Predictive Mathematical Models of the Short-Term and Long-Term Growth of the COVID-19 Pandemic

The prediction of the dynamics of the COVID-19 outbreak and the corresponding needs of the health care system (COVID-19 patients’ admissions, the number of critically ill patients, need for intensive care units, etc.) is based on the combination of a limited growth model (Verhulst model) and a short-term predictive model that allows predictions to be made for the following day. In both cases, the uncertainty analysis of the prediction is performed, i.e., the set of equivalent models that adjust the historical data with the same accuracy. This set of models provides the posterior distribution of the parameters of the predictive model that adjusts the historical series. It can be extrapolated to the same analyzed time series (e.g., the number of infected individuals per day) or to another time series of interest to which it is correlated and used, e.g., to predict the number of patients admitted to urgent care units, the number of critically ill patients, or the total number of admissions, which are directly related to health needs. These models can be regionalized, that is, the predictions can be made at the local level if data are disaggregated. We show that the Verhulst and the Gompertz models provide similar results and can be also used to monitor and predict new outbreaks. However, the Verhulst model seems to be easier to interpret and to use.

A Novel Multivariable MGM (1, m) Direct Prediction Model and Its Optimization

With regard to the traditional MGM (1, m) model having jumping error in solving process, an MGM (1, m) direct prediction model (denoted as DMGM (1, m) model) is proposed and its solution method is put forward at first. Second, considering the inherent time development trend of system behavior sequence is ignored in the DMGM (1, m) model, the DMGM (1, m) model is optimized by introducing a time polynomial term, and the optimized model can be abbreviated as TPDMGM (1, m, φ ) model. Subsequently, it is theoretically proved that the TPDMGM (1, m, φ ) model can achieve mutual transformation with the traditional MGM (1, m) model and the DMGM (1, m) model by adjusting the parameter values. Finally, two case studies about predicting the deformation of foundation pit and Henan’s vehicle ownership have been carried out to validate the effectiveness of proposed models. Meanwhile, the MGM (1, m) model and Verhulst model are established for comparison. Results show that the modeling performance of four models from superior to inferior is ranked as TPDMGM (1, m, φ ) model, DMGM (1, m) model, MGM (1, m) model, and Verhulst model, which on the one hand testifies the correctness of defect analysis of the MGM (1, m) model and on the other hand verifies that the TPDMGM (1, m, φ ) model has advantages in predicting the system variables with mutual relation, mutual restriction, and time development trend characteristic.

Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups

Pandemic curves, such as COVID-19, often show multiple and unpredictable contamination peaks, often called second, third and fourth waves, which are separated by wide plateaus. Here, by considering the statistical inhomogeneity of age groups, we show a quantitative understanding of the different behaviour rules to flatten a pandemic COVID-19 curve and concomitant multi-peak recurrence. The simulations are based on the Verhulst model with analytical generalized logistic equations for the limited growth. From the log–lin plot, we observe an early exponential growth proportional to . The first peak is often τgrow @ 5 d. The exponential growth is followed by a recovery phase with an exponential decay proportional to . For the characteristic time holds: . Even with isolation, outbreaks due to returning travellers can result in a recurrence of multi-peaks visible on log–lin scales. The exponential growth for the first wave is faster than for the succeeding waves, with characteristic times, τ of about 10 d. Our analysis ascertains that isolation is an efficient method in preventing contamination and enables an improved strategy for scientists, governments and the general public to timely balance between medical burdens, mental health, socio-economic and educational interests.

Forecasting the Number of the Wounded after an Earthquake Disaster Based on the Continuous Interval Grey Discrete Verhulst Model

Earthquake disaster causes serious casualties, so the prediction of casualties is conducive to the reasonable and efficient allocation of emergency relief materials, which plays a significant role in emergency rescue. In this paper, a continuous interval grey discrete Verhulst model based on kernels and measures (CGDVM-KM), different from the previous forecasting methods, can help us to efficiently predict the number of the wounded in a very short time, that is, an “S-shape” curve for the numbers of the sick and wounded. That is, the continuous interval sequence is converted into the kernel and measure sequences with equal information quantity by the interval whitening method, and it is combined with the classical grey discrete Verhulst model, and then the grey discrete Verhulst models of the kernel and measure sequences are presented, respectively. Finally, CGDVM-KM is developed. It can effectively overcome the systematic errors caused by the discrete form equation for parameter estimation and continuous form equation for simulation and prediction in classical grey Verhulst model, so as to improve the prediction accuracy. At the same time, the rationality and validity of the model are verified by examples. A comparison with other forecasting models shows that the model has higher prediction accuracy and better simulation effect in forecasting the wounded in massive earthquake disasters.

