The foundation for all prior analyses in this work has been the fundamental MSIR model shown by Figure 3.1. In this instance, the model is considered with the addition of maternal (passive) and potentially interacting childhood (active birth
targeted) immunisation as described in Section 3.3.3. The system equations are derived from the full incomplete immunity model (3.33)-(3.38) by removing the state variables corresponding to secondary infection (i.e. S2(t) andI2(t)) and set-
ting the parameterωRequal to zero. It should be recalled that the proportionsPm
andPv correspond to the application of maternal and birth targeted immunisation
respectively, which are applied in terms of the net birth rate,µN. The interaction between neonatal immunity derived from the two intervention types and natural infection experience of the mother is described by the parameter ϑ.
The system has a single, stable equilibrium point corresponding to either eradica- tion or endemic persistence of the infectious agent. In the absence of any interven- tion, these equilibrium points are delineated by the bifurcation point,R0 = 1. For
R0 >1 and under suboptimal intervention, an endemic level of infection transmis-
sion is observed, which is subsequently dependent on the proportionsPm, Pv and
ϑ. Evaluating this equilibrium using nominal values for pre-vaccine measles in the UK (Table 3.1), a series of relations between endemic prevalence of infection and various combinations of intervention effort, with and without antibody interaction, are elicited and shown in Figure 6.1.
(a) Primary infection profiles for ϑ= 0. (b) Primary infection profiles forϑ= 1.
Figure 6.1: Endemic prevalence of infection for specific proportions of birth tar- geted vaccination,Pv, and antibody interaction,ϑ, with respect to maternal immu-
nisation, Pm, and evaluated at nominal parameter values for pre-vaccine measles
in the UK.
Chapter 6 6.1 Time Domain Steady State Analysis 148
interventions would have a positive impact on reducing the transmission of the pathogen. In the absence of antibody interaction, i.e. ϑ = 0, Figure 6.1(a) shows that in this example it is possible to force the disease free equilibrium to become stable and eradicate the infectious agent. However, for high values ofϑ, the efficacy of birth targeted vaccination is significantly reduced. In Figure 6.1(b), high levels of antibody interaction are seen to prevent the eradication of the infection by means of the considered intervention strategy.
Following the analysis in Section 2.2.1, an eradication threshold forϑcan be found, above which, too few newborns successfully respond to the active vaccine and herd immunity can no longer be achieved. The system Jacobian matrix is evaluated at the disease free equilibrium, with maximum maternal and birth targeted immuni- sation (i.e. Pm,Pv = 1). This gives rise to a set of eigenvalues:
λ1 = −2µ, λ2 = −(ωM +ϑµ), (6.1) λ3 = −v−2µ+ ϑβωM 2(ϑµ+ωM) ,
which are always negative and real provided the interaction parameter adheres to the following inequality:
ϑ < 2(v+ 2µ)ωM βωM −2vµ−4µ2
. (6.2)
Evaluating (6.2) with respect to the UK measles parameter set (Table 3.1), it is found that for eradication of the infection to be possible, ϑ <0.11.
From Figure 6.1(b) it can be seen that forϑ = 1, the minimum prevalence of infec- tion that can be achieved is 9675 cases (a reduction of 611 cases or 5.94%). This limit in efficacy occurs since only newborns born to fully susceptible mothers will benefit from either of the two immunisations. As population immunity is raised, the number of successful birth targeted vaccinations quickly diminish, restricting
the impact of the intervention. If maternal immunisation is successfully applied to all pregnant women, i.e. Pm = 1, then all newborns will be protected by MAb and
potentially none will respond to the active vaccine. This result emphasizes the ef- forts of Williams et al. [1995] and Nicoara et al. [1999] who discuss the importance of targeting childhood vaccination beyond the average duration of MAb.
An expression approximating the average age at primary infection predicted by the model may be derived similarly to those of the MSIR (3.19) and MSIR with maternal immunisation only (3.61), and given in the following form:
Av ≈ ˆ µM ˆ µM + ˆµS (ω−1+λ−1) + µˆM ˆ µM + ˆµS (λ−1), (6.3) where ˆ µM = µ(1−(1−ϑ)Pv) ˆ Ab++P mAbˆ− , ˆ µS = µ(1−Pv)(1−Pm) ˆAb−.
The function is based on a weighted average of the corresponding proportions of seropositive (Ab+) and seronegative Ab− newborns, excluding those who success-
fully respond to the active (birth targeted) vaccine and avoid primary infection. The resulting relations between the average age at primary infection, the two forms of intervention and their interactions are illustrated by Figure 6.2, which is evalu- ated again with respect to the UK measles parameter set (Table 3.1). It should be noted, with respect to Figure 6.2(a), that for values of Pm, Pv and ϑ that result
in the eradication of the infectious agent, an average age at primary infection can no longer be calculated.
It can be seen from the two graphs in Figure 6.2 that again both interventions have a positive impact on the average age at primary infection. However, in cases where there are high levels of antibody interaction the benefits of active immunisation are substantially limited.
Chapter 6 6.1 Time Domain Steady State Analysis 150
(a) Average age of infection forϑ= 0. (b) Average age of infection forϑ= 1.
Figure 6.2: Average age at primary infection for specific proportions of birth tar- geted vaccination, Pv, and antibody interaction, ϑ, with respect to maternal im-
munisation,Pm; evaluated at nominal parameter values for pre-vaccine measles in
the UK.
All results in this section suggest that if the detrimental effects of antibody interac- tion are reduced then active immunisation is by far the most effective intervention with respect to wider population infection. However, the characteristics that have been discussed in this section are potentially accentuated by the solid and lifelong immunity assumed by the MSIR model. In this case, a particularly high propor- tion of the population is seen to reside in a seropositive state, leading to naturally high levels of MAb among newborns.