Modeling the waning and boosting of immunity from infection or vaccination

Abstract

Immunity following natural infection or immunization may wane, increasing susceptibility to infection with time since infection or vaccination. Symptoms, and concomitantly infectiousness, depend on residual immunity. We quantify these phenomena in a model population composed of individuals whose susceptibility, infectiousness, and symptoms all vary with immune status. We also model age, which affects contact, vaccination and possibly waning rates. The resurgences of pertussis that have been observed wherever effective vaccination programs have reduced typical disease among young children follow from these processes. As one example, we compare simulations with the experience of Sweden following resumption of pertussis vaccination after the hiatus from 1979 to 1996, reproducing the observations leading health authorities to introduce booster doses among school-aged children and adolescents in 2007 and 2014, respectively. Because pertussis comprises a spectrum of symptoms, only the most severe of which are medically attended, accurate models are needed to design optimal vaccination programs where surveillance is less effective.

Keywords: Age- and immunity-structured population; Immuno-epidemiology; Mathematical epidemiology; Vaccination; Waning and boosting of immunity.

Fig. D.1.

Population age distribution. The observed distribution (blue circles) is determined from information in Table D.1. The simulation age distribution is also shown (red squares).

Fig. D.2.

Distribution of individuals in each age class by immune status. The proportion of susceptible (A) or infected (B) individuals from the total population of each status with no vaccination (A1)-(B1), with primary vaccination alone (A2)-(B2), with primary vaccination plus one booster dose (A3)-(B3), with primary vaccination plus two booster doses (A4)-(B4). (Column (A)) Colors represent the level of susceptibility: fully susceptible S₁ (red), low partial immunity S₂ (orange), medium partial immunity S₃ (yellow), vaccinated immunity S₄ (light blue), and complete immunity S₅ (blue). (Column (B)) Colors represent the level of symptoms and transmissibility: severe symptoms and full transmissibility I₁ (red), moderate symptoms and transmissibility I₂ (orange), mild symptoms and low transmissibility I₃ (yellow), and neither symptoms nor transmissibility I₄ (light blue). The height of the bars in the top row indicates the total proportion in each age class while the bottom row is normalized by age group. Colors from Brewer (2013).

Fig. D.3.

Distribution of individuals in each age class by immune status. The proportion of susceptible (A) or infected (B) individuals from the total population of each status with no vaccination (A1)-(B1), with primary vaccination only (A2)-(B2), with primary vaccination plus one booster dose (A3)-(B3), with primary vaccination plus two booster doses (A4)-(B4). (Column (A)) Colors represent the level of susceptibility: fully susceptible S₁ (red), low partial immunity S₂ (orange), medium partial immunity S₃ (yellow), vaccinated immunity S₄ (light blue), and complete immunity S₅ (blue). (Column (B)) Colors represent the level of symptoms and transmissibility: severe symptoms and full transmissibility I₁ (red), moderate symptoms and transmissibility I₂ (orange), mild symptoms and low transmissibility I₃ (yellow), and neither symptoms nor transmissibility I₄ (light blue). The height of the bars in the top row indicate the total proportion in each age class while the bottom row is normalized by age group. Colors from Brewer (2013).

Fig. D.4.

Relative change in incidence by age. Comparisons of the incidence of infection by age group under different vaccination strategies: Scenario 1 - primary relative to no vaccination (blue); Scenario 2 - primary vaccination with a single booster dose relative to no vaccination (red); and Scenario 3 - primary vaccination with both booster doses relative to no vaccination (orange). The large panel is a composite of the smaller ones, which are for individual S classes. Negative values on the y-axis indicate a reduction in incidence. In contrast to Fig. 3, the baseline of comparison is absence of vaccination.

Fig. D.5.