Application of GA-BP Neural Network Optimized by Grey Verhulst Model around Settlement Prediction of Foundation Pit

Due to the limitation in the prediction of the foundation pit settlement, this paper proposed a new methodology which takes advantage of the grey Verhulst model and a genetic algorithm. In the previous study, excavation times are often the only factor to predict the settlement, which is mainly because the correspondence between real-time excavation depth and the excavation time is hard to determine. To solve this issue, the supporting times are precisely recorded and the excavation depth rate can be obtained through the excavation time length and excavation depth between two adjacent supports. After the correspondence between real-time excavation depth and the excavation time is obtained, the internal friction angle, cohesion, bulk density, Poisson’s ratio, void ratio, water level changes, permeability coefficient, number of supports, and excavation depth, which can influence the settlement, are taken to be considered in this study. For the application of the methodology, the settlement monitoring point of D4, which is near the bridge pier of the highway, is studied in this paper. The predicted values of the BP neural network, GA-BP neural network, BP neural network optimized by the grey Verhulst model, and GA-BP neural network optimized by the grey Verhulst model are detailed compared with the measured values. And the evaluation indexes of RMSE, MAE, MSE, MAPE, and R 2 are calculated for these models. The results show that the grey Verhulst model can greatly improve the consistency between predicted values and measured values, while the accuracy and resolution is still low. The genetic algorithm (GA) can greatly improve the accuracy of the predicted values, while the GA-BP neural network shows low reflection to the fluctuation of measured values. The GA-BP neural network optimized by the grey Verhulst model, which has taken the advantages of GA and the grey Verhulst model, has extremely high accuracy and well consistency with the measured values.

THE DIFFERENTIATED ECONOMETRIC APPROACH FOR THE IDENTIFYING WAYS TO OPTIMIZE THE TRANSPORT LOGISTICS

The paper is dedicated to the scientific approaches to calculating the cost trends of transport companies as well as the definition of the essence of optimization of transport logistics and an assessment of possible options for the possible improving the existing process of logistics support (due to the fulfillment in particular). The authors investigated the economic essence of transport logistics as a type of activity. The research was based on the popular logistic model for calculating volume changes simultaneously with limited resources commonly known as the Verhulst model. Initially, the Verhulst model was valid only for a short time interval taking into account the fact that the natural environment growth is necessarily limited by a number of factors: depletion of resources, natural conditions etc. The proposed model describes growth passing into the stabilization phase which is much better suited to the real conditions. Within the framework of using the model to solve the problem of logistics optimization it was proposed to find the unknown coefficients of the model. It was noted that by providing fulfillment services that a logistics company receives a number of advantages: optimization of processes, reduction of storage costs, additional benefits. Reducing logistics supply chains and reducing logistics costs today are strategic tasks for optimizing logistics systems as well as for the ensuring just-in-time deliveries and consolidating complex supplies. Using a differentiated econometric approach to determining ways to optimize transport logistics, a procedure for calculating the feasibility of providing additional services using the Verhulst logistic equation was proposed. Logistics operations should be carried out with minimal costs and minimum time for the customer. As it was shown by the applied differentiated econometric approach to determining ways to optimize transport logistics: the higher the costs of designing logistics activities of the transport company are – the lower the logistics costs will result from additional services. It was also revealed that the provision of fulfillment services is the main direction of optimization of the logistics infrastructure.

Research on short-term traffic flow prediction based on the tensor decomposition algorithm

In view of the uncertainties in short-time traffic flows and the multimode correlation of traffic flow data, a grey prediction model for short-time traffic flows based on tensor decomposition is proposed. First, traffic flow data are expressed as tensors based on the multimode characteristics of traffic flow data, and the principle of the tensor decomposition algorithm is introduced. Second, the Verhulst model is a classic grey prediction model that can effectively predict saturated S-type data, but traffic flow data do not have saturated S-type data. Therefore, the tensor decomposition algorithm is applied to the Verhulst model, and then, the Verhulst model of the tensor decomposition algorithm is established. Finally, the new model is applied to short-term traffic flow prediction, and an instance analysis shows that the model can deeply excavate the multimode correlation of traffic flow data. At the same time, the effect of the new model is superior to five other grey prediction models. The predicted results can provide intelligent transportation system planning, control and optimization with reliable real-time dynamic information in a timely manner.