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Edited by: Krasimira Tsaneva-Atanasova, University of Exeter, UK

Reviewed by: Maria Vittoria Barbarossa, Heidelberg University, Germany; Mauricio Lima, Pontifical Catholic University of Chile, Chile

*Correspondence: Dalila Hamami

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Vaccination programs for childhood diseases, such as measles, mumps and rubella have greatly contributed to decreasing the incidence and impact of those diseases. Nonetheless, despite long vaccination programmes across the world, mumps has not yet been eradicated in those countries: indeed, large outbreaks continue. For example, in Scotland large outbreaks occurred in 2004, 2005, and 2015, despite introducing the MMR (Measles-Mumps-Rubella) vaccine more than 20 years ago. There are indications that this vaccine-preventable disease is re-emerging in highly vaccinated populations. Here we investigate whether the resurgence of mumps is due to waning immunity, and further, could a booster dose be the solution to eradicate mumps or would it just extend the period of waning immunity? Using mathematical modeling we enhance a seasonally-structured disease model with four scenarios: no vaccination, vaccinated individuals protected for life, vaccinated individuals at risk of waning immunity, and introduction of measures to increase immunity (a third dose, or a better vaccine). The model is parameterised from observed clinical data in Scotland 2004–2015 and the literature. The results of the four scenarios are compared with observed clinical data 2004–2016. While the force of infection is relatively sensitive to the duration of immunity and the number of boosters undertaken, we conclude that periodic large outbreaks of mumps will be sustained for all except the second scenario. This suggests that the current protocol of two vaccinations is optimal in the sense that while there are periodic large outbreaks, the severity of cases in vaccinated individuals is less than in unvaccinated individuals, and the size of the outbreaks does not decrease sufficiently with a third booster to make economic sense. This recommendation relies on continuous efforts to maintain high levels of vaccination uptake.

To prevent, control and eradicate childhood diseases, vaccination programs have been adopted throughout the world. For example the trivalent measles-mumps-rubella vaccine (MMR) (Harling et al.,

A second hypothesis is to link vaccination status and age, e.g., proposing that outbreaks continue in the older population but die out in the increasingly vaccinated population. However, while age structure has shown to be informative in many models of traditionally childhood diseases (Andreasen,

Having rejected the first two hypotheses, the arguments used lead to the third and more plausible hypothesis: MMR vaccine efficacy against mumps reduces over time (van Boven et al.,

We investigate the hypothesis of waning immunity using mathematically-based computational modeling. The basic model is a seasonal compartmental SEIR model (Anderson et al., ^{1}

During the period 1988–2015, Health Protection Scotland (HPS), the national surveillance center for Scotland, reported 10943 mumps cases. 10486 of these cases were between 2004 and 2015. Vaccination was introduced in 1988, with a second dose introduced in 1996. Figure

We consider a compartmental structure for a model of mumps formulated as an extended SEIR (Anderson and May,

Our goal is to provide as simple a model as is necessary to demonstrate the impact of waning immunity, therefore we have ignored features which others have chosen to include. For example, the models of Glass and Grenfell (

This model is general and could be parameterised for any seasonal disease with up to two vaccinations. We use data from Health Protection Scotland (HPS) from 2004 to 2016^{2}

Demographic estimation

Birth and death rate (μ) estimated from Scottish demographic data^{3}

Immigration rate estimation (λ)

As the net migration to Scotland is insignificant (typically 15,000 per year), the model has been simplified by having neither mass emigration nor immigration of susceptible individuals. A small constant rate of immigration of infected individuals is required to prevent the disease dying out entirely. This is justified by the knowledge that there is immigration, and there are many populations in the world where mumps is more prevalent and the global population is more mobile, transmitting disease between countries. A small rate of immigration of infectious individuals is estimated as in Finkenstädt et al. (

Vaccination rates estimation (μ_{1}, μ_{2}, μ_{3})

According to vaccination data^{4}

Waning immunity estimation (τ, δ)

