Table 1 Summary of notation

${\overline{M}}_k$

Passively immune infants in k th group

${\overline{S}}_k$

Susceptibles in the k th group

${\overline{I}}_k$

Infectives in the k th group

${\overline{R}}_k$

Recovered people with immunity in the k th group

$N_k$

Total population size in the k th group

$b_k$

Fraction at which new-borns of group k have the passive immunity

${\beta }_{kj}$

Rate of disease transmission between susceptible individuals in the k th group and infectious individuals in the j th group

$d_k^M$, $d_k^S$, $d_k^I$, $d_k^R$

Mortality rates of susceptible, infectious and recovered individuals in the k th group, respectively

${\tau }_k$

The rate of immunity loss in the k th group

${\delta }_k$

Rate of the transfer out of the passively immune class in the k th group

${\gamma }_k$

Recovery rate of infectious individuals in the k th group