Analysis of the yearly transition function in measles disease modeling

Abstract

Globally, there were an estimated 9.8 million measles cases and 207 500 measles deaths in 2019. As the effort to eliminate measles around the world continues, modeling remains a valuable tool for public health decision-makers and program implementers. This study presents a novel approach to the use of a yearly transition function that formulates mathematically the vaccine schedules for different age groups while accounting for the effects of the age of vaccination, the timing of vaccination, and disease seasonality on the yearly number of measles cases in a country. The methodology presented adds to an existing modeling framework and expands its analysis, making its utilization more adjustable for the user and contributing to its conceptual clarity. This article also adjusts for the temporal interaction between vaccination and exposure to disease, applying adjustments to estimated yearly counts of cases and the number of vaccines administered that increase population immunity. These new model features provide the ability to forecast and compare the effects of different vaccination timing scenarios and seasonality of transmission on the expected disease incidence. Although the work presented is applied to the example of measles, it has potential relevance to modeling other vaccine-preventable diseases.

1 |. INTRODUCTION

Globally, there were an estimated 9.8 million measles cases and 207 500 measles deaths in 2019.¹ After many years of advances in the fight against measles due to increased coverage of measles-containing vaccine (MCV), global measles vaccination coverage reached a plateau in 2016. During 2017–2019, annual measles cases reported globally consistently increased, reaching a number not seen since 1996. The COVID-19 pandemic introduced additional challenges in 2020, disrupting immunization services worldwide and leaving millions of children exposed to potential measles epidemics.² However, a reduction in measles cases was observed in 2020, possibly due to the non-pharmaceutical interventions adopted to prevent COVID-19, such as school closures and social distancing. As these non-pharmaceutical interventions begin to ease in many areas globally and measles vaccination coverage remains suboptimal in many countries, measles outbreaks are likely to occur.^3,4

As global efforts to eliminate measles continue, modeling remains a valuable tool for public health decision-makers and program implementers, as evidenced by the dramatic increase in mathematical modeling of COVID-19 during the recent pandemic.^5–14 Together with a more localized approach to finding sub-populations with low coverage,¹⁵ which can be obscured by coverage estimates in larger population groups,¹⁶ modeling can help inform routine immunization (RI) policy (eg, recommended vaccination ages) and interventions to improve RI coverage, as well as optimization of the timing of supplemental immunization activities.¹⁷ There have been extensive efforts to model measles disease transmission and vaccination impacts over time,^18–34 spanning from the use of deterministic and stochastic models, time series analyses, and logit functions to metapopulation, agent-based, and network approaches. Literature surveys have followed these developments that started in the 1960s.^35–37 These models are used for the continued assessment of the global measles situation,³⁸ including some, in particular, in low- and middle-income countries (LMICs).³⁷ A Vaccine Impact Modelling Consortium (VIMC) was created to inform resource allocation decisions and vaccination strategies by major global health partners, such as Gavi, the Vaccine Alliance, the Bill and Melinda Gates Foundation, the World Health Organization, and others. The VIMC includes two models for each of the 10 diseases in the portfolio of vaccines supported by Gavi for use in 98 LMICs. The two models used for measles have been described by Verguet et al.³⁰ and Eilertson et al.³⁹ The first is an age-stratified dynamic compartmental model of measles transmission, while the second is a dynamic meta-population model, using particle filtering to fit an attack rate function at a country level. Notwithstanding the recent increased attention to disease transmission modeling during the COVID-19 pandemic, these two models continue to be used at global level to inform measles vaccination policy and resource allocation across countries.

This article expands upon the Eilertson et al. model by presenting a novel approach to the use of a yearly transition function that formulates mathematically the vaccine schedules for different age groups while accounting for the effects of the timing of vaccination and disease seasonality on the yearly number of measles cases in a country. This model was selected because it uses the annual reports of measles cases by country to adjust its infection function for that population, so that the one-year time-steps in the model are compared to the existing yearly case reports. This analysis extends the model presented by Eilertson et al. and explores additional considerations for the practical implementation of this transition function, making its utilization more adjustable for the user and contributing to its conceptual clarity. This article also addresses the temporal interaction between vaccination and exposure to the pathogen and applies adjustments for the estimates of yearly counts of cases and the number of vaccines administered that increase population immunity. Incorporating these new concepts into the model provides the ability to forecast and compare the effects of different vaccination timing scenarios and seasonality of transmission on the expected disease incidence. The application of this modeling approach to measles helps illustrate these concepts, given that either natural measles infection or two doses of measles vaccine provide long-lasting, possibly lifelong immunity.⁴⁰

The analysis is divided into three age categories that represent the overall susceptible population demarcated by the annual birth cohort measures used in reporting measles first- and second-dose vaccination: infants less than 12 months old ($a=0$), 1-year-olds ($a=1$), and ages 2 and older $(a\ge 2)$. The measles vaccine is not 100% effective. Its effectiveness varies with the age when it is administered, especially in the first year of life when effectiveness increases between 6 and 12 months of age.⁴⁰

Assuming statistical distributions for the times of vaccine receipt and exposure to infection and based on the country’s history of vaccination and reported cases, we estimate the probability that the vaccine reached susceptible individuals before they were exposed to the virus. Using this calculated probability, the transition numbers reflect the timing effects of both disease transmission and vaccination in the adjustment of modeling parameters as described below. In consequence, the resulting forecasting of disease incidence and its effects account for both the timing decisions in vaccination and the seasonality of the disease. Although this work is applied to the example of measles, it has potential relevance to other vaccine-preventable diseases. This study contributes towards improved mathematical modeling methods that better reflect programmatic realities on the ground, which may facilitate the use of modeled results in informing public health interventions.

2 |. METHODS

2.1 |. The original yearly transition function

The model developed by Eilertson et al. considers a discrete time-step of 1 year. This model uses the following transition function for the number of susceptible individuals $S_t^a$ in a population of age-group $a$ in year $t$ as follows:

$$S_t^a=S_{t-1}^{a-1}+V^a\left(B_t,S_{t-1}^{a-1}\right)-I_{t-1}^{a-1},$$ (1)

where $I_t^a$ is the number of newly infected individuals of age group $a$ in year $t$, and $V^a$ represents the effects of vaccination as a function of births $B_t$ and the number of susceptible individuals from the previous year. Because vaccination is expected to remove individuals from the susceptible population by conferring vaccine-induced immunity, a negative sign before the vaccination function would be more intuitive; however, we reproduce here the original equation as presented in Eilertson et al., which is written in general terms allowing the sign of the $V^a$ term to vary freely. Eilertson et al. provide little guidance about the composition of the vaccination function or how this transition function deals with death. Each of the terms in this transition function represents functions that depend on several parameters.

2.2 |. The effects of vaccination timing

During a given year, a population may be both exposed to measles virus circulation and may also have the opportunity to be vaccinated against measles, requiring specification of the temporal sequence between vaccination and exposure. In the context of this article, we use the term ‘exposed’ in its more general definition meaning ‘having contact with the virus’, rather than in the more particular meaning in SEIR modeling of someone who is infected and incubating the virus but is not yet infectious.

