Splines and Compartment Models

Abstract

Printed in Singapore PrefaceThe application of mathematics in life sciences first requires the formulation of adequate models of biological processes that allow the quantitative evaluation of life processes by means of observations and experiments. With regard to this, the knowledge with reference to the observed biological processes, the preconditions and characteristics of the applied mathematical models as well as the conditions surrounding data collection, need to be taken into account. In this entire context it is effective to develop specific quantitative methods for the evaluation of data and to characterize attributes mathematically, thereby justifying conclusions and interpreting results comprehensively. This synopsis of problems is a characteristic of biometric work, which due to its formulation is interdisciplinary. Typical questions continue to be brought to light, for example:• Can the conditions required by the mathematical method be seen as fulfilled in the observed examples? • Does a solution exist for the mathematical problem associated with the general problem definition? • Is there exactly one such solution?• Can it be proven that this solution possess desirable characteristics? • Are the evaluation model and possibilities for data acquisition consistent? • Do numerical problems arise when applying processes?Under such general points of views, two themes are discussed in this book: splines and compartment models. Their application can be seen in different areas of the sciences and technology. Why does one deal with such different mathematical concepts in this book? Examples are given in the context of life sciences with the appropriate typical terminology. At the same time, mathematical terms are needed that cannot be explained in detail here. The reader would find it useful to be familiar with basic knowledge of analysis, algebra, statistics and probability calculus as well as the theory of ordinary differential equations. Detailed knowledge of these areas however is not assumed. In this book we took care to include the history of the presented ideas and include references with regard to this. The historical comparison is not to be seen as just a reference to the scientists of the past. It helps to enforce the relativization of the own work, motivates students and further allow the reader to research the sources themselves. To begin, narrowing in on simple models seems to be advisable for the solid application of mathematical models in the life sciences. This simplifies the detailed clarification of their conditions of application, reduces the demand on the extent of observation and in many cases satisfies the purpose. The literature listed in the references confines itself to the titles quoted in the text. Numerous additional publications which were evaluated but did not explicitly contribute to results are not listed. This concerns standard pharmacology or pharmacokinetics textbooks, publications concerning computer programs, applications of pharmacokinetical methods (inclusively about meth...

Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines

“…These improvements are considerable even in the cases of R and R 2 , given the already good fitting performance of the cubic and randomly calibrated spline models. Additionally, it provides the area under the ROC curve metric (AUC) in order to assess the model performance [43]. A ROC curve (receiver operating characteristic curve) is a graph showing the performance of a model at all thresholds.…”

“…This complex-network-based definition (i.e., community detection based on modularity optimization) of the knot vector offers the missing conceptualization to the splines knots, defining them as borderline points of connectivity of the modularity-based communities. According to this approach, the visibility graph of the COVID-19 infection curve is divided into five modularity-based communities, which correspond to the periods Q1 = [1,4] ∪ [9,19], Q2 = [5,8], Q3 = [20,26], Q4 = [27,32], and Q5 = [33,43], as it is shown in Figure 9, where positive integers in these intervals are elements of variable X 1 .…”

“…Consequently, the spline knot vector can be defined by the knots t = (1,4,8,19,26,32,43) in the body of the time-series COVID-19 infection curve. This partition facilitates the application of the spline regression algorithm and comparison of the determination ability of the spline model with the cubic regression models previously shown.…”

“…According to this approach, the visibility graph of the COVID-19 infection curve is divided into five modularity-based communities, which correspond to the periods Q1 = 1,4 ⋃ 9,19 , Q2 = [5,8], Q3 = [20,26], Q4 = [27,32], and Q5 = [33,43], as it is shown in Figure 9, where positive integers in these intervals are elements of variable X1. Consequently, the spline knot vector can be defined by the knots t = (1,4,8,19,26,32,43) in the body of the time-series COVID-19 infection curve. This partition facilitates the application of the spline regression algorithm and comparison of the determination ability of the spline model with the cubic regression models previously shown.…”