The potential impact of the demographic transition in the Senegal-Gambia region of sub-Saharan Africa on the burden of infectious disease and its potential synergies with control programmes: the case of hepatitis B

Background Sub-Saharan Africa (SSA) continues to suffer high communicable disease burdens as its demographic transition (DT) proceeds. Although the consequent changes in population structures influence age-specific contact patterns relevant for transmission, the age distribution of immunity, and the disease burden, investigation of the potential of DT to affect infectious disease epidemiology in regions of SSA has hitherto been overlooked. With a substantial disease burden and complex epidemiology, hepatitis B virus (HBV) represents a prime example of an infection whose epidemiology may be significantly influenced by the DT. Methods An age-structured mathematical model for HBV in the Senegal and Gambia (SG) region was set within a demographic framework with varying vital rates mirroring the entire course of the DT there over 1850–2100, to investigate the effects of the DT on HBV epidemiology, with and without the combined action of vaccination. The model was run from its reconstructed ancien régime (old order) demo-epidemiologic equilibrium and calibrated against SG 1950 age-distribution estimates and Gambian pre-vaccination HBV age-prevalence data. Results The model, which reproduced well demographic and HBV age-prevalence data, predicted a complex transition of HBV epidemiology over the course of the DT. This included a prolonged epoch of expansion alongside population growth and rejuvenation until 1990–2000, followed by a dramatic retreat, mainly reflecting projected fertility decline during the twenty-first century. This transitional pattern was mostly explained by the underlying demographically driven changes in horizontal transmission resulting from the changes in the age structure of the population. During 2000–2150 the HBV burden is predicted to decline by more than 70% even in the absence of vaccination. Conclusions Demographic change alone may strongly affect HBV disease burden and shape HBV endemicity. The onset of the demographically driven decline in HBV prevalence, aligned with the expansion of HBV vaccination, forms a synergy potentially boosting effectiveness of control. Such a synergy currently appears to be presenting a “window of opportunity” facilitating HBV elimination which it would be important to exploit and which underlines the importance of taking demographic change into account when assessing the potential longer term impact of vaccination and other control measures. Electronic supplementary material The online version of this article (10.1186/s12916-018-1100-0) contains supplementary material, which is available to authorized users.

where (i=S,G,SG) represents the corresponding average female population during the same time period in the same age group in Senegal, in The Gambia, and in SG respectively.
The overall (estimated and projected) trend of the TFR for SG over 1950-2100 based on the underlying UN estimates 1950-2015 and the "medium" projection variant for 2015-2100 is displayed in Fig. S1.1.

SG 1950-2100 mortality
Similarly, life tables for both sexes for SG were generated for each quinquennial period during 1950-2100, by averaging the underlying life tables for Senegal and Gambia. This was done by retaining the estimated (or projected) life tables for Gambia and Senegal, and then averaging them using the total number of deaths of observed in each country in the same period.