Not considering deaths, a person who is susceptible at the beginning of year $t-1$ can experience one of the following possible scenarios during the year: the individual is (i) neither vaccinated nor exposed; (ii) exposed and infected but not immunized by a vaccine (ie, including breakthrough infections); (iii) vaccinated and immunized but not exposed to the virus; (iv) vaccinated and immunized and then exposed to the virus later in the year but without being infected because of the protection conferred by vaccination; or (v) exposed to the virus and infected and then vaccinated later in the year without being immunized from the vaccination because of the prior infection (Figure 1).

FIGURE 1.

Measles-susceptible population at the beginning of the year will experience one of the following scenarios along the year: (i) neither vaccinated, nor exposed; (ii) infected but never vaccinated; (iii) vaccinated but never exposed to the virus; (iv) vaccinated and later exposed to the virus, without being infected; or (v) infected and later received a vaccine.

Individuals in scenario (i) remain susceptible at the beginning of next year, while scenarios (ii) and (iii) remove individuals from the susceptible pool via immunity conferred by natural infection or vaccination, respectively. In scenario (iv), vaccination will prevent an infection from occurring, so the effect of vaccination should be counted towards the population immunity achieved; hence, the potential infection is not counted. By contrast, in scenario (v), the individual’s natural infection will contribute to the population immunity, and the vaccination should be discounted from the effect of vaccination on population immunity because the individual was already immune from natural infection. Breakthrough infections among vaccinated persons are included in scenario (ii).

2.3 |. Effects of the timing of the vaccine and the seasonality of the disease

Of all the cases that are exposed and vaccinated in the same year, how many are vaccinated first? This question can be answered if the probability density functions of the timings of vaccination, $V\left(t_1\right)$, and exposure, $i\left(t_2\right)$, are estimated for the given year. Here $t_1$ represents the time of the year when the vaccine is administered, and $t_2$ represents the time of the year when the infectious exposure occurs; $0\le t_1\le 1$ and $0\le t_2\le 1$. Since both events are independent, the joint distribution is given by $z\left(t_1,t_2\right)=V\left(t_1\right)i\left(t_2\right)$. Therefore, the probability $q$ of receiving the vaccine before being exposed to the infection is given by:

$$q=P\left(t_1<t_2\right)={\int }_0^1{\int }_0^{t_2}v\left(t_1\right)i\left(t_2\right)d{t}_{1}d{t}_2,$$ (2)

With this approach, an example to illustrate the concept would be if both vaccination and infections occur uniformly throughout the year. In such a case, the probability of being vaccinated first would just be $q=0.5$.

Another example is if the vaccination occurs mostly during the first months of the year. If a portion $\theta$ of the vaccine is administered uniformly during the first ($1-\alpha$) part of the year, $0<\theta <1$, and the rest of the vaccine, $1-\theta$, is uniformly spread for the remaining part, $\alpha$, of the year, $0<\alpha <1$, then the probability density function for vaccination is given by:

$$v\left(t_1\right)=\left\{\begin{array}{ll}\frac{\theta }{1-\alpha }& 0\le t_1\le 1-\alpha \\ \frac{1-\theta }{\alpha }& 1-\alpha <t_1\le 1\end{array}\right..$$ (3)

Assuming infections occur uniformly throughout the year, it can be computed that, for this vaccination scenario, the probability of vaccinated before being exposed is $q=\frac{\theta +\alpha }{2}$.

2.4 |. An alternative formulation of the transition function

To account for the effects of variable timing of vaccination and exposure, as well as to include the effect of death on the population of age group $a$, we reformulate the equation presented by Eilertson et al. The term $S_t^a$ is more precisely defined as the number of susceptible individuals of age group $a$ at the beginning of year $t$. We can also define the age group $a$ as all the individuals whose age is in the interval $[a,a+1)$ at the beginning of year $t$. Notice, then, that this age group will transition, during calendar year $t$, between ages $a$ and $a+1$ years old.

If ${\epsilon }_t^a$ is defined as the proportion of the susceptible population of age group $a$ that would be exposed in year $t$, and ${\omega }_t^a$ as the proportion of the susceptible population of age group $a$ that was vaccinated that year, then ${\epsilon }_t^a{\omega }_t^a$ represents the proportion of the susceptible population age group $a$ that were both exposed and vaccinated that calendar year (scenarios (iv) and (v) in Figure 1). The expected proportion who die for an age group $a$ during the calendar year is ${\gamma }_{a,t}^{*}$, which is computed using age-specific death rates and other age-specific considerations, as explained in the following sections.

Using these definitions, the transition function can be more precisely written as:

$$S_t^a=\left(1-{\epsilon }_{t-1}^{a-1}\right)\left(1-{\omega }_{t-1}^{a-1}\right)S_{t-1}^{a-1}\left(1-{\gamma }_{a,t-1}^{*}\right),a\ge 2.$$ (4)

The equations used for age groups 0 and 1 $\left(aϵ\right\{\mathrm{0,\; 1}\left\}\right)$ are presented in separate sections below, and include considerations to account for births, temporary maternal immunity, and minimum vaccination age. We simulate the number of a susceptible population of age group $a$ that would be exposed in year $t$ following Eilertson et al.,³⁹ Equation (2). Additionally, we calculate the exposed proportion as ${\epsilon }_{t-1}^{a-1}=I_{t-1}^{a-1}/S_{t-1}^{a-1}$.

The calculation of ${\omega }_{t-1}^{a-1}$ requires other considerations depending on the age group and will be discussed below for each of these groups. Because measles vaccination prevents infections both in the year it is administered and in the following years, the simulated number of cases $Y_{t-1}^{a-1}$ for age group $a-1$ in year $t-1$ can be generally calculated as:

$$Y_{t-1}^{a-1}=\left(1-q_{t-1}^{a-1}{\omega }_{t-1}^{a-1}\right)I_{t-1}^{a-1},a\ge 2,$$ (5)

where $q_{t-1}^{a-1}$ represents the probability that the vaccine is administered before a person is exposed to infection in year $t-1$ for age group $a-1$. This quantity is used to calculate the number of reported cases using Eilertson et al.³⁹ equation 3 with $Y_{t-1}^{a-1}$ values instead of $I_{t-1}^{a-1}$. As before, the equations used for age groups $a=0$ and $a=1$ are presented in separate sections below.

