Summary and Future Directions

The different features of the COVID-19 series in Mauritius builds the foundations of this book. Several models under the family of high-ordered integer-valued autoregressive processes, most precisely, INAR( p ) processes with different Poisson-mixtures innovations and their corresponding zero-inflated parts under different thinning mechanisms, were considered thoroughly while taking into account the effect of time-varying covariate specifications and non-stationarity. As discussed earlier in Chapter 2 , such processes are highly suitable to model over-dispersed time series with higher orders. In Chapter 3 , the INAR( p ) processes with different Poisson mixtures innovations were mounted and tested via simulation experiments. The simulation experiments allowed us to conclude that ZI-NB, ZI-PWE, and ZI-PGLD, under the GB thinning, were able to produce less biased estimates and satisfactory standard errors. On the other, we also observe smooth execution of the binomial thinning under any scenarios, without any simulation failures and this allowed us to further explore the best suited innovation distribution under Chapters 5 to 8 . Moreover, both CLS and CML approaches are shown to provide fairly consistent estimates, but with CML yielding slightly lesser biased estimates under both Binomial and Generalized Binomial thinnings. Some computational failures were noticed with the CML approach in particular with the CMP, PT, PIG, PL and WCG innovations. The results under Chapter 3 , motivated us to move to Chapter 4 whereby different ZI-models under the GB and NB thinnings were applied to the COVID-19 series in Mauritius. Here, the non-stationary INAR (7) with ZI-NB, ZI-PWE and ZI-PGLD innovations under the GB thinnings were shown to provide lower AICs though its complex algorithm were really time consuming to execute. Note that working under the NB thinning proved not only to be computationally challenging but the AIC that it reaps were not comparatively very satisfactory. Conversely, the estimates from the Generalized binomial thinning show less biasedness, but this may be subject to the quality of the data as well thus GB should be used restrictively. Given the satisfactory performance of the ZI-PWE and ZI-PGLD distributions after the ZI-NB, under Chapter 5 , an attempt to further explore them, while catering for periodic aspect observed in the COVID-19 new infection and death series of South Africa and Mauritius, were conducted. Different periodic INAR processes with Poisson-mixtures models and their ZI versions were implemented and assessed under the non-stationary setting. Here, again the periodic INAR processes with ZI-NB outperformed the other models. As mentioned earlier, PWE and PGLD were recently emerged and considering their performance under univariate framework, we explored the bivariate processes of the PWE and PGLD under Chapters 6 and 7 , respectively. We explored extensively two models namely the bivariate of order one with PWE innovation and the bivariate of order one with PGLD innovation under the stock markets data series and the monthly crime data of Pittsburgh, respectively. These two series worked well for bivariate INAR (1) processes but this also motivated us to extend it to BINAR( p ) processes so that high-ordered integer-valued series can be studied too. Chapter 8 brings forward the properties of BINAR( p ) processes and assess its applicability on the COVID-19 new infection and death series of Mauritius under both the stationary and non-stationary setting. Here, various time varying covariates were used namely the COVID-19 stringency index which has proved to be highly significant in reducing the number of new COVID-19 infection cases in Mauritius and in South Africa, the age factor and comorbidities like the cardiovascular diseases, diabetes and smoking status of COVID-19 patients. All these proved to be significant however, the take-aways from this study is the utmost need to widen the list of metrics considered for the computation of the COVID-19 stringency index. For instance, latent variable like behavior of the population with regard to the newly established COVID-19 specific legislations, adherence to vaccination roll-outs, can be induced into the computation of this index. This work also came up with reliable forecasts results which will surely help authorities to come up with more preparedness and remedial strategies.