Health
Integer time series models for tuberculosis in Africa
Models based on the negative binomial distribution gave the better fit for Sao Tome and Principe, Eritrea, Seychelles, Tanzania, Cabo Verde, Djibouti, Eswatini, Kenya, Lesotho, Malawi, Namibia, South Africa and Zimbabwe. Models based on Poisson distribution gave the better fit for Chad, Comoros, the Democratic Republic of Congo, Mauritius, Morocco, Nigeria, the Central African Republic, Benin, Botswana, Burkina Faso, Burundi, Cameroon, Cote d’Ivoire, Egypt, Equatorial Guinea, Ethiopia, Ghana, Guinea, Liberia, Madagascar, Mali, Mauritania, Niger, Senegal, Sudan, the Gambia, Uganda, Zambia, Gabon, Guinea-Bissau, Mozambique, the Republic of Congo, Rwanda, Sierra Leone, Somalia, Tunisia, Angola, Algeria and Togo.
In the following countries, the newly reported cases of TB exhibits a decreasing trend over 2000 to 2021: Benin, Botswana, Burkina Faso, Burundi, Cameroon, Cote d’Ivoire, Egypt, Ethiopia, Ghana, Guinea, Madagascar, Mali, Mauritania, Niger, Senegal, Sudan, The Gambia, Uganda, Zambia, Eritrea, Seychelles and Tanzania. The decreasing trend appears sharpest for Botswana and shallowest for the Gambia. Equatorial Guinea and Liberia exhibit positive trends in the newly reported cases of TB over 2000 to 2021. Equatorial Guinea has the sharper positive trend.
The number of new cases immediately preceding has a positive impact on the rate of new cases for Gabon, Guinea-Bissau, Mozambique, the Republic of Congo, Rwanda, Sierra Leone, Somalia, Tunisia, Angola, Algeria, Togo, Cabo Verde, Djibouti, Eswatini, Kenya, Lesotho, Malawi, Namibia, South Africa, Zimbabwe, and Sao Tome and Principe. The second to the previous number of new cases has a negative impact on the rate of new cases for Sao Tome and Principe.
Figures 3, 4, 5, 6, 7, 8, 9, 10 and 11 plot the predicted new number of cases of TB for the period 2022 to 2031. Also plotted in these figures are the median of the new number of cases and 95% confidence intervals of the new number of cases. For the identity Poisson model, log Poisson model, log Poisson model regressed linearly versus year, identity Poisson model regressed on the previous observation, log Poisson model regressed on the previous observation, log negative binomial model regressed linearly versus year, log negative binomial model regressed on the previous observation and the log negative binomial model regressed on the two previous observations, the predicted new number of cases, say P(t), was estimated by \(\widehat{\beta _0}\), \(\exp \left( \widehat{\beta _0} \right)\), \(\exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right)\), \(\widehat{\beta _0} + \widehat{\beta _1} Z_{t – 1}\), \(\exp \left( \widehat{\beta _0} + \widehat{\beta _1} Z_{t – 1} \right)\), \({\widehat{\phi }} \left( 1 – {\widehat{\phi }} \right) ^{-1} \exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right)\), \({\widehat{\phi }} \left( 1 – {\widehat{\phi }} \right) ^{-1} \exp \left( \widehat{\beta _0} + \widehat{\beta _1} Z_{t-1} \right)\) and \({\widehat{\phi }} \left( 1 – {\widehat{\phi }} \right) ^{-1} \exp \left( \widehat{\beta _0} + \widehat{\beta _1} Z_{t-1} + \widehat{\beta _2} Z_{t-2} \right)\), respectively, for \(t = 2022, 2023, \ldots , 2031\). The corresponding median of the new number of cases, say M(t), was estimated as the roots of the following equations
$$\begin{aligned}{} & {} \displaystyle \sum _{k = 0}^{M
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