2.5 |. Vaccination and number of cases for newborns (age group 0)

Three factors make the first group, all infants born the previous calendar year and aged 0–11 months at the beginning of year $t$, unique: (1) births, (2) temporary immunity from maternal antibodies, and (3) age of initial vaccination. It is assumed that the number of births per year is known and distributed uniformly in time within the year. Infants are not born directly into a susceptible group but rather into a special group with temporary immunity, which wanes over time until the infant is left without antibody protection and becomes susceptible to measles infection.⁴¹ The infant’s immune system develops during the first year, which causes the vaccine to induce different immune responses at different ages.⁴² The age of the initial vaccination is set by each country under these and other epidemiological considerations while research continues to define an optimal age for the first dose of measles-containing vaccine (MCV).⁴³ In countries with a high risk of measles mortality among infants, the WHO recommends that the first dose of MCV (MCV1) be administered at age 9 months and a second dose (MCV2) at age 15–18 months, and also recommends certain Supplemental Immunization Activities (SIA) from age 6 months. In countries with low levels of measles transmission, MCV1 may be administered at 12 months of age, with MCV2 administered at 15–18 months or at school entry, based on programmatic considerations to achieve high coverage.⁴⁴

For year $t$, let $e_t$ represent the age in months of the first routine measles vaccination (MCV1), $u_t$ represent the minimum age in months for SIA vaccination, and ${\mu }_t$ represent the expected age in months at which maternal antibody protection is lost. At the beginning of year $t$, we can define the number of susceptible individuals in age group $a=0\left(S_t^0\right)$ and the temporally immune group with maternal antibody protection $\left(M_t^0\right)$ in terms of births during the previous year $\left(B_{t-1}\right)$, vaccination activities, infections, and deaths. Figure 2 represents a visualization of the transition from births, appearing on the horizontal axis, to the susceptible group or the temporarily immune group at the respective vertical line segments on $t$. For infants, age and time increase simultaneously; therefore, their progression is represented by the 45°-angled shaded areas. Each shaded area represents a group of children born in certain months of the year.

FIGURE 2.

Transition from births to susceptible group $S_t^0$ or to the temporally immune group $M_t^0$. Notice that immunity is lost at age ${\mu }_{t-1}$ and that routine vaccination starts at age ${\text{e}}_{t-1}$. Also note that this group’s immunity is not affected by SIAs carried out early in the year, but only by MCV1 if the age of administration is set at less than 1 year of age (assumed at age ${\text{e}}_{t-1}$).

As shown in Figure 2, the group of infants born in the last ${\mu }_t$ months of the year will not be affected by any vaccination, assuming none is recommended before age ${\mu }_t$ months. The size of the group of temporarily protected infants at time $t$ is calculated as:

$$M_t^0={\mu }_{t-1}{B}_{t-1/12}\left(1-{\mu }_{t-1}{\gamma }_{0,t-1}/24\right).$$ (6)

This quantity reduces the number of births that occurred in the last ${\mu }_{t-1}$ months of the calendar year by an expected proportion who die for this sub-group: ${\mu }_{t-1}{\gamma }_{0,t-1}/24$. This expected proportion who die reflects only the deaths that will be reported in the calendar year $t-1$ and is proportional to the average age in this sub-group (${\mu }_{t-1}/24$), assuming deaths in this group are uniformly distributed in age and time during the calendar year. Parameter ${\gamma }_{0,t-1}$ is the age-specific death rate divided by 1000 for age 0 in year $t-1$. This expression and the analogous expressions that follow for older age groups are convenient approximations to quantities derived from demographic principles. Details are in a Supplementary file.

Infants above age ${\mu }_{t-1}$ are assumed susceptible to measles infection, while only the population at or above $u_{t-1}$ may be vaccinated during SIAs, and only the population at age $e_{t-1}$ is scheduled to receive routine immunization with MCV1. An SIA will affect the population above age ${\mu }_{t-1}$ and below age $e_{t-1}$ only when $u_{t-1}<e_{t-1}$ and the SIA is implemented in a later-month of the calendar year $\left({\sigma }_{t-1}^0>12+u_{t-1}-e_{t-1}\right)$; otherwise, those in this specific sub-group (ages between ${\mu }_{t-1}$ and $e_{t-1}$) may not be old enough to receive vaccination through the SIA (or may not have been born yet if the SIA was applied at an earlier month in the year). When that is the case, the proportion of the susceptible population that was vaccinated by an SIA that year for this group can be computed as:

$${\omega }_{t-1}^{\mu ,e}=\frac{{\sigma }_{t-1}^0+e_{t-1}-u_{t-1}-12}{e_{t-1}-{\mu }_{t-1}}{k}_{t-1}^{u,e}{\phi }_{\text{SIA}}^{u,e},$$ (7)

where $k_{t-1}^{u,e}$ is the SIA coverage for this age group for the specific SIA, which is assumed known, ${\sigma }_{t-1}^0$ represents the month of application of the SIA, and ${\phi }_{\text{SIA}}^{\text{u},\text{e}}$ represents the average effectiveness of MCV between the ages $u_{t-1}$ and $e_{t-1}$. Different SIAs during the same year are assumed to be independent and their effects on the same group can be added. If no SIA is implemented at a month ${\sigma }_{t-1}^0$ such that ${\sigma }_{t-1}^0>12+u_{t-1}-e_{t-1}$, then ${\omega }_{t-1}^{\text{u},\text{e}}=0$.

The susceptible population at the beginning of year $t$ that is below routine immunization age can then be calculated as:

$$S_t^{\mu ,e}=\left(1-{\epsilon }_{t-1}^{\mu ,e}\right)\left(1-{\omega }_{t-1}^{\mu ,e}\right)B_{t-1}^{\mu ,e}\left(1-\left({\mu }_{t-1}+e_{t-1}\right){\gamma }_{0,t-1}/24\right).$$ (8)

The details of the computation of ${\epsilon }_{t-1}^{\mu ,e}$ are discussed below, near the end of this section. $B_{t-1}^{\mu ,e}$ is the proportion of births between months $\left(12-e_{t-1}\right)$ and $\left(12-{\mu }_{t-1}\right)$ in year $t-1$ and is computed as $B_{t-1}^{\mu ,e}=\left(e_{t-1}-{\mu }_{t-1}\right)B_{t-1}/12$, assuming births are uniformly distributed during the calendar year.

Additionally, the population born in the first $12-e_{t-1}$ months of the year is assumed to be freely subject to infection after age ${\mu }_{t-1}$. This population is also simultaneously subject to vaccination by SIA and infection after age $u_{t-1}$ and becomes a candidate for MCV1 at age $e_{t-1}$ months. The susceptible population above routine vaccination age can be calculated as:

$$S_t^{e+}=\left(1-{\epsilon }_{t-1}^{e+}\right)\left(1-{\omega }_{t-1}^{e+}\right)B_{t-1}^{e+}\left(1-\left(12+e_{t-1}\right){\gamma }_{0,t-1}/24\right).$$ (9)

The details of the computation of ${\epsilon }_{t-1}^{e+}$ are discussed below, near the end of this section. $B_{t-1}^{e+}$ is the proportion of births in the first $12-e_{t-1}$ months of the previous year $(t-1)$ and is computed as $B_{t-1}^{e+}=\left(12-e_{t-1}\right)B_{t-1}/12$, assuming births are uniformly distributed during the calendar year.