Our basic assumption is individuals vaccinated with MMR1 and MMR2 (resp. only MMR1) are temporarily protected and that immunity wanes toward susceptibility at constant rate δ (resp. τ). LeBaron et al. (

Transmission rate estimation (β1, β2, β3)

In our model, the transmission rate depends on two features: seasonality (High, Low) and type of susceptible (native susceptible, modified susceptible) giving four rates: β1 (High season and native susceptible), β2 (high season and modified susceptible, β3 (low season and native susceptible), β4 (low season and modified susceptible). For seasonality, data report higher number of cases October to May, and fewer between June and September^{5}^{6}_{0} (see Table _{0} is ranged [4–11]. See Sensitivity Analysis for sensitivity analysis of the particular choices of these rates.

Incubation rate α and recovery rate γ

Established empirical studies (Anderson et al.,

Initial conditions

The initial mix of susceptible, vaccinated, exposed, infected and recovered is calculated for 1996 according to the above assumptions about population based on vaccination beginning in 1988. See Appendix

The description of the model and parameters above can be summarized by seven ordinary differential equations:

This model is coded in Bio-PEPA (see Appendix

To capture the impact of vaccination efficacy and the effect of waning immunity on the population of Scotland for future projection of epidemics, the history of mumps epidemics (from pre-vaccine to post-vaccine era) are reproduced where four strategies are considered:

According to observed mumps data in Scotland in Figure

We begin by checking model performance without vaccine. Figure

We point out that predicted cycles do not damp out during 100 years of simulations. By varying seasonality parameter of the model, including removing seasonality altogether, we observed that after a long period the model reaches an endemic state. To further reinforce the suitability of the model we considered R_{0} ranging from [7 to 14]. Figure _{0} leads to decreasing the inter-epidemic period from 5 to 3 years.

Turning to the successful post-vaccine era (and assuming life-long immunity), Figures

Figure

Figure

Vaccination coverage dips in this period, but this is not the main factor leading to the resurgence and sustainability of mumps, nor is seasonality on its own (as above). We investigate the variability of vaccination coverage by ranging its value from [75 to 95], where 75% is the minimum value related to the threshold level and 95% is the maximum value of applied vaccine coverage in Scotland. Figure ^{7}

To further investigate the impact of waning immunity Figure

Further, we consider scenario 4: the impact of increasing the period of immunity by applying an additional dose of MMR (R and. Cameron, 2016). This could be similarly done by increasing immunity by increasing the efficacy of the vaccination (Public Health England,

The results above depend on precise parameter values, therefore we used sensitivity analysis to show that the qualitative results of periodic large outbreaks hold across the range. We identify significant parameters reproducing first the observed data, and second leading to the low level endemic state. Table

For all analysis we used ANOVA as implemented in Minitab (Minitab 17 Statistical Software, _{0} leads to a decrease in period between large outbreaks and therefore an increase in the number of oscillations (see Figure

Our analysis shows that mumps epidemics will continue, with larger outbreaks of ~1200 every 10 years as shown in Figure ^{8}

In this paper, we have presented the results of mathematical modeling using Bio-PEPA, identifying the impact of vaccination and waning immunity in the mumps component of the MMR vaccine. Even though vaccination has been ongoing since 1988, thus largely preventing mumps in children, our results show that waning immunity is the main factor in a repeated pattern of outbreaks. Simulations and analysis undertaken showed that waning immunity over 10 years leads to the highest number of infected and to the longest inter-epidemic period for larger outbreaks.

The first part of this study was to build a seasonal model which reproduces the patterns of the observed data in three scenarios: no vaccination, initial post-vaccine period with immunity for life, and with waning of vaccine-induced immunity as suggested by several sources (Dayan et al.,

