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To explore the outcomes of intervention implemented within systems where active immunity is incomplete, the analysis is applied to the MSIRS2 model structure proposed in Section 3.3.3. The model is described by the set of system equations (3.33)-(3.39), where it can be recalled that the additional state variablesS2(t) and

I2(t) correspond to characteristics associated with altered secondary infections.

The parametersωR,σ andγ describe, respectively, the rate of loss of active immu-

As discussed in Section 3.3.1, in cases such as hRSV it is often unclear whether the more significant source of transmission is from primary or secondary infectivity. In the MSIRS2 model, higher values of σ and γ result in a greater prevalence of reinfection, which may subsequently dominate the behaviour of the system and the overall force of infection. In contrast to the MSIR example, rapid waning of active immunity yields lower numbers of seropositive individuals throughout the childbearing population, leading also to fewer numbers of children born with pro- tective levels of MAb. The objective of this analysis is therefore to explore whether the static influence of maternal immunisation over wider population transmission is affected by the balance of primary and secondary infection.

The MSIRS2 system is found to have a single, stable equilibrium point correspond- ing to either eradication or endemic persistence of the infection. Provided that the invasion threshold is met, an endemic level of transmission is observed, which is subsequently dependent on the intervention parameters, Pm, Pv and ϑ. The equi-

librium is evaluated at nominal parameter values for hRSV (Table 3.2), where the secondary infection parameters,ωR,σ andγ, and the transmission parameter,

β, are varied to encompass a spectrum of infectivity biases, whilst maintaining a constant force of infection.

The resulting relations between the endemic prevalence of primary infection and various combinations of intervention effort, for primary and reinfection dominated transmission, with and without antibody interaction, are shown in Figure 6.3. For Figures 6.3(a) and 6.3(b) the secondary infection parameters are set to: ωR =

1.8 yr−1 _{(average duration of immunity of 6.67 months), σ = 1 and γ = 1, which}

correspond to the upper interval values given by Table 3.3(C). In order to maintain a typical force of infection for hRSV of λ= 0.84 yr−1_{, the transmission parameter}

is set toβ = 152 yr−1.

For comparison, Figures 6.3(c) and 6.3(d) show the relations that result from parameterising the model such that it emulates an MSIR type structure, where

Chapter 6 6.1 Time Domain Steady State Analysis 152

only primary infection contributes to the overall force of infection. In this instance,

ωR = 0 yr−1 (lifelong immunity), σ = 0, γ = 0, and subsequently, β = 2490 yr−1

(see Table 3.3(A)).

(a) Primary infection profiles evaluated at ωR= 1.8 yr−1, σ= 1,γ= 1 &ϑ= 0.

(b) Primary infection profiles evaluated at ωR= 1.8 yr−1, σ= 1,γ= 1 &ϑ= 1.

(c) Primary infection profiles evaluated at ωR= 0 yr−1,σ= 0, γ= 0 &ϑ= 0.

(d) Primary infection profiles evaluated at ωR= 0 yr−1,σ= 0, γ= 0 &ϑ= 1. Figure 6.3: Endemic prevalence of primary infection for specific proportions of birth targeted vaccination,Pv, and antibody interaction,ϑ, with respect to mater-

nal immunisation,Pm. Evaluated at parameter values corresponding to reinfection-

dominant, (a)-(b), or primary infection-dominant, (c)-(d), transmission.

The results indicate that, when applied independently, both the active and passive immunisation methods yield a positive impact on the prevalence of primary infec- tion. For high values ofϑ, the efficacy of the active vaccine is once again inhibited through its interaction with MAb. However, in the absence of maternal immuni- sation, i.e. Pm = 0, it can be seen that the childhood vaccine is considerably more

This is likely to be a result of waning population immunity, which subsequently leads to a reduction in the number of seropositive newborns. It should also be noted that since the transmission parameter, β, is reduced in order to maintain a constant force of infection in the presence of greater secondary infectivity, the basic reproduction number,R0, is subsequently decreased.

When a combination of the two interventions is applied with high levels of in- teraction, it can be seen from Figure 6.3(d) that the MSIR-type system displays qualitatively similar results to that of the measles example shown in Figure 6.1(b). In this instance, birth targeted vaccination is predominantly limited by naturally occurring levels of MAb, such that it is still beneficial to implement the passive vaccine. However, in the reinfection example (Figure 6.3(b)), where natural levels of MAb are low, maternal immunisation has a significantly detrimental effect on the efficacy of the active vaccine. This implies that in situations where mater- nal immunisation is likely to interact with an existing or prospective childhood vaccine, its application should be considered with caution.

(a) Average age of infection forϑ= 0. (b) Average age of infection forϑ= 1.

Figure 6.4: Average age at primary infection for specific proportions of birth tar- geted vaccination,Pv, and antibody interaction,ϑ, with respect to maternal immu-

nisation,Pm; evaluated at nominal parameter values for hRSV whereωR= 1.8yr−1,

σ= 1 and γ = 1.

The resulting relations between the average age at primary infection and the ap- plied interventions are calculated for the reinfection example using equation (6.3) and shown in Figure 6.4. In contrast to the results shown in Figure 6.3(b), it can