If an SIA is implemented after month $u_{t-1}\left({\sigma }_{t-1}^0>u_{t-1}\right)$, and assuming that the probability is low that this SIA and a routine MCV1 dose or other SIAs will vaccinate the same children in this group more than once in the same year (“random-case scenario”), then the proportion of the susceptible population that was vaccinated by routine immunization or SIA that year will be estimated by the sum of the MCV1 rate and the SIA rate for this group as follows:

$${\omega }_{t-1}^{e+}=\left\{\begin{array}{ll}{m}_{t-1}^1{\phi }_{1,e_{t-1}}+\frac{{\sigma }_{t-1}^0-u_{t-1}}{12-e_{t-1}}{k}_{t-1}^{u+}{\phi }_{\text{SIA}}^{u+}& \hfill \text{when}{u}_{t-1}<{\sigma }_{t-1}^0\le 12+u_{t-1}-e_{t-1}\hfill \\ m_{t-1}^1{\phi }_{1,e_{t-1}}+k_{t-1}^{u+}{\phi }_{\text{SIA}}^{u+}& \hfill \text{when}{\sigma }_{t-1}^0>12+u_{t-1}-e_{t-1}\hfill \end{array},\right.$$ (10)

where $m_{t-1}^1$ is the vaccination coverage for MCV1 at year $t-1,{\phi }_{1,e_{t-1}}$ is the vaccine effectiveness of MCV1 administered at age $e_{t-1}$ months, $k_{\text{t}-1}^{\text{u}+}$ is the SIA coverage for this age group when implemented after age $u_{t-1}$, and ${\phi }_{\text{SIA}}^{\text{u}+}$ is the mean effectiveness of the measles vaccine for those vaccinated in this group. If other SIAs are implemented in the same year to the same group, then the assumption of independence will allow the addition of these terms. If no SIAs are implemented after month $u_{t-1}$ in the calendar year, then the second term of the vaccination rate is zero, and the vaccination in this group is uniquely MCV1 $\left({\omega }_{t-1}^{e+}=m_{t-1}^1{\phi }_{1,e_{t-1}}\right)$.

If the probability is not low of receiving both an SIA dose of MCV and a routine MCV1 dose (eg, when populations who receive MCV1 doses also receive SIA doses, resulting in a considerable proportion of children receiving both vaccines in the same year), then a “dependent-case scenario” can be assumed in which every child who can potentially get both vaccines will indeed get them. Even in this scenario, SIAs are considered independent from each other when implemented in the same year. In this case, the smaller number between MCV1 coverage and SIA coverage is assumed to be coverage with a second dose. A “selective vaccination scenario” can also be computed by considering settings in which an accurate registry of vaccinated children exists (eg, high utilization and retention of vaccination record cards, electronic immunization registries) such that SIAs can be effectively targeted to individuals missing vaccine doses and the probability of giving unnecessary repeated doses is low. In this case, all the vaccines above the estimated number of vaccinated individuals with a first dose would first be directed to the unvaccinated population, and then would be directed to the vaccinated population as a second dose.

From the two susceptible sub-populations above, the susceptible population at age 0 can be computed as ${S_t^0=S}_t^{\mu ,e}+S_t^{e+}$. Note that the computation of susceptible populations requires the seasonality (or timing) of infection and the proportion of the susceptible population that was vaccinated that year, for newborns, whose age group at time $t-1$ will transitionally be either between ${\mu }_{t-1}$ and $e_{t-1}$ (marked as $\mu ,e$), or above $e_{t-1}(e+)$.

Following the conceptual framework of the Eilertson et al. model, we can define the susceptible population of newborns at time $t-1$ that will be exposed to the virus between ages ${\mu }_{t-1}$ and $e_{t-1}$ as: $S_{t-1}^{\mu ,e}=B_{t-1}^{\mu ,e}$, while the susceptible population of newborns at time $t-1$ that will be exposed to the virus between ages $e_{t-1}$ and 12 months is: $S_{t-1}^{e+}=B_{t-1}^{e+}$.

This susceptible population born after month $12-e_{t-1}$ and before the end of month $12-{\mu }_{t-1}$ that will be exposed to the virus only between ages ${\mu }_{t-1}$ and $u_{t-1}$ has a rate of infection of ${\pi }_{\mu ,e}=\left(e_{t-1}-{\mu }_{t-1}\right){\pi }_{t-1}/24$ without the intervention of vaccine. On the other hand, the susceptible population that will be exposed to the virus only between ages $e_{t-1}$ and 12 months (born in the first $12-e_{t-1}$ months of year $t-1$) has a rate of infection of ${\pi }_{e+}=\left(12+e_{t-1}-2{\mu }_{t-1}\right){\pi }_{t-1}/24$ without the intervention of vaccine. In both of these rates, ${\pi }_{t-1}$ represents the rate of infection in the absence of vaccination multiplied by the average time that each susceptible group is exposed to the infection. Then, in the absence of vaccination, the infections for these subgroups are simulated using $I_{t-1}^{\mu ,e}~Bin\left(B_{t-1}^{\mu ,e},{\pi }_{\mu ,e}\right)$ and $I_{t-1}^{e+}~Bin\left(B_{t-1}^{u+},{\pi }_{e+}\right)$. The realization of the proportion of susceptible exposed is calculated by ${\epsilon }_{t-1}^{\mu ,e}=I_{t-1}^{\mu ,e}/S_{t-1}^{\mu ,e}$ and ${\epsilon }_{t-1}^{e+}=I_{t-1}^{e+}/S_{t-1}^{e+}$.

Computing the adjusted number of infected in age group −1, born in year $t-1$, under the effect of vaccination, and with proportions $q_{t-1}^{\mu ,e}$ and $q_{t-1}^{e+}$ of vaccines administered before susceptible individuals, for each respective age sub-group, are exposed to infection, gives:

$$Y_{t-1}^{-1}=\left({\epsilon }_{t-1}^{\mu ,e}-q_{t-1}^{\mu ,e}{\epsilon }_{t-1}^{\mu ,e}{\omega }_{t-1}^{\mu ,e}\right)S_{t-1}^{\mu ,e}+\left({\epsilon }_{t-1}^{e+}-q_{t-1}^{e+}{\epsilon }_{t-1}^{e+}{\omega }_{t-1}^{e+}\right)S_{t-1}^{e+}.$$ (11)

2.6 |. Vaccination and number of cases for 1-year-olds (age group 1)

The age group where $a=1$ (ie, those in a birth cohort that will be 1 year of age or more, but less than two, at the beginning of year $t$) is also unique because it must incorporate the previous year’s age 0 susceptible group—children who are still covered by maternal immunity for part of the year—into the current year’s starting susceptible group, removing those who have been immunized through natural infection, have been immunized through the application of a measles containing vaccine, or have died. Figure 3 shows a representation of the different factors involved.

FIGURE 3.

Transition from age group 0 at year $t-1$ to age group 1 at year $t$. The group protected by maternal immunity becomes susceptible, MCV1 and MCV2 are available to the individuals of adequate age, and SIAs can reach some part of the population, depending on age and timing (here, SIA is implemented in month ${\sigma }_{t-1}^1$).

