## Appendix C Arima Models of the Incidence of Malaria  
Models for single time series in epidemiology and infectious disease research have been widely used [11; 15; 16; 17; 18; 35; 46; 47; 48; 49; 50; 51; 52; 53]. In this section, malaria time series will be analysed from a different perspective, involving simulations in two different human-to-mosquito transmission efficiency \(w_{h}\) settings (\(w_{h}=0.420\) and \(w_{h}=0.733\)), based upon our malaria transmission agent-based stochastic model [3]. Those results will be compared with two empirical time series with different transmission patterns from separate world regions where malaria is still endemic, \"Okech\" and \"Landoh\" [25; 27]. The Okech time series reveals a pattern consistent with an unstable epidemic state, with a decreasing trend from high levels of malaria incidence, evolving to imminent disease elimination. The other empirical time series reveal an unstable epidemic behavior, but, with a non-stationary increasing trend and an average level of malaria incidence by African standards of between 10 and 50 malaria cases per 100 inhabitants per year. See Table 12.  
As our model shows annual seasonality approximately, we choose a SARIMA model with a period of 12 months:  
\[\Phi^{P}(B^{s})\phi^{P}(B)\
abla_{s}^{D}\
abla^{d}X_{t}=\Theta^{Q}(B^{s})\theta ^{q}(B)Z_{t}\,, \tag{12}\]where \(Z_{t}\) represents a Gaussian white noise time series (i.e., \(Z_{t}=N(0,\sigma^{2})\)) and the ordinary \(\
abla\) operator represents the one-difference operator, \(\
abla X_{t}=X_{t}-X_{t-1}\), \(\
abla^{d}X_{t}=(1-B)^{d}X_{t}\) is the ordinary differencing level \(\
abla^{d}\) and \(\
abla_{S}^{D}X_{t}=(1-B^{s})^{D}X_{t}\) is the differencing level of the seasonal \(\
abla^{D}\) with periodicity \(s\). Operator \(\phi^{p}(B)\) is the ordinary autoregressive operator \(\phi^{p}(B)=1-\phi_{1}B-\phi_{2}B^{2}-\phi_{3}B^{3}-...-\phi_{p}B^{p},\theta^ {q}(B)\) the ordinary moving average operator \(\theta^{q}(B)=1+\theta_{1}B+\theta_{2}B^{2}+\theta_{3}B^{3}+...+\theta_{q}B^{q}\), \(\Phi^{p}(B^{s})\) is the seasonal autoregressive operator \(\[PERSON]^{p}(B^{s})=1-\Phi_{1}B^{s}-\[PERSON]_{2}B^{2s}-\[PERSON]_{3}B^{3s}-...-\Phi_{p}B^{p _{s}}\), \(\Theta^{Q}(B^{s})\) is the seasonal moving average operator \(\Theta^{Q}(B^{s})=1+\Theta_{1}B^{s}+\Theta_{2}B^{2s}+\Theta_{3}B^{3}+...+ \Theta_{Q}B^{qs}\). Here \(B^{s}(X_{t})=X_{t-s}\) is the seasonal - or \(s\)-order - backshift operator. In particular \(B(X_{t})=X_{t-1}\). Table 14 shows all parameters of the SARIMA model for the two simulations, while Table 14 shows all parameters of the SARIMA model for the two simulations. From the present model simulations, the SARIMA linear models have been defined according to the level of gametocytemia and disease transmission efficiency. There are small differences in the type of the best SARIMA model between stochastic simulations at the same level of disease transmission intensity. The presented results have been obtained from the first-tested simulation at every transmission level. Close to the phase transition at the lower transmission level (63 days gametocytemia duration; \(w_{h}=0.420\)) the most reliable SARIMA model has been defined as \((2,1,2)\times(2,1,2)^{12}\), corresponding to a combination of auto-regressive(\(p=2\) and \(P=2\)), and moving average (\(q=2\) and \(Q=2\)) procedures, with the need of time series differencing (once, \(d=1\) and \(D=1\)), both in the non-seasonal and the 12-month seasonal model fractions. At the higher transmission level (110 days gametocytemia duration; \(w_{h}=0.733\)) the best SARIMA model has been defined as \((2,1,1)\times(2,1,1)^{12}\), with a predominantly auto-regressive component (\(p=2\) and \(P=2\)) and a lower-order moving average (\(q=1\) and \(Q=1\)), with the need for time series differencing (once, \(d=1\) and \(D=1\)), both in the non-seasonal and in the 12-month seasonal model fractions. In both cases, the best model has been defined according to Akaike information criterion optimization.