Corporation engaged in the research, development, manufacturing and distribution of solar cell chips. In additoin, the company is involved in the trading of electronic parts and components. The company's products are mainly used in space application, solar power generation, power generation for civilian use and industry use, as well as consumer products, among others.
Latest Arima News
Jul 4, 2023
’s are parameters and ’s are unknown residuals assumed to be independent random variables with an identical standard normal distribution. When the time series is not stationary, one needs to integrate differencing into ARIMA(p, d, q). Let B denote Box-Jenkins backshift operator given by . Then from ARMA(p, q) one can derive ARIMA(p, d, q) model given by We note that ARIMA(p, d, q) model with nontrivial differencing has p+q+1 parameters. In order to find the order (p, d, q) of ARIMA that best fits to the given data, we first need to determine minimal differencing d that leads to stationary time series. This can be done using the Augmented Dickey–Fuller (ADF) test. Then, p and q can be selected by minimizing the Akaike Information Criterion corrected (AICc) score given by 13 where k is the number of model parameters, n is the sample size, and MSE is the mean square error term. AICc score is preferred to the original AIC when the sample size is small as in our case. 13 For ARIMA(p, d, q), we have k=p+q+1, when d is positive. Nonlinear Epidemic SI Model The mathematical model that we study is adapted from Zhao et al, where a two-compartment model is considered. More specifically, the population of sexually active individuals (ages 15–49) N is divided as Susceptible (S) and HIV Infectious (I). The flow diagram is given in Figure 1 , where is the annual rate of the population becoming sexually active, β is the transmission rate, µ is the sum of the annual natural death rate and rate of departure from the sexually active state, and is the annual death rate due to the disease. Figure 1 Flow diagram for the epidemic model SI. It gives the following SI epidemic model There are essentially three different kinds of models for that we may consider in this study, namely, constant, exponential, and logistic growth models. The constant model assumes that annually individuals become sexually active. The exponential model assumes that the rate of change of population is proportional to the current population N with the rate of growth r. Finally, the logistic growth assumes that there is a carrying capacity K that the population can never exceed. Adding the equations in our model and recalling yield It follows from the data presented in Table 1 that the population of individuals aged between 15 and 49 can be assumed to be constant; that is, where we took the average between the years 2010 and 2020. It means , which leads to and To estimate parameters, we use the constraint least square optimization. Since we let , we are left with three variables to estimate. We may estimate from HIV mortality data for the period 2014–2019, see Table 2 . Table 2 Mortality Due to HIV in Kazakhstan The HIV mortality rate for a given year can be obtained by dividing the number of deaths in Table 2 by the population size in Table 1 . Since we have δ constant, the point estimator can then be obtained by taking the average of these mortality rates leading to It remains to estimate β and µ. This is done with the built-in function curve_fit in the Python SciPy library that utilizes the Trust Region Reflective algorithm. Since all parameters are positive, we considered a lower bound constraint of 0.01 for both. To measure the goodness of fit we reported the coefficient of determination, r2 given by where is the estimated number of infectious individuals and is the average number of infectious individuals. To compare the SI model with ARIMA we report AICc scores for both. HIV Control Strategies Aside from comparing the models and forecasting the prevalence we also consider possible outcomes of HIV control strategies, more specifically, the effect of possible pre-exposure prophylaxis (PrEP) introduction. In the presence of PrEP we split the Susceptible compartment as Susceptible PrEP users and Susceptible individuals not using PrEP resulting in three-compartment model Figure 2 Flow diagram for the nonlinear epidemic model with PrEP. Here, k is the proportion of PrEP users becoming sexually active, and α is the efficacy of PrEP in reducing susceptibility per act. The corresponding epidemic model is given by Here, disease-free equilibrium is For calibration we barrow , and δ from the parameters estimates of the SI model above. We assume that α=0.5 and simulate how the prevalence changes with varying k in [0,1]. Results Order Estimates for ARIMA In the analysis, we utilized the Augmented Dickey–Fuller (ADF) test to determine the appropriate value for the parameter d in the ARIMA model. 