The vaccine may be administered via routine immunizations (MCV1 and MCV2) or an SIA. Notice that the SIA is assumed to occur in month ${\sigma }_{t-1}^1$. It is assumed that children who receive MCV1 are the only children eligible to receive MCV2. It is also assumed that vaccination administered during SIAs targets both susceptible and non-susceptible individuals without distinction. Alternatives to these assumptions are discussed below.

This age group is exposed to infection after age ${\mu }_{t-1}$ and may receive the benefits of SIA vaccination after age $u_{t-1}$. The number susceptible at the beginning of year $t$ can be calculated as:

$$S_t^1=\left(1-{\epsilon }_{t-1}^0\right)\left(1-{\omega }_{t-1}^0\right)\left(M_{t-1}^0+S_{t-1}^0\right)\left(1-\frac{{\gamma }_{0,t-1}+{\gamma }_{1,t-1}}{2}\right).$$ (12)

Notice that the age-group-specific expected proportion who die in the calendar year is approximated by the average of the age-specific death rates (divided by 1000) that span the transition ages of this group (0–11 and 12–23 months old). Without any vaccination, the modeled number of infected during year $t-1$ would be distributed as $I_{t-1}^0~Bin\left(M_{t-1}^0+S_{t-1}^0,{\pi }_{0,t-1}\right)$, where ${\pi }_{0,t-1}=\left(1-{\left({\mu }_{t-1}/12\right)}^2/2\right){\pi }_{t-1}$. Then, the proportion ${\epsilon }_{t-1}^0$ is calculated as ${\epsilon }_{t-1}^0=I_{t-1}^0/\left(M_{t-1}^0+S_{t-1}^0\right)$.

On the other hand, the proportion of the susceptible population that was vaccinated that year for the “random-case scenario” (ie, when the probability is low that a child vaccinated during an SIA would also receive a routine MCV1 or other SIA dose in the same year) can be calculated as:

$${\omega }_{t-1}^0=e_{1,t-1}{m}_{t-1}^1{\phi }_{1,e_{1,t-1}}/12+\left(2-e_{2,t-1}/12\right)m_{t-1}^2\left({\phi }_{2,t-1}-{\phi }_{1,e_{1,t-1}}\right)+{\omega }_{t-1}^{SIA},$$ (13)

where $m_{t-1}^1$ is the vaccination coverage for MCV1 at year $t-1,{\phi }_{1,e_{-}(1,t-1)}$ is the effectiveness of MCV1 administered at age $e_{1,t-1}$ months, $m_{t-1}^2$ is the vaccination coverage for MCV2 at year $t-1$, and ${\phi }_{2,t-1}$ is the effectiveness of MCV2. On the other hand, ${\omega }_{t-1}^{\text{SIA}}$ can be calculated as:

$${\omega }_{t-1}^{\text{SIA}}=\left\{\begin{array}{lll}\left(\frac{12+{\sigma }_{t-1}^1-u_{t-1}}{12}\right)k_{t-1}^1{\phi }_{\text{SIA},t-1}& \text{if}& {\sigma }_{t-1}^1\le u_{t-1}\\ k_{t-1}^1{\phi }_{\text{SIA},t-1}& \text{if}& {\sigma }_{t-1}^1>u_{t-1}\end{array}.\right.$$ (14)

Here, $k_{t-1}^1$ is the SIA coverage for age group 1 on year $t-1$, and ${\phi }_{\text{SIA,t-1}}$ is the average effectiveness of the vaccine for SIAs in this age group.

As is the case for the age group 0, a “dependent-case scenario” can be constructed for age group 1 by considering the lower coverage between MCV1 and SIAs as second doses. A “selective vaccination scenario” can also be computed for this age group 1 by considering that all the vaccines above the estimated number of vaccinated individuals with a first dose would first be directed to the unvaccinated population, and then would be directed to the vaccinated population as a second dose.

The adjusted number of infected in age group 0 at year $t-1$ (ie, those who will turn 1 year old in the course of the year) under the effect of vaccination, and with a proportion $q_{t-1}^0$ of vaccines administered before susceptible individuals are exposed to infection is:

$$Y_{t-1}^0=\left({\epsilon }_{t-1}^0-q_{t-1}^0{\epsilon }_{t-1}^0{\omega }_{t-1}^0\right)\left(M_{t-1}^0+S_{t-1}^0\right).$$ (15)

2.7 |. Vaccination and number of cases for 2-year-olds and older (age groups ≥2)

Finally, when $a\ge 2$, considerations about births, maternal antibodies, and routine MCV1 can be set aside. Figure 4A shows a depiction of the elements affecting these ages. In panel (A), susceptible individuals transition from a previous period and age ($t-1,a-1$) into the current period and age $(t,a)$, while potentially affected by vaccination activities through routine immunization services and/or SIAs. As before, MCV2 is represented with a horizontal line, administered at a specific age $e_{2,t-1}$ in months, and SIAs by vertical lines. SIAs may occur more than once and at different times, and children’s eligibility may end at different ages. Panel (B) depicts how, during this transition, individuals are also exposed to the virus, with a transmission rate that may vary seasonally (as shown by the varied vertical shading over time, with darker shading indicating the period of more intense transmission), and some will be infected. Finally, some individuals will die from one period to the next, which is expected to occur uniformly by age and time within the calendar year (as for younger age groups above, this transition is not depicted in the figure).

FIGURE 4.

(A) Graphical representation of the transition function showing vaccination elements. (B) Measles seasonal infectiousness represented by shaded area with varying intensity.

Following Eilertson et al.’s methodology, the number of infections each year depends only on the proportion of susceptible individuals in the population (computed as the number of susceptible individuals divided by the population) and a parameter for the average proportion of susceptibles who get infected $\left(\pi \right)$. However, when vaccination is available, susceptible individuals may also be vaccinated before being infected and hence be removed from the pool of susceptibles before being exposed to the virus. The larger the vaccination activity is in reaching susceptible individuals, the more susceptibles are removed from that pool through vaccination (note that by construction, the model function assumes that vaccination may reach individuals who are not already immune due to prior infection or vaccination). In the same way, some people could become infected before receiving the vaccine that year, affecting the real count of those successfully vaccinated. The transition function for the susceptible population $S_t^{\text{a}}$ for age $a$ at the beginning of year $t$ is given by:

$$S_t^a=\left(1-{\epsilon }_{t-1}^{a-1}\right)\left(1-{\omega }_{t-1}^{a-1}\right)S_{t-1}^{a-1}\left(1-\frac{{\gamma }_{a-1,t-1}+{\gamma }_{a,t-1}}{2}\right),a\ge 2.$$ (16)