Reconstruction of SG demographic "ancien-régime "
Over period 1950-2015 Senegal and Gambia have been characterized by rapid population growth fueled by declining mortality in presence of persistently high fertility. Mortality decline surely initiated prior to 1950 but unfortunately, to the best of our knowledge, this early phase is poorly documented. In particular, the total fertility rate (TFR) increased in Senegal (Gambia) from about 6.5 (5.5) in 1950 to 7.5 (6.5) in 1980 before initiating its decline (Fig S1.1). This state of affairs implied that since 1950 onward the Senegal and The Gambia (total) population has been increasing at a large growth rate, in the region of 2.5-3.0 % per year (Fig S1.2). In particular, in the case of Senegal (which represents the large majority of the SG population), the growth rate has been fairly constant. Assuming negligible perturbation by migration this suggests that its population experienced a phase of approximately stable growth (Keyfitz and Caswell 2008), where a combination of persistent high fertility and not-too-fast declining mortality were promoting a coarsely time invariant age distribution. This is also documented by the substantial stability of the underlying age specific growth rates (not reported) which represent the most reliable indicators of the presence of stable growth (Preston et al 2000). The corresponding trends for the Gambia are much more erratic, possibly due to the very small population size which possibly made it very sensitive to migrations, but still the effects of the gap between high fertility and declining mortality are very clear, with an average growth rate about 3.5% during 1950-2015. An implication of this trend is that already in 1950 Senegal and Gambia populations were far from the ideal state of ancien-régime stationary equilibrium 1 that we expect to have been broadly prevailing at the beginning of the demographic transition i.e., before the destabilization that occurred when mortality decline, along the mortality transition, initiated. Since our main hypotheses here are that also HBV was in stationary equilibrium during the demographic ancien-régime, and that the early mortality transition had the potential to perturb the equilibrium of HBV, in order to initialize the model from a condition of full demo-epidemiologic stationarity, we supplied a coarse reconstruction of SG demographic ancien-régime.
The concept of a stationary demographic equilibrium implies a stationary (over time) mortality regime (i.e., a stationary life table) combined with a stationary reproduction, where the typical female individual produces on average one female offspring during her entire fertile period given prevailing mortality conditions. The latter condition is expressed by requiring that the ( As a preliminary step towards the reconstruction of the ancien-régime mortality we made the simplifying hypothesis that our estimate of SG fertility in 1950 based on UN 1950 estimates was representative of the true SG ancien-régime fertility. As previously pointed out, according to UN data both Senegal and Gambia exhibited a marked increase in the total fertility rate which increased in 1 In demographic jargon a stationary population is one characterized by time-invariant total size and age distribution. Senegal (Gambia) from about 6.5 (5.5) in 1950 to 7.5 (6.5) in 1980 before plateauing and initiating to decline. Such initial phase of increase in TFR parallel to mortality decline is a well-documented fact of the fertility transition in the developing world (e.g., Dyson and Murphy 1985). Note also that the slope of the trend was already markedly positive (especially in Senegal) at 1950. This therefore suggests that some increase in fertility was likely already in place prior to 1950. Therefore, our hypothesis is to be considered as a departure point representing a useful baseline. Nonetheless the results reported in the manuscript are not sensitive to small departures from this baseline. The resulting ancien-régime age-specific fertility schedule ( , ) for SG is reported below in

SG ancien-regime mortality
As for mortality, according to the UN, West Africa countries were characterized by a large heterogeneity in mortality profiles in 1950, with some countries e.g., Sierra Leone, showing a much higher mortality compared to SG. Therefore, as a first step, we borrowed the life tables estimated by the UN for Sierra Leone in 1950-1980 by assuming that they adequately represented pre-1950 mortality in SG. This allowed us to have estimates of SG mortality going back to quinquennium 1915-1920. Combining the resulting figures of SG age-specific mortality at 1915-1920 with ancient-regime SG fertility led to a (demographic) net reproductive rate NRRin the region of 1.4, therefore still inconsistent with full population stationarity which requires NRR=1. As a next step we therefore supplied a number of simple reconstructions of the Sierra Leone ancien-regime life-table, meant as a life-table which, combined with SG age-specific fertility rates ( , ), allowed the attainment of a stationary population. The simplest approach was based on a scaling of the ( , ) function by means of a single proportionality factor q (0<q<1) that -once applied to all ages but zero yielded a NRR equal to one. This procedure essentially assumes that all progress in mortality prior to 1915-1920 was concentrated on infant mortality, yielding an unaltered profile of life expectancy at ages different from birth. The resulting female (male) life table implied roughly 50% of each birth cohort eliminated during the first year of life, and 60% before age 5, with a female (male) life expectancy at birth was 20.9 (18.6) yr, while life expectancy at age 5 was about 43 (38).
The SG ancien-régime life tables for females and males reconstructed by this approach are reported in Tables S1.2 & S1.3 below.  Table S1.2 The reconstructed SG "ancien-régime" female life table. Column 2: survivor function l(x), representing the number still alive at each exact age x. Column 3: a(x,n), representing the average number of person-years lived by those dying in each age group (x,x+n). Column 4: d(x,n), representing the number dying in each age group. Column 5: L(x,n), representing the number of person-years lived in each age group. Column 6: m(x,n), representing the mortality rate in each age group. Column 7: e(x), the life expectancy at each exact age x.  Table S1.3. The reconstructed SG "ancien-régime" male life table. Column 2: survivor function l(x), representing the number still alive at each exact age x. Column 3: a(x,n), representing the average number of person-years lived by those dying in each age group (x,x+n). Column 4: d(x,n), representing the number dying in each age group. Column 5: L(x,n), representing the number of person-years lived in each age group. Column 6: m(x,n), representing the mortality rate in each age group. Column 7: e(x), the life expectancy at each exact age x.

