Model formulation and analysis for rubella infection in Zambia

This study used a deterministic compartmental model to describe the transmission dynamics of rubella infection in Zambia. Compartmental models are widely used to study the dynamics of viral infections among other infections that are transmitted in populations [Reference Anderson and May13]. The total population (N) is divided into the following epidemiological classes: susceptible, S (individuals who are not yet infected but may be infected with rubella over time). Some of these susceptible populations may be vaccinated or will be exposed when they come into contact with either an infectious pregnant woman or an infectious individual in the general population. The exposed (E) or latent class are individuals (pregnant women or the general population) who have been infected with rubella but are not infectious to others. Within days, this group of exposed individuals becomes infectious and move to the infectious (I) class. The infectious sub-classes are pregnant women and the general population that can transmit rubella infection to others in the susceptible class due to the high accumulation of viral load. Both the exposed and the infectious classes may recover from rubella infection at different rates and will be classified under the recovery sub-class. Infectious pregnant women can either host an infected foetus or an uninfected foetus due to the possibility of vertical transmission and the infected foetus may be either aborted or they will be born with CRS. This study emphasised that children may recover from the viral infection and join the recovery class.

The model adjusts for seasonality by [Reference Anderson and May13] including seasonality-dependent force of infection λ(t) = β ₀ + β ₁sin(2πt/12) + β ₂cos (2πt/12) where t is time in months. The study distinguished between different levels of vaccination by assuming that individuals who are vaccinated only receive one dose and modelled the possibility of receiving full doses of vaccination. The transmission rate was estimated from the observed number of cases using the non-linear least square estimation via Levenberg–Marquardt algorithm that minimises the sum of squared error between the observed cumulative number of rubella cases and predicted cumulative cases. Other parameters were carefully chosen to fit our definition to fit the model. The time step for the model is monthly and we modelled counterfactual scenario of no vaccination post 2016. The systems of differential equations were analysed using the ‘desolve’ package in R [Reference Soetaert, Petzoldt and Setzer14]. Using the approach of Cori et al. [Reference Cori15], we estimated the time-dependent effective reproduction number Rt using monthly incidence time series rubella data between 2005 and 2016. The Cori et al. approach assumes that once infected, individuals have an infectivity profile given by a probability distribution W _s, and is dependent on time since infection of the case, s, but independent of calendar time, t. The authors estimated the instantaneous reproductive number as the ratio of the number of new infections generated at time step t, I _t, to the total infectiousness of infected individuals at time t, given by $\sum\nolimits_{s = 1}^t {I_{t-s}W_s} $, the sum of infection incidence up to time step t  −  1, weighted by the infectivity function W _s. R _t is the average number of secondary cases that each infected individual would infect if the conditions remained as they were at time t. The incidence at time t is Poisson distributed with mean$R_t\sum\nolimits_{s = 1}^t {I_{t-s}w_s} $.

A description of variables and parameters is given in Table 1 below.

Table 1. Description of variables and parameters

Figure 1 is a compartmental model showing the conceptualised dynamics of rubella between 2005 and 2016 in Zambia

Fig. 1. A compartmental model showing the dynamics of rubella infection in Zambia.

Systems of differential equations

$$\displaystyle{{dS_G} \over {dt}} = {-}\displaystyle{{\lambda _{IG}S_GI_PI_G} \over N}-\mu S_G-\varepsilon S_G$$

$$\displaystyle{{dS_P} \over {dt}} = {-}\displaystyle{{\lambda _{IG}S_PI_PI_G} \over N}-\mu S_P-\varepsilon S_P$$

$$\displaystyle{{dE_P} \over {dt}} = \displaystyle{{\lambda _{IG}S_PI_PI_G} \over N}-\mu E_P-\sigma _PE_P-\eta _PE_P$$

$$\displaystyle{{dE_G} \over {dt}} = \displaystyle{{\lambda _{IG}S_GI_PI_G} \over N}-\mu E_G-\sigma _GE_G-\eta _GE_G$$

$$\displaystyle{{dI_G} \over {dt}} = \sigma _GE_G-\eta _GI_G-\mu _RI_G-\;\mu I_G$$

$$\displaystyle{{dI_P} \over {dt}} = \sigma _PE_P-\eta _PI_P-\mu _RI_P-\;\mu I_P$$

$$\displaystyle{{dV_1} \over {dt}} = \varepsilon S_T-\mu V_1-\theta V_1-kV_1$$

$$\displaystyle{{dV_2} \over {dt}} = \theta V_1-\mu V_2-kV_2$$

$$\displaystyle{{dQ} \over {dt}} = \displaystyle{{\lambda _{IG}S_PI_PI_G} \over N} + \displaystyle{{\lambda _{IG}S_GI_PI_G} \over N}$$

$$\displaystyle{{dR} \over {dt}} = \eta _PE_P + \eta _GE_G + \eta _GI_G + \eta _PI_P + kV_1 + kV_2-\;\mu R$$

$$\displaystyle{{dS_T} \over {dt}} = \displaystyle{{dS_G} \over {dt}} + \displaystyle{{dS_P} \over {dt}}-\varepsilon S_T + \rho V_1 + \pi N$$

$$S_T( t ) = S_G( t ) + S_P( t ) $$

The total population size N(t) is assumed to be constant and homogenous:

$$N( t ) = S_G( t ) + S_P( t ) + E_G( t ) + E_P( t ) + I_G( t ) + I_P( t ) + V_1( t ) + V_2( t ) + R( t ) $$

We did not provide detailed analytic expression of rubella infection since the focus of the study was to track the incidence of rubella.

Ethical consideration

The data used in this study are de-identified secondary data from the Government-run measles surveillance programme. Ethical clearance was sought from the Ministry of Health, ERES Coverage IRB (00005948) and the National Health Research Authority.