The age-group-specific expected proportion who die in the calendar year is approximated by the average of the age-specific death rates (divided by 1000) that span the transition of this group ($a-1$ and $a$). The proportion ${\epsilon }_{t-1}^{a-1}$ is calculated as ${\epsilon }_{t-1}^{a-1}=I_{t-1}^{a-1}/S_{t-1}^{a-1}$. For the computation of ${\omega }_{t-1}^{a-1}$, a general assumption is that the probability of MCV2 and SIA vaccines being both given to the same child in the same year is low (ie, “random-case scenario”). However, it is possible that a vaccine may be given to individuals who were already immune from a previous year, either through vaccination or infection. Since it is programmatically difficult to determine the immunity status of an individual at the time of vaccination, a plausible scenario is that the vaccine will be given uniformly to all the population in an age group, independent of their immunity status. For the purpose of this analysis, we will consider, as with the previous age group, that MCV2 is fully dependent on MCV1 and that SIAs are implemented proportionally to all the population (whether susceptible or not). The proportion of the vaccinated susceptible population for this “random-case scenario” is then calculated as:

$${\omega }_{t-1}^{a-1}=\left(e_{2,t-1}/12-a+1\right)m_{t-1}^{a-1}\left({\phi }_2-{\phi }_1\right)+p{s}_{t-1}^{a-1}{\phi }_{SIA},a\ge 2,$$ (17)

where $m_t^a$ is the proportion of the age-specific population that received MCV2 through routine delivery in year t (depending on target age), $S_t^a$ is the proportion of the age-specific target population that received a vaccine through SIA, and ${\phi }_1,{\phi }_2$, and ${\phi }_{\text{SIA}}$ are the mean vaccine effectiveness of the MCV1, MCV2, and SIA vaccines. The parameter $p$ will be 1 if the SIA covers all the specific age group, and will be $\left(12-{\sigma }_{t-1}^a\right)/12$ for age group $a+1$ when the age limit of the SIA is $a$.

As for younger age groups, a “dependent-case scenario” can be built by considering the proportion of all SIAs below the current total coverage for the age group to be of no additional benefit. The “selective vaccination scenario” can also be computed for this age group following the assumptions stated above for younger age groups.

2.8 |. Computational model and case study application to Ethiopia

To validate the presented concepts, a computational model was constructed using the R 4.0.2 programming software,⁴⁵ with packages readxl,⁴⁶ tidyverse,⁴⁷ and matrixStats⁴⁸ following the methods described in Eilertson et al.³⁹ and incorporating the extensions described above. The model was applied to estimate the true incidence of measles cases in Ethiopia from 1960 to 2019 as an illustrative case study of the transition functions described above. In Ethiopia, MCV1 was introduced in 1980, MCV2 in 2019, and SIAs in 2000. The data from 1960 to 1979 were used on the first pass to improve the model’s initial estimation of the proportion of the susceptible population by age group, which is started as a geometric sequence of initial random values as proposed by Eilertson et al.³⁹ On the second pass, the same data were used to generate estimates of the number of cases in the years before vaccination. The data from 1980 to 2019 were used to generate estimations for the years with vaccination. The model also includes a function for measles surveillance performance, that estimates the proportion of true cases that are reported in the surveillance system each year, which offers a time adjustment for the static proportion of reported cases proposed originally by Eilertson et al.³⁹ Two illustrative surveillance performance function scenarios were assessed: (i) assuming Ethiopia’s surveillance system (as in the Eilertson et al. model); and (ii) assuming Ethiopia’s surveillance system had a linear increase of 3% in its reporting proportion between 2000 and 2019. For each of these scenarios, the model estimates the reporting proportion as an output of the model as in Eilertson et al. Yearly $q$ probabilities of receiving immunization from a measles containing vaccine before being exposed to infection were calculated using Equation (2) with the summarized disease seasonality from monthly cases reported, and with the monthly considerations of vaccination coverage including the supplemental immunization activities as recorded in Appendix 1, and the routine immunization coverage for MCV1 and MCV2. All data sources are cited in the next section.

2.9 |. Data

For the computation of the expected number of cases from 1960 to 2019, we used yearly estimates for population size by age,⁴⁹ birth rates,⁵⁰ crude death rates,⁵¹ death rates by age group (0–1, 1–4, 5–9, 10–14, 15–19, and 20–24 years old),⁵² and the number of measles cases as reported to the World Health Organization.⁵³ For vaccination estimates, we included administration of MCV1 from 1980 and MCV2 from 2019,⁵⁴ and all SIAs since 2000 based on a list that provided the age ranges and size of populations reached by each campaign⁵⁵ (see list of SIAs in Appendix 1). Although the measles vaccine was available at one hospital and some non-governmental organizations in Ethiopia before 1980, there is no record of considerable use⁵⁶; therefore, none of the effects of those limited vaccination efforts were considered in the model. For this run, the vaccine effectiveness for MCV1 was set to 58% from 6 to 8 months, 84% from 9 to 11 months, and 92.5% at 12 months and older, and for MCV2, 95% (Table 1). The effectiveness of measles vaccination administered through SIAs was set at 58% from 6 to 8 months, 84% from 9 to 11 months, and 95% at 12 months and older. The infant’s duration of protection from maternal immunity was parametrized as 6 months, after which children were assumed to be fully susceptible unless vaccinated. Vaccine coverage for any specific vaccination activity was assumed to achieve up to 95% of the targeted age-group population, reflecting the regional and global coverage targets for achieving interruption of endemic measles transmission as well as the coverage threshold for a “high quality” SIA, and taking into consideration country-level estimates of measles second-dose coverage achieved in low- and lower middle-income countries over the past two decades, almost all of which are less than 95%.⁵⁷

TABLE 1.

Parameters, input values, and data sources used in the case study application of the model to Ethiopia.

Parameter	Value	Source
Vaccine effectiveness MCV1 at
6–8 months	58%	59
9–11 months	84%	60
12 months and above	92.5%	60
Vaccine effectiveness MCV2	95%	Assumption
Vaccine effectiveness SIA at
6–8 months	58%	59
9–11 months	84%	60
12 months and above	95%	60
Maternal immunity age limit	6 months	60
Upper coverage limit for vaccination activity	95%	Assumption
Age for MCV1 per immunization schedule	9 months	60
Age for MCV2 per immunization schedule	15 months	60

Running the computational model with these inputs, we let the program produce 100 retrospectively simulated instances of the history of the disease.

3 |. RESULTS

The model estimated the number of true measles cases in Ethiopia from 1960 to 2019, which is several orders of magnitude larger than the number of cases reported to Ethiopia’s public health disease surveillance system based on available surveillance data (Figure 5A), as expected given that reporting to disease surveillance systems typically underestimate true disease burden in the population, especially in settings in which such systems are weaker. Estimated true measles incidence rises until the advent of MCV1, after which it declines steadily through the period when SIAs were introduced (Figure 5B).

FIGURE 5.

(A) Computationally modeled estimates of the number of measles infections compared to measles cases reported in Ethiopia (1960–2019) from WHO⁴⁵; the blue line depicts the mean annual number of modeled measles infections from 100 simulations, while the red line depicts the number of measles cases actually reported in national surveillance data. (B) Estimated measles incidence per 100 000 population in Ethiopia (1960–2019); the gray line depicts the mean annual number of modeled measles infections per 100 000 population from 100 simulations.

