Z \right>\)

The \(R_0\) versions of this are:

  • \(X = \tilde X\)

  • \(Z = \tilde Z\)

At the endemic equilibrium, \(X = fqZ\) and \(Z = \kappa\)

Forcing

When VC varies from day-to-day, we need functions to describe the temporal dispersion of a parasite generation in mosquitoes and in humans.

  • \(V(t)\) is the VC on day \(t.\) The number of infective bites arising at time \(t+\ell\) is \(V_\ell(t,\ell)\) such that \[V(t) = \int_t^\infty V(t ,\ell) d \ell\] and the annual average VC is: \[V_A = \int_0^{365} V(t) \; dt\]

  • \(D(t)\) is the HTC on day \(t.\) The infectiousness on day \(\ell\) of an infection that started on day \(t\) is: \(D_\ell(t,\ell)\) such that \[D_\ell(t) = \int_0^\infty D(t, \ell) d \ell\] and the annual average HTC is: \[D_A = \int_0^{365} D(t) \; dt\]

For one complete generation, we need to compute convolutions. The number of infections at lag \(\ell\) after the infection: \[R_{x,\ell}(t, \ell) = \int_0^\infty V_\ell(t, \ell-\zeta) D_\ell(t, \zeta) d\zeta\] so the number arising from infections at time \(t\) is \[R_x (t) = \int_0^\infty R_{x, \ell}(t, \ell) d \ell\] or \[R_z(t, \ell) = \int_0^\infty D_\ell(t, \ell-\zeta) V_\ell(t, \zeta) d\zeta\] and \[R_z(t) = \int_0^\infty R_{z, \ell}(t, \ell) d \ell\]

Fig 1: A diagram of computation for \(R_{x,\ell}(t, \ell)\) as a convolution .

We want to deal with canonical seasonal patterns where \[V(t+365) = V(t)\]

We want to wrap offspring generated over many years in the future by time of year, so we write

\[R(d) = \sum_{d+i\; 365} R(t)\] where \(i\) is all the counting numbers.

Let the number of infections occurring on day \(d\) be \(X(d)\) and \(Z(d),\) and we compute \[R_x(t) \tilde X(d)\] and after wrapping, for all \(d\):
\[R_x(d) \tilde X(d) = \lambda R_x(d)\] or we compute \[R_z(t) \tilde Z(d)\] and after wrapping, for all \(d\):
\[R_z(d) \tilde Z(d) = \lambda R_z(d)\] then \(\tilde X\) and \(\tilde Z\) are called the temporal eigenfunctions, and the largest value of \(\lambda\) is called the spectral average.

General

Now, we put it all together:

  • \([V](t)\)

  • \([D](t)\)

  • The spatial-temporal next-generation matrix is:

    • \([R](t)\)

    • \([Z](t)\)