Waning immunity is expressed in our model by including an additional compartment of modified susceptible, which is increased by vaccinated individuals (MMR1 and MMR2) losing their immunity. We find that assuming 5 years of MMR1 vaccine-induced immunity (resp. 10 years of MMR2 vaccine-induced immunity) generates simulation results consistent with more recent mumps post-vaccine data from Scotland (2004–2015). In addition, as our model suggests a 10-year-long gradually damping oscillation, the following trajectory of mumps disease would show a decrease in 2016 and so on, building back up from 2020 to another high peak in the year 2025. The most recent data provided by HPS has confirmed this prediction, where the year 2016 depicts 215 cases compared to 2015 which defines 836 cases. Although our estimates of the amplitude of mumps epidemics are higher than observed data, we conjecture that this can be explained by a low level of reporting. Anecdotally, cases of mumps in vaccinated individuals have much milder symptoms and therefore may be undetected (Public Health England,

By considering different values of immunity duration (scenario 4) we can estimate the time needed to reverse the epidemic trend and eliminate mumps. This models the situation that, for example, a new, more effective, vaccine is introduced, or a third vaccine dose is introduced into the national programme. This is shown in Figure

It is worth noting that the basic reproductive number R_{0} for the pre-vaccine era is estimated at 10.5 which falls in the range [7–14] as cited in literature (Anderson and May, _{0} is estimated at 6 where in the literature it is quoted at [4–7] (Anderson and May, _{0} indicates the number of secondary infections, clearly showing that the number of doses of vaccination and immunity duration has a great impact on decreasing infectious contacts.

Cumulatively, our findings suggest that the more “unprotected” individuals (who were either never vaccinated or lost their immunity), the shorter the period between two high peaks of epidemic outbreak (note the number of cycles in Table _{0}). In addition, in both cases related to scenarios 1 and 3 (No vaccination and waning immunity), an earlier high peak of mumps is expected. This occurs because the pool of susceptibles is increasing faster as those vaccinated lose their immunity and move to the susceptible state (scenario 3), or the pool of susceptibles is decreasing faster when no vaccination is applied and R_{0} is higher (scenario 1). Clearly, controlling the number of susceptible individuals has a great impact on controlling disease. As argued by Gay (_{0}) to be <1. We do this here by adjusting immunity duration.

These conclusions illustrate an enhanced understanding of mumps disease in response to mass immunization gained through mathematical modeling. Further, our multi-disciplinary team could explore the potential impact of further vaccination on cyclic outbreaks. Our conclusion for public health services is that they should urge vaccine uptake in those eligible since a high degree of protection is offered by the vaccine overall for those under 18. Considering the possible economic cost/benefit of a third vaccine dose, it seems that while there would be an increased period of immunity, the cyclic outbreaks would continue at about 2/3 the current level, therefore this would not offer significant advantages over the present situation. The Joint Committee on Vaccination and Immunization^{9}

We suggest further study with this model could include vaccination programmes targeted to those subject to waning immunity or at higher risk due to social mixing in a diverse population (as in higher education). Such a model might also include economic factors to allow the effect of targeted programmes to be more precisely evaluated. Another interesting facet would be to bring more attention to the level of immunity by analyzing the vaccine/virus content and detect eventual discrepancy between vaccine strain and mumps outbreak. This might also be linked with a data science approach to analyzing serology of confirmed cases. There are further opportunities to use data science to analyse other features, such as geographic distribution. These developments would allow an enhanced version of Figure

The Conception or design of the work: CS, DH, and KP. Data collection: RC, KP. Data analysis and interpretation: CS, DH, and KP. Drafting the article: DH, CS. Critical revision of the article: CS, KP, DH, and RC. Final approval of the version to be published: CS, KP, DH, and RC.

DH is grateful to the Algerian government, the University of Mostaganem and to the University of Oran for supporting her series of research visits to the University of Stirling.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank the reviewers for their many helpful comments to improve this paper.

The Supplementary Material for this article can be found online at:

^{1}Department of Health Protection Scotland, National surveillance center.

^{2}Department of Health Protection Scotland, National surveillance center.

^{3}Department of Health Protection Scotland, National surveillance center.

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^{5}Department of Health Protection Scotland, National surveillance center.

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^{7}Average number of infected corresponds to the average of the highest peaks during 100 years of simulations.

^{8}Department of Health Protection Scotland, National surveillance center.

^{9}UK body advising government health policy on vaccination and immunisation.