We also used the model to simulate 100 times the number of cases and visually compared these simulations to the actual number of cases reported from Ethiopia’s surveillance data (Figure 6A). Additionally, Figure 6A shows the median of the number of cases simulated and their 2.5 and 97.5 percentiles, all assuming that Ethiopia’s surveillance system produced a constant reporting fraction of true cases throughout all the simulated years (estimated by the model to be 1% in this scenario). In this figure, we can see that the result is influenced by the existence of an outlier number of cases reported from actual surveillance data in 1985. Figure 6B shows the effect of running the model after substituting the 1985 outlier value with the mean number of actual cases reported for the years 1983 and 1987. Under these conditions, the model estimates that about 3% of true cases get reported every year. Although this improves the congruence between the median of the simulated runs and the actual number of cases reported, Figure 6B shows that the number of actual reported cases from Ethiopia’s surveillance data increases in the last years of the graph (2018–2019), which is not captured by the modeled estimates of the number of cases that would be expected to be reported by the surveillance system in that year; however, note that both the actual number and the estimated number of reported cases both fall well below the modeled estimate of the number of true infections shown in Figure 5A. As this trend may reflect improvements in Ethiopia’s surveillance capacity over time, we explored an alternative scenario for the surveillance performance function. Figure 6C presents results assuming a linear improvement of 3% in the reporting rate of true cases applied incrementally from years 2000 to 2019. The model run under these conditions estimates a reporting proportion of 2% of the true cases before the year 2000, and a linearly increased reporting proportion of true cases from 2% to 5% between 2000 and 2019 (eg, due to surveillance quality improvements). Across all three scenarios explored, only 1–2 empirical data points fall outside of the 95% range across the 100 simulation model runs (1985 in Figure 6A; 2014 and 2015 in Figure 6B; 2015 in Figure 6C). While the median across model runs overestimates the actual number of reported cases in some years and underestimates the actual number of reported cases in others, the 95% range consistently captures the historical empirical data.

FIGURE 6.

Simulated vs actual number of cases reported when (A) full data is used and assuming 1% reporting rate, (B) outlier in 1985 is removed and assuming 3% reporting rate, and (C) when the outlier in 1985 is removed and assuming a 3% reporting rate before 2000 that linearly increases to 5% from 2000 to 2019. In each panel, the gray lines show the estimated number of cases expected to be reported in one of the 100 simulated runs; across the 100 simulation runs, the green line shows the median, the red line shows the 97.5 percentile, and the dark blue line shows the 2.5 percentile of the number of cases expected to be reported.

The $q$ values were computed using a numeric implementation for Equation (2) and are listed in Appendix 2.

4 |. DISCUSSION

We constructed an approach to a yearly transition function to account for the effects of measles vaccination timing among different age groups and the effects of disease seasonality on the number of measles cases in a country. Our methodology expands on the existing framework by Eilertson et al.,³⁹ which presents a dynamic meta-population model, using particle filtering to fit an attack rate function at the country level. We developed and showed explicit functional expressions for each underlying component of the transition function to adjust for the temporal interaction between vaccination and exposure to disease. The computational model developed based on these functional expressions has the advantage of being relatively parsimonious in its parameter requirements and faster to run, making it well-suited to explore measles transmission and vaccination dynamics in low- and middle-income settings in which data may be lacking to solve more traditional ordinary differential equation compartmental disease transmission models, while incorporating within-year immunity dynamics not considered in the yearly runs of the original Eilertson et al. model. The proposed model qualitatively reproduces results found in other measles disease transmission models, such as estimating a much higher number of true cases compared to reported cases in low-income countries with weaker surveillance infrastructure.¹ The simulation model applied to the case study of Ethiopia illustrates the feasibility of this modeling approach to use publicly available data for low- and middle-income country settings to estimate true measles incidence, the impact of real-world vaccination policies, and the interplay between disease transmission and surveillance reporting systems over time.

Some of the limitations intrinsic to other models are still present in our approach. For LMICs particularly, variability in the proportion of cases that are reported over time (ie, what proportion of cases get reported during a certain period vs the proportion reported in other periods) may be difficult to characterize to understand when anomalous increases or decreases in reported cases reflect real changes in disease burden or surveillance quality or simply other idiosyncratic dynamics. For example, the illustrative results shown for Ethiopia reveal the potential for outlier points of data to considerably influence the model estimates. In this specific case, the dramatic increase in the number of measles cases reported in 1985 for Ethiopia could be related to the famine suffered locally that year.⁵⁸ Although the illustrative results shown here use the mean of the number of cases in two nearby years (1983 and 1987) to replace this outlier value, alternative approaches to handling outliers are possible within the model framework. The results show that considering potential variation in surveillance quality over time, including improvements that lead to a higher percentage of the cases being reported, benefits the overall model fit. Due to lack of access to the code and data used in Eilertson et al., we do not present a formal model comparison between the two approaches, although this could be a direction for future work. As in Eilertson et al., we do not present a forecast of future measles incidence in a specific country setting in this article. Another anticipated limitation will be accounting for the effects of the COVID-19 pandemic when using the model to forecast measles incidence in the upcoming years.

Although this article illustrates the model’s performance in estimating retrospective measles cases in a single country setting, our methodology could also be applied to forecast the effects of vaccination coverage by age and the timing of vaccination activities on the expected number of measles cases by year. All data inputs to the model are adjustable to explore scenario analyses. The model is also flexible to accommodate other distributional assumptions in lieu of the distributions used in the exposition of methods here. For example, this model could be used to estimate the impact on population immunity and future outbreak trajectories of timing an SIA earlier or later in the year for various age ranges, or to compare the relative health impacts of investing in improving routine MCV1 or MCV2 coverage vs SIAs for various age ranges. Importantly, this modeling approach also permits explicit consideration of surveillance quality and the relationship between reported cases and true incidence, which could lend itself to assessment of the value of investing in improved surveillance as well as better calibrating disease burden estimates based on assumed reporting rates. We expect that this novel approach will benefit the planning of programmatic activities while contributing to answering outstanding questions.³⁷