14 , 18 Initially, we examined stationarity in the time series, and the ADF test with d = 0 and d = 1 yielded non-significant p-values, indicating the persistence of non-stationarity. However, upon increasing d to 2, we conducted the ADF test again and obtained a significant p-value, suggesting that the differenced series achieved stationarity. This finding is further supported by the 2nd order differencing and autocorrelation plots depicted in Figure 3 , where a clearer pattern of stationarity is observed. Thus, based on the ADF test results and visual analysis, we concluded that a value of d = 2 should be utilized for differencing in the ARIMA model. Figure 3 Differencing and autocorrelation when the parameter d=2, 2nd order differencing plot and autocorrelation plot. After selecting d = 2 based on the Augmented Dickey–Fuller (ADF) test, we proceeded to determine the optimal values for the parameters p and q in the ARIMA model. By minimizing the Akaike Information Criterion with correction (AICc) score, we found that the best fit was achieved with p = 1 and q = 0, resulting in an AICc score of −65.94. The results of the ARIMA(1,2,0) model are presented in Figure 4 , where the density plot of residuals indicates a normal distribution with a mean of 0.0078. These findings suggest that the ARIMA(1,2,0) model adequately captures the underlying patterns in the data. Figure 4 Normality of residuals for ARIMA(1,2,0) where mean equals 0.0078. Figure 5 presents the graphical representation of the actual HIV prevalence data in Kazakhstan alongside the fitted values obtained from the ARIMA model. Overall, the model demonstrates a satisfactory fit, closely aligning with the actual data, except for the initial two-year period. Table 3 provides a summary of the fitting details, including relevant statistics and parameters, shedding further light on the model’s performance. Table 3 The Goodness of Fit Metrics for ARIMA and SI Figure 5 ARIMA prediction of prevalence. Parameter Estimates for SI By fitting the nonlinear dynamical SI model to the HIV prevalence data, we were able to estimate the model parameters as follows: β=0.09072 and, µ=0.01. The coefficient of determination r2 for the fitted model is calculated to be 0.9971, which corresponds to an impressive 99.71% accuracy in explaining the variance in the data. Figure 6 illustrates the comparison between the fitted values and the actual values for the number of infectious individuals, providing a visual representation of the model’s performance in capturing the underlying dynamics of the HIV prevalence. Figure 6 SI prediction of prevalence. Based on the estimated model parameters (β=0.09072 and, µ=0.01), the basic reproduction number (R0) for the HIV prevalence can be calculated as 9.06. This value represents the average number of new infections caused by a single infectious individual in a completely susceptible population. Using these parameters, we conducted a forecast for the number of HIV active cases up to 2030, as shown in Table 4 . The forecast indicates that by 2030, the number of HIV active cases is expected to double compared to the current level. This projection highlights the potential growth and impact of the HIV epidemic if no significant interventions or prevention measures are implemented. It emphasizes the importance of continued efforts in HIV prevention, treatment, and awareness to address the challenges posed by the disease. Table 4 SI Predicted Active Cases Model Comparisons and Prevalence Forecasting Table 3 provides the goodness of fit metrics for both ARIMA and SI models. According to all three metrics we see that while both models have two parameters, the nonlinear SI model outperforms statistical ARIMA. Table 5 and Figure 7 provide HIV prevalence forecasting results for both models. While ARIMA predicts a linear trend, SI forecasts a nonlinear increase with higher prevalence. Table 5 Forecasted Prevalence (%) Controlling HIV with PrEP In our analysis, we explored the effects of PrEP (Pre-Exposure Prophylaxis) on controlling HIV transmission. Our approach is supported by previous research conducted by Zhao et al, who investigated the impact of population recruitment on HIV epidemics and the effectiveness of HIV prevention interventions using the SI model. 11 Specifically, we examined the impact of varying the parameter k, which represents the proportion of PrEP users becoming sexually active. It is important to note that we did not determine specific values for the parameters used in this study. Instead, we conducted a sensitivity analysis by varying the value of k from 0 to 1 to assess its influence on the predicted prevalence of HIV. Figure 8 presents the results of the SI model, showcasing the predicted effects of PrEP. In the absence of PrEP, the model yielded a basic reproduction number of 9.06, indicating the potential for rapid transmission. However, by incorporating PrEP and adjusting the value of k, we observed a reduction in the basic reproduction number to as low as 4.53. This finding suggests that PrEP usage can significantly decrease the transmission rate of HIV. 15 , 19 Furthermore, our analysis indicated a notable reduction in the predicted prevalence of HIV, further highlighting the potential effectiveness of PrEP in controlling the spread of the disease. Figure 8 HIV prevalence forecasting with PrEP for various scenarios. Discussion In this paper, we study the dynamics of HIV in Kazakhstan using statistical and mathematical models. In particular, we use the so-called Autoregressive Integrated Moving Average (ARIMA) statistical tool and a Susceptible-Infected (SI) compartmental model of nonlinear differential equations to forecast the HIV prevalence rate in Kazakhstan. 8 , 11 , 12 We estimated the parameters of the models using open data on the prevalence of HIV infection among women and men (aged 15–49 years) in Kazakhstan provided by the Kazakhstan Bureau of National Statistics. Based on this data, the ARIMA model suggests that the prevalence of HIV infection in Kazakhstan will increase from 0.29 in 2021 to 0.47 by 2030. On the other hand, the SI model suggests that this parameter will increase to 0.60 by 2030 based on the same data. As a result, we have the Akaike Information Criterion corrected (AICc) score for the ARIMA(1,2,0) and SI models are equal to −65.94 and −118.26, respectively. Moreover, the mean square error (MSE) for the estimated parameters were 0.0016 and 0.000009 for the ARIMA(1,2,0) and the SI models, respectively. Finally, to see the goodness of fit of the models we calculate the R-squared which turned out to be 32.54% and 99.60% for the ARIMA(1,2,0) and SI models, respectively. From these analyses, we conclude that the nonlinear mathematical model fits the given data better than the ARIMA model. Furthermore, with the SI model it is easy to compute the basic reproduction number R, which plays a vital role in taking public health decisions. In particular, the basic reproduction number of HIV in Kazakhstan is estimated to be 9.06. We further studied the SI model for HIV prevention under the pre-exposure prophylaxis (PrEP) strategy. Depending on the coverage of PrEP among susceptible individuals, we provided different scenarios for HIV prevalence forecasting. For instance, if k=0.1, which means the proportion of PrEP users is 10%, then R is equal to 8.15, whereas the prevalence of HIV infection in Kazakhstan will be around 0.25. If k =0.6, then R and the prevalence drop to 6.34 and 0.21, respectively. Therefore, we conclude that prevention strategies such as PrEP can significantly help to mitigate HIV prevalence. From the perspective of healthcare providers, since HIV prevalence is expected to increase, it will require more healthcare needs, which, consequently, leads to more costs for the country. Therefore, it is recommended for healthcare providers and policymakers to use this model to calculate the cost required for the regional allocation of healthcare resources. Moreover, this model can be used for planning effective healthcare treatments. Further, this model can be used for the study of designing disease-modifying vaccines and also for the prevention of HIV infection in the future. While ARIMA is a statistical model that can be used in most of the time series analysis, it does not necessarily take into account various compartments involved in the population. In that sense, the SI model is more likely to provide better insight into the epidemic dynamics of HIV. However, with only two compartments, the SI model can be considered a simple dynamical system that does not necessarily reflect other aspects of the disease. To this end, it is possible to design a more complex system that includes age compartments, gender compartments, ethnicity compartments, regional compartments, and so on. However, having a more complex model does not necessarily lead to better forecasting. Another limitation of the SI model considered is being an autonomous system that assumes all epidemiological parameters are time-independent. Since our data has only 11 points, this assumption does not affect the predictions significantly. Conclusion This study revealed that ARIMA (1,2,0) predicts a linear increasing trend, while SI forecasts a nonlinear increase with a higher prevalence of HIV. It is predicted that the prevalence rate will rise almost two times from the results of both models (from 0.29 to 0.47 in ARIMA (1,2,0) and from 0.29 to 0.6 in the SI model). Moreover, HIV control strategies such as PrEP can significantly reduce the predicted prevalence. Since these models can provide us with different scenarios for HIV prevalence forecasting, the results of this study will be beneficial for the prevention and control of HIV infection spread. Data Sharing Statement The data that support the findings of this study are available from Republican Center for Electronic Health of the Ministry of Health of the Republic of Kazakhstan, but restrictions apply to the availability of these data, which were used under the contract-agreement for the current study, and so are not publicly available. Data are however available from the A.G., is a PI of the project, upon reasonable request and with permission of Ministry of Health of the Republic of Kazakhstan. Informed Consent Statement Patient consent was waived due to the retrospective nature of the study in which only anonymous data were analyzed. Institutional Review Board Statement The study was approved by the institutional review and ethic committee of the Nazarbayev University (NU-IREC 315/21092020 on 23/09/2020) with exemption from informed consent. Acknowledgments We thank all staff from Republican Center of Electronic Healthcare for provided data and consultancy. We would like to express our appreciation for all healthcare professionals for their continuous effort to fight against HIV infection. Author Contributions All authors made a significant contribution to the work reported, whether that is in the conception, study design, execution, acquisition of data, analysis and interpretation, or in all these areas; took part in drafting, revising or critically reviewing the article; gave final approval of the version to be published; have agreed on the journal to which the article has been submitted; and agree to be accountable for all aspects of the work. Funding This study was supported by grant from Ministry of Education and Science of the Republic of Kazakhstan 2021–2023 (Funder Project Reference: AP09259016). The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. A.G. is a PI of the project. Disclosure References
Arima has made 2 investments. Their latest investment was in Avalent Technologies as part of their Series B on March 20, 2002.
Arima Investments Activity
1 Team Member
Arima has 1 team member, including current Chief Financial Officer, Fangcheng Lu.