Relative change in incidence by age. Comparisons of the incidence of infection by age group under different vaccination strategies: Scenario 1 - primary relative to no vaccination (blue); Scenario 2 - primary vaccination with a single booster dose relative to no vaccination (red); and Scenario 3 - primary vaccination with both booster doses relative to no vaccination (orange). The large panel is a composite of the smaller ones, which are for individual S classes. Negative values on the y-axis indicate a reduction in incidence. In contrast to Fig. 3, proportionate mixing is assumed.

Fig. D.6.

Infectious population by symptomatic class under the assumption of proportionate mixing. The proportion of infectious individuals with severe symptoms (A), severe and moderate symptoms (B) or any symptoms (C) under no vaccination (blue), primary vaccination alone (red), primary vaccination with the first booster dose (yellow) and primary vaccination with both booster doses (purple). Note the y-axis log scale. In contrast to Fig. 5, proportionate mixing is assumed rather than the observed mixing matrix.

Fig. 1.

Schematic of the PDE system given in Eq. (2) - Eq. (4) for one age group. S₁, S₂, S₃, S₄, and S₅ (blue shaded boxes) represent susceptible individuals who are immunologically naive, have some immunity, are moderately immune, were recently vaccinated, and are fully immune, respectively. I₁, I₂, I₃, and I₄ (red shaded boxes) represent infected individuals with typically severe symptoms who are maximally infectious, moderate symptoms and reduced infectiousness, mild symptoms and even less infectiousness, and neither symptoms nor infectiousness, respectively (we set I₅ = 0 in the text for ease of notation). Recovery from disease leads to a fully immune state (orange dash-dotted line). As individuals age, susceptible ones with incomplete immunity, including naive (S₁), some (S₂), moderate (S₃), and vaccine-induced (S₄) immunity, can be infected (red solid line) and become infectious. After infection, they recover (dot-dashed orange lines) fully immune (S₅). However, as individuals age, their immunity wanes (black wavy lines). The immunologically naive group (S₁) can become immune (S₄) through primary or re-vaccination (black solid line). Groups with some (S₂), moderate (S₃), and vaccine-induced immunity (S₄) can become fully immune (S₅) through re-vaccination (green dotted lines).

Fig. 2.

Diagram showing the total number of secondary infections generated by an infectious person who became infected while in the (j, k) group. The horizontal progressions indicate that infectious people may age to the next group (infectious and alive) with probability π_jk.

Fig. 3.

Relative change in incidence by age. Comparisons of incidence by age group under different vaccination strategies: Scenario 1 - primary relative to no vaccination (blue); Scenario 2 - primary vaccination plus a single booster dose relative to primary vaccination alone (red); and Scenario 3 - primary vaccination plus two booster doses relative to primary vaccination with one (orange). The large panel is a composite of the smaller ones, which are for individual S classes. Negative values on the y-axes indicate that vaccination strategies reduce incidence.

Fig. 4.

Proportion of individuals shift between statuses. The relative change in proportion, normalized within age groups, is shown for (A) fully susceptible S₁ (red), low immunity S₂ (orange), medium immunity S₃ (yellow), vaccinated S₄ (light blue), and completely immune S₅ (blue) and for (B) severe symptoms and full infectivity I₁ (red), moderate symptoms and infectivity I₂ (orange), mild symptoms and low infectivity I₃ (yellow), and neither symptoms nor infectivity I₄ (light blue). (A1)-(B1) shows the difference between primary vaccination and no vaccination (Scenario 1 from Fig. 3). (A2)-(B2) shows the difference between primary vaccination with a single booster dose and primary vaccination alone (Scenario 2 from Fig. 3). (A3)-(B3) shows the difference between primary vaccination with both booster doses compared to primary vaccination with a single booster dose (Scenario 3 from Fig. 3). Colors from Brewer (2013).

Fig. 5.

Infectious population by symptomatic class. The proportion of infectious individuals with severe symptoms (A), severe and moderate symptoms (B) or any symptoms (C) under no vaccination (blue), primary vaccination alone (red), primary vaccination with the first booster dose (yellow) and primary vaccination with both booster doses (purple). Note that the y-axis is log scale.