Model-Based Estimation of Expected Time to Cholera Extinction in Lusaka, Zambia

Abstract

The developing world has been facing a significant health issue due to cholera as an endemic communicable disease. Lusaka was Zambia’s worst affected province, with 5414 reported cases of cholera during the outbreak from late October 2017 to May 12, 2018. To explore the epidemiological characteristics associated with the outbreak, we fitted weekly reported cholera cases with a compartmental disease model that incorporates two transmission routes, namely environment-to-human and human-to-human. Estimates of the basic reproduction number show that both transmission modes contributed almost equally during the first wave. In contrast, the environment-to-human transmission appears to be mostly dominating factor for the second wave. Our study finds that a massive abundance of environmental vibrio’s with a huge reduction in water sanitation efficacy triggered the secondary wave. To estimate the expected time to extinction (ETE) of cholera, we formulate the stochastic version of our model and find that cholera can last up to 6.5–7 years in Lusaka if any further outbreak occurs at a later time. Results indicate that a considerable amount of attention is to be paid to sanitation and vaccination programs in order to reduce the severity of the disease and to eradicate cholera from the community in Lusaka.

Appendix A: Results with Normal Error Structure

Fig. 7

Fitting the model (1) to the weekly reported cholera cases in Lusaka, Zambia, for the $1\textrm{st}$ wave. The empirical distributions of each estimated parameter are illustrated by the histograms using 1000 bootstrap replicates with normal error structure. The location of the true values of the parameters is marked as vertical dotted yellow lines. Here reported discrete data points are represented by blue circles, while the best fit of the model to the data is depicted by the solid red line in the bottom-left panel. The dashed red lines demonstrate 95% confidence bands around the best-fit line. Also, 1000 epidemic curves considering normal error structure (with mean the best-fit solution and deviation 20% of the mean) exhibited by the sky-blue lines (Color figure online)

Fig. 8

Fitting the model (1) to the weekly reported cholera cases in Lusaka, Zambia, for the $2\textrm{nd}$ wave. The empirical distributions of each estimated parameter are illustrated by the histograms using 1000 bootstrap replicates with normal error structure. The location of the true values of the parameters is marked as vertical dotted yellow lines. Here reported discrete data points are represented by blue circles, while the best fit of the model to the data is depicted by the solid red line in the bottom-left panel. The dashed red lines demonstrate 95% confidence bands around the best-fit line. Also, 1000 epidemic curves considering normal error structure (with mean the best-fit line and deviation 20% of the mean) exhibited by the cyan lines. We observe that during the second wave, the rate of environment-to-human transmission increases drastically (about 52%) due to the high abundance of environmental vibrio’s. Also, water sanitation efficiency ($\rho$) decreases significantly (about 44%) during this period (Color figure online)

We consider bootstrap realizations assuming normal error structures in the data and estimate parameters for both waves to assure that results revealed by our study are robust in the context of chosen error structure. For the normal error structure, we consider 1000 bootstrap replicates from the best-fitted curve with the best-fit solution as the mean and a deviation of 20% of the mean. The empirical distribution and 95% confidence interval for each of the estimated parameters associated with both waves are depicted in Figs. 7 and  8, respectively. In addition, estimated $\mathcal {R}_0$ with its partial partitions ($\mathcal {R}_e$, $\mathcal {R}_h$) is provided in Table 2. It can be seen that during the second wave, $\rho$ decreased by about 44%, and transmission from the environment was mostly responsible (estimated $\mathcal {R}_e$ was 90% of $\mathcal {R}_0$). Although the mean values of the estimated parameters are almost the same with both error structures, the uncertainty ranges of the estimated parameters with a normal error structure are wider than those obtained using the Poisson error structure in the data. The estimated ETE of cholera in Lusaka with normal error structure is found to be about 7 years which assures the robustness of the results in the context of chosen error structure in the data.

Table 4 Mean values with the 95% confidence intervals of the estimated parameters assuming normal error structure are illustrated in Figs. 7 and  8, respectively. Also, the value of the basic reproduction number ($\mathcal {R}_0$) with its partial partitions $\mathcal {R}_e$, $\mathcal {R}_h$ corresponds to both waves