APPENDIX 1. LIST OF SIAS IN ETHIOPIA SINCE YEAR 2000.⁵⁵

Year	Country	Activity	Intervention	Start date	End date	Age group	Extent	Implementation status	Target population	Reached population	% Reached	Coverage survey done	Survey results
2000	Ethiopia	FollowUp	Measles	6/23/00		9–59 M	Sub-National	Done	3,800,000	3,610,000	95
2000	Ethiopia	FollowUp	Measles	12/29/00		9–59 M	Sub-national	Done	3,026,147	2,346,464	78
2001	Ethiopia	FollowUp	Measles	11/1/01	12/30/01	9–59 M	Sub-national	Done	2,166,232	1,646,336	76
2002	Ethiopia	CatchUp	Measles	11/25/02	12/4/02	9 M–14 Y	Rollover-Nat	Done	2,316,214	2,277,988	98
2003	Ethiopia	CatchUp	Measles	4/1/03	12/1/03	6 M–14 Y	Rollover-Nat	Done	5,605,502	5,101,007	91
2004	Ethiopia	CatchUp	Measles	3/20/04	6/4/04	6 M–14 Y	Rollover-Nat	Done	8,835,802	7,422,074	84
2005	Ethiopia	CatchUp	Measles			6 M–14 Y	Sub-national	Done	198,456	136,935	69
2005	Ethiopia	FollowUp	Measles	9/1/05		9–59 M	Rollover-Nat	Done	1,073,066	987,221	92
2006	Ethiopia	FollowUp	Measles	3/1/06	6/1/06	9–59 M	Rollover-Nat	Done	11,688,720	10,169,187	87
2007	Ethiopia	FollowUp	Measles	11/1/07	11/30/07	6–59 M	Sub-national	Done	1,117,345	1,072,701	96
2008	Ethiopia	FollowUp	Measles	10/31/08	12/7/08	6–59 M	Sub-national	Done	11,791,819	10,848,474	92
2009	Ethiopia	FollowUp	Measles	1/1/09	1/31/09	6–59 M	Sub-national	Done	773,910	662,168	86
2009	Ethiopia	FollowUp	Measles	1/1/09	1/31/09	6–59 M	Sub-national	Done	279,102	264,134	95
2009	Ethiopia	FollowUp	Measles	1/1/09	1/31/09	6–59 M	Sub-national	Done	62,504	57,762	93
2009	Ethiopia	FollowUp	Measles	6/10/09	6/17/09	6–59 M	Sub-national	Done	285,644	266,621	93
2010	Ethiopia	FollowUp	Measles	2/18/10	4/30/10	6–59 M	Sub-national	Done	1,057,327	961,798	91
2010	Ethiopia	FollowUp	Measles	10/22/10	10/25/10	9–47 M	Rollover-Nat	Done	7,656,367	8,171,534	107
2011	Ethiopia	FollowUp	Measles	2/18/11	2/21/11	9–47 M	Rollover-Nat	Done	774,658	757,421	98
2011	Ethiopia	Campaign	Measles	10/1/11	11/1/11	6 M–14 Y	Unknown	Done	7,326,463	7,034,264	96
2013	Ethiopia	CatchUp	Measles	5/29/13	6/5/13	9–59 M	National	Done	11,873,928	11,609,484	98	Yes	91
2015	Ethiopia	Outbreak Response	Measles	10/19/15	1/31/16	6–59 M	Sub-national	Done	4,865,893	5,145,284	106
2016	Ethiopia	CatchUp	Measles	4/22/16	4/28/16	6 M–<15 Y	Sub-national	Done	25,706,550	24,986,589	97	Yes	94
2017	Ethiopia	FollowUp	Measles	2/23/17	3/10/17	9 M–14 Y	Sub-national	Done	22,035,787	21,225,199	96	Yes	93
2017	Ethiopia	FollowUp	Measles	7/29/17	8/4/17	6–179 M	Sub-national	Done	2,579,178	2,524,841	98
2020	Ethiopia	FollowUp	Measles	6/30/20	7/24/20	9–59 M	National	Done	14,135,353	13,970,822	99

APPENDIX 2. Q VALUES COMPUTED FOR ETHIOPIA, BY YEAR AND AGE GROUP.

Year	mu_e	e_12	‘0’	‘1’	‘2+’	Vaccination policy
1980	0	0.374	0.576	0	0	RI MCV1 only
1981	0	0.374	0.576	0	0	RI MCV1 only
1982	0	0.374	0.576	0	0	RI MCV1 only
1983	0	0.374	0.576	0	0	RI MCV1 only
1984	0	0.374	0.576	0	0	RI MCV1 only
1985	0	0.374	0.576	0	0	RI MCV1 only
1986	0	0.374	0.576	0	0	RI MCV1 only
1987	0	0.374	0.576	0	0	RI MCV1 only
1988	0	0.374	0.576	0	0	RI MCV1 only
1989	0	0.374	0.576	0	0	RI MCV1 only
1990	0	0.374	0.576	0	0	RI MCV1 only
1991	0	0.374	0.576	0	0	RI MCV1 only
1992	0	0.374	0.576	0	0	RI MCV1 only
1993	0	0.374	0.576	0	0	RI MCV1 only
1994	0	0.374	0.576	0	0	RI MCV1 only
1995	0	0.374	0.576	0	0	RI MCV1 only
1996	0	0.374	0.576	0	0	RI MCV1 only
1997	0	0.374	0.576	0	0	RI MCV1 only
1998	0	0.374	0.576	0	0	RI MCV1 only
1999	0	0.374	0.576	0	0	RI MCV1 only
2000	0	0.292	0.362	0.179	0.179	RI MCV1; SIA for ages 9–59 months
2001	0	0.349	0.398	0.0675	0.0675	RI MCV1; SIA for ages 9–59 months
2002	0	0.385	0.483	0.1	0.1	RI MCV1; SIA for ages 9 months 14 years
2003	0.772	0.441	0.492	0.234	0.234	RI MCV1; SIA for ages 9 months 14 years
2004	0	0.374	0.543	0.405	0.405	RI MCV1; SIA for ages 9 months 14 years
2005	0	0.374	0.516	0.192	0.192	RI MCV1; SIA for ages 9–59 months
2006	0	0.374	0.524	0.405	0.405	RI MCV1; SIA for ages 9–59 months
2007	0.719	0.391	0.494	0.1	0.1	RI MCV1; SIA for ages 6–59 months
2008	0.772	0.521	0.322	0.127	0.127	RI MCV1; SIA for ages 6–59 months
2009	0	0.374	0.593	0.771	0.771	RI MCV1; SIA for ages 6–59 months
2010	0	0.449	0.371	0.183	0.183	RI MCV1; SIA for ages 6–59 months
2011	0.931	0.459	0.498	0.321	0.321	RI MCV1; SIA for ages 6 months-14 years
2012	0	0.374	0.576	0	0	RI MCV1 only
2013	0	0.374	0.497	0.318	0.318	RI MCV1; SIA for ages 9–59 months
2014	0	0.374	0.576	0	0	RI MCV1 only
2015	0.525	0.403	0.404	0.096	0.096	RI MCV1; SIA for ages 6–59 months
2016	0	0.374	0.544	0.456	0.456	RI MCV1; SIA for ages 9–59 months
2017	0	0.386	0.602	0.652	0.652	RI MCV1; SIA for ages 6 months-14 years
2018	0	0.374	0.576	0	0	RI MCV1 only
2019	0	0.374	0.548	0.659	0	RI MCV1, MCV2

Abbreviations: ‘mu_e’, born current year, may only get MCV1 due to age; ‘e_12’, born current year, may get MCV1 and SIA due to age; ‘0’, born previous year but turns 1 year old in current year; ‘1’, turns 2 years old in current year; ‘2+’, turns 3 years old or more in current year.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study were obtained from the following resources available in the public domain, as described in the methods and listed in the reference list. References to data sources used: 49–55,59,60.