Age l(x) a(x,n) d(x,n) L(x,n) m(x,n) e(x)
A summary overview of the evolution of the SG female life tables as depicted by the survivor function is reported in  Also alternative approaches were used to reconstruct an ancien-régime life table for SG, for example by projecting into the past the time series of probabilities of dying at each age estimated by the UN during . Though the results could differ somewhat between themselves, the epidemiological results on the prediction of the course of HBV during the DT reported in the main text are fairly robust with respect to the reconstructed ancien-régime life table.

Text S2: Model equations
The age-structured model of population and HBV transmission dynamics is based on the following system of partial differential equations (PDE) and related boundary conditions.
dX a t λ a t τ a t μ a t X a t at where: X g (a,t) = number of people of gender g and age a, who at time t are susceptible to HBV infection H g (a,t) = number of people of gender g and age a, who at time t have latent HBV infection; Y g (a,t) = number of people of gender g and age a, who at time t have acute HBV infection; C g (a,t) = number of people of gender g and age a, who at time t have chronic HBV infection; Z g (a,t) = number of people of gender g and age a, who at time t are in the recovered state; V g (a,t) = number of people of gender g and age a, who at time t are immune to HBV infection thanks to successful immunization; The population of gender g and age a at time t is given by: In particular the only non-zero boundary conditions are those for the compartments of susceptible individuals and of those suffering perinatal infection N f (a,t) = number of women at age a, time t; ν(a,t) = fertility rate per annum at age a and time t; (a 3 ,a 4 )= fertile age span π g = proportion of births of gender g; θ Y = proportion of vertically infected births to acutely infected mothers at time t; θ C = proportion of vertically infected births to chronically infected mothers at time t; Post natal force of infection: This is gender and age-specific and it is defined at each time point t as the sum of the age-dependent (but gender-independent) horizontal force of infection and of the age-gender -specific sexual force of infection (Garnett & Anderson, 1993 where βʹ(a,α) represents the per-capita age-specific transmission rates, and κ the infectiousness of persons chronically infected relative to that of persons with acute infection. We follow the standard assumption that the horizontal force of infection is piecewise constant (Anderson and May 1991) so that the transmission rates βʹ(a,α) can be represented into the form of a WAIFW ("who acquires infection from whom") matrix of elements . We used the following age groups (in completed years) relevant for horizontal transmission: 0, 1-4, 5-9, 10-14, 15+ (Edmunds et al 1996).

