When Shape Matters: Using a Simple Mathematical Model to Estimate Critical Area Sizes in Conservation

In the analysis of anthropogenic impact on the environment arises the question of whether the shapes of preserved habitat fragments play an important role in the conservation of wild species. In this work we use a very simple mathematical model based on a reaction-diffusion equation to analyze the effects of geometric shape and area on the permanence of populations in habitat fragments. Our results indicate that a dimensionless quantity calculated from a combination of biological variables is the main component that determines if the species survives in the preserved fragment and whether its geometric shape is important. We provide a methodology to calculate critical area sizes for which population size is most affected by fragment shape. The calculation is based on four quantitative variables: maximum per capita reproduction rate, per capita mortality rate outside the fragment, carrying capacity in the conserved environment and mobility in the disturbed environment. The methodology is illustrated by a preliminary study, in which the model is used to estimate threshold area sizes for habitat fragments for the threatened species Sapajus xanthosternos .

Stability analysis of a simple mathematical model for school bullying

<abstract><p>School bullying is a highly concerned problem due to its effect on students' academic achievement. The effect might go beyond that to develop health problems, school drop out and, in some extreme cases, commit suicide for victims. On the other hand, adolescents who continuously bully over time are at risk of becoming involved in gang membership and other types of crime. Therefore, we propose a simple mathematical model for school bullying by considering two variables: the number of victims students and the number of bullies students. The main assumption employed to develop the mathematical model is that school policy bans bullying and expels students who practice this behavior to maintain a constructive educational environment within the school. We show that the model has two equilibrium points, and that both equilibrium points are locally and globally asymptotically stable under certain conditions. Also, we define a threshold parameter with a new criterion called the bullying index. Furthermore, we show that the model exhibits the phenomena of transcritical bifurcation subject to the bullying index. All the findings are supported with numerical simulations.</p></abstract>

Mathematical model of the influence of the time of start reaction in 100 m sprint run

The purpose of this study is to explain the influence of the time of start reaction tR to the results of sprinters in a 100 m run, using a new simple mathematical model based on the measured values for distance s, corresponding time t, and tR. The research is based on IAAF data obtained by measuring the segment length, the time of start reaction, transient times in 100 m run, and final times for the top sprinters C. Lewis (1988); M. Green (2011), and U. Bolt (2009) (men) and F. Griffith-Joyner (1988); E. Ashford (1988), and H. Drechsler (1988) (women). The values of the start reaction tR for both male and female top sprinters indicate that there appear no substantial differences in the values of tR based on gender which would directly favor male or female sprinters in achieving the top results in the 100m run. The influence of the time of start reaction tR decreases exponentially with the time t during the run (t>tR) and ends up at about 30 m, influencing the initial velocity vR although it is not directly related to the result of the run. Due to its applicative simplicity, the presented mathematical model and related conclusions can represent a solid basis for future studies concerning sprint running.

How many relevant SARS-CoV-2 variants might we expect in the future?

Objectives: The emergence of new SARS-CoV-2 variants is a major challenge in the management of Covid-19 pandemic. A crucial issue is to quantify the number of variants which may represent a potential risk for public health in the