Vaccination pulse

This model is somewhat different from the others presented in this section.

Rather than modelling the births with a pulse, they are assumed to be continuous and pulse is used to model vaccination. Every year a proportion of susceptibles are moved directly into the recovered class, mimicking a proportion of the population being vaccinated on a single day. This is not unrealistic, large numbers of individuals are in fact frequently vaccinated together. A good example is the mass vaccination of United Kingdom school children against measles (and rubella) in November 1994 [Ramsay et al.,

1994]. Modelling had predicted a large epidemic would occur in 1995. The

mass pulse vaccination of more than 90% of school children [Shulgin et al., 1998] successfully prevented the epidemic.

Quite a lot of work has been done in the field of pulsed vaccinations as

part of the study of vaccination strategies in general driven by the need to make best use of the available resources. In their review Nokes and Swinton [1997] noted how pulsed vaccination was used in America to great success in particular against polio. They highlight the mixed success of

pulsed vaccination in other areas of the world and suggest that part of the

problem is due to the time intervals and age ranges for vaccination being decided based on “intuitive reasoning rather than quantitative epidemiological understanding”.

A slight generalisation of this model is used by Stone et al. [2000], where rather than applying the vaccination pulse every year, it is applied every T years. He examines the model’s “infection free” solution. That is a solution where I ≡ 0 and S oscillates driven by the pulsing. An explicit expression for this solution is found. The criteria for the stability of this “infection free” equilibrium are determined, and the maximum number of years between pulses for which the stability is sustained is computed. This gives an estimate for the maximum advisable separation between vaccination pulses if the strategy is to be effective in eliminating the disease. This work is continued in D’Onofrio [2002b].

Using an age structured SIR model, they evaluate pulsed vaccination as a strategy for controlling measles in Israel. They show that pulsed vaccination could be a more effective strategy than the cohort vaccination strategy in place at the time.

D’Onofrio [2002a] applies the same vaccination pulse to an SEIR model.

He revisits his work [D’Onofrio, 2004] by adding gamma distributed infectious and latent periods (see Subsection 3.3.7 on page 131) to the basic SEIR model.

Lu et al. [2002] considers another variant of the SIR model. This time

adding vertical transmission, that is a proportion of those born to infected

parents are themselves infected. Thus births come into both the susceptible and infected classes. Two different vaccination strategies are considered in conjunction with this system: a constant rate and pulsed vaccination.

For the pulsed vaccination strategy, the existence of both infection-free and

endemic solutions is observed. A similar analysis is performed on the SIS model with both constant rate an pulsed vaccination by Zhou and Liu [2003].

Here, in contrast to much of the work above, only the endemic equilibria are considered so as to be comparable to the results from other systems in this section.

S I R

Susceptible Infected Recovered

xS gI

B βSI

Figure 3.17: Box diagram for the vaccination pulse model.

This model takes the simple SIR model from the previous subsection, but with a constant birth rate:

˙ S = −βSI +B ˙ I = +βSI −gI ˙ R = gI. (3.5)

The system is forced by

S(t+) =S(t−)(1−x) R(t+) =R(t−) +S(t−)x

)

fort=nτ,n∈N

xof the susceptibles at the start of every year. For comparability with the

pulsed SIR model the same range of proportions x is used. Hence 0.01≤ x≤0.5. For simplicity the birth rate is tied toxso thatB =x/τ. Although the frequency of vaccination pulses is fixed to be annual for comparison with

other models, it is possible to simulate varying the interval between pulses

by rescaling time. This would also rescale the infectious period, hence to consider the effect of a biannual vaccination pulse the value of gwould need to be doubled.

Figure 3.18 on the next page shows a selection of attractors for the system. They appear very different from the previous attractors studied. Most strikingly they appear “upside down”. The pulses of vaccination lead to a drop in the level of infection, rather than a rise with the birth pulses, this is of course just what a vaccination pulse should achieve. The second feature not seen before is the apparent spiralling to a fixed point in the blue and magenta attractors. It is suspected that this is not seen in the other systems because the perturbations here are smaller because they are a proportion of the value of S, rather than a fixed size. This means that the system is perturbed less from its un-forced fixed point.

The periods of the resulting attractors over the whole of the parameter

space, after pushing out, for g = 0.075 and g = 0.500 are displayed in Figure 3.19 and Figure 3.20 (on pages 113–115) respectively. The dynamics for g = 0.075 are very simple. Period one dominates, with only a small region of period two in the top left. Attractors from both regions are shown in Figure 3.18 on the following page. However, when g is raised to 0.500 the dynamics become much more complex. As seen before multiple tongues come down from the top left, although the pattern is subtly different, rather than overlapping the tongues lie next to each other.

Clearly much more work could be done here, and much has been done. Nevertheless, this model provides an interesting comparison to the other

models with birth pulses. Showing that similar kinds of complex dynamics

do occur and that they are perhaps a general feature of models with where seasonality is modelled though pulsing. Much more work would need to be done to demonstrate this.

10−2 10−1 100 Susceptibles,S 10−7 10−6 10−5 10−4 10−3 10−2 10−1 In fe ct ed s, I

Figure 3.18: A selection of attractors of the vaccination pulse model (Equations 3.5 on page 111). Parameters are given as (β/g, x, g). Red: (2.5,0.48,0.075) period 2, green: (1.3,0.4,0.075) period 1, blue: (7,0.42,0.500) period 2, ma- genta: (20,0.5,0.500) period 1.

1 5 10 15 20

Approximate reproductive ratio, β/g

0.01 0.02 0.05 0.10 0.20 0.50 P u ls e si ze , x Chaos 1 2 3 4 5 6 7 8 9+ Period

Figure 3.19: Attractors found for the vaccination pulse model (Equations 3.5 on page 111) after pushing out for g= 0.075.