Sexual force of infection of HBV:
This represents the age-gender specific rate at which susceptible individuals of gender g and age acquire HBV infection through heterosexual sexual contacts, per unit of time: Where 1 , 2 = age at onset and cessation of sexual activity, respectively   , g c αt  = average numbers of new sexual partners per annum by people of gender g' at age α, time t ρ g (a,α,t) = contact or mixing matrix, representing the proportion of sexual partners at time t which people of age a, gender g have with people of the other gender having age α; β 1 g΄ , β 2 g΄ = risk of transmission of HBV in a partnership with an infected person of gender g΄ in, respectively, the stages of acute and chronic infection.
In particular, the mixing matrix is specified according to the so-called preferential rule (Garnett and Anderson 1993), according to which individuals of given gender and age choose their sexual partners in proportions which are an average of the limit cases of perfect ageassortativeness (where all partners are chosen in the same group as own) and of proportionate mixing (where partners are chosen at random): where ε is a weighting factor ranging between 0 and 1, determining the degree of assortative sexual mixing by age group (actually here we took ε as age-independent). δ a,a΄ : is equal to one for = ′ and zero elsewhere and corresponds to the case of a fully ageassortative mixing matrix, defined as follows (Garnett and Anderson 1993): (1.14) P g (a',t) represents a proportionate (or random) sexual mixing matrix, defined as follows (Garnett and Anderson 1993)  with Latin hypercube sampled parameters mode, Μ, corresponding to age group, and baseline value, θʹ.  To assess levels of uncertainty in the results, a series of model runs was carried out using the best 10% of the parameter constellations sampled with LHS, i.e. the 10% giving rise to the lowest values of the least squares function when fitting the HBV data, In analysing these results the maximum and minimum values of the appropriate model outputs at each time point were selected as the upper and lower uncertainty bounds at that time point (see Figures S3.1-S3.3); it is important to note therefore that in each figure the successive points on the upper and lower uncertainty bounds do not necessarily correspond to a single trajectory.

Text S4 Vaccination at birth
Lack of availability of monovalent HBV vaccine and issues relating to infrastructure and the logistics of delivering birth doses of vaccine remain substantial barriers to the implementation in the region of effective programmes of vaccination at or shortly after birth, even though this has proved to be an effective tool for the prevention of perinatal HBV infection. Nevertheless, in order to assess the potential impact of a programme of vaccination at birth the modelling also investigated the effectiveness of vaccination at birth in place of infant vaccination but with the same baseline coverage of 46% ( Figure S4.1). In the case using the UN medium projections vaccination at birth provides a small but significant advantage compared with infant vaccination, achieving similar levels of reduction in prevalence of chronic infection a decade or two earlier after 50 years or so ( Figure S4.1 upper panel). However in the worst case scenario of fertility remaining at 1990 levels the difference is much more marked and while by 2150 infection nears elimination with vaccination at birth, infant vaccination achieves no more by 20150 than prevalence beginning to stabilises at just over 3% ( Figure S4.1 lower panel) Figure S4.1 HBV disease burden over the DT comparing impact of vaccination at 3.5 months of age with vaccination at birth each with effective coverage of 46% starting in 2005 Text S5 Incidence ratios Figure 5 in the main text shows model results for temporal change in predicted HBV incidence by transmission route over the entire course of the DT under the UN medium variant. Figure S5.1 shows the ratios between incidence results for horizontal and vertical and horizontal and sexual transmission in order to clarify the relationships between incidence arising via these transmission routes Figure S5.1 Ratio relative to incidence by horizontal transmission of (a) incidence by vertical and (b) incidence by sexual transmission (corresponds to ratios of results shown in Figure 5 in the main text for incidence arising from horizontal, vertical and sexual tranmission ).

Text S6 Influence of individual parameters
For each of the model parameters in turn shown in Table S6.1 model runs were undertaken using the values of this parameter found in the 2,400 parameter sets (10% of the total) resulting in the lowest least squares values; at the same time the remaining parameter values were kept at the single best fit value. For each set of model runs varying a single parameter least squares values were calculated for the fit to the Gambian HBsAg seroprevalence data. The distribution of resulting least squares values for each parameters are reported in Figure  S6,1 which shows that the parameters with most influence on the fit to HBsAg data were the (i) risk of transmission from someone with chronic infection relative to transmission from someone with acute infection (L), (ii) risk of vertical transmission from mother with chronic infection to unvaccinated newborn (O), and (iii) the scaling factor for the WAIFW matrix (P). Less influential were the durations of latent (A), acute (B) and chronic (C) infection; parameters relating to sexual transmission were the least influential Note that a logarithimic scale is used for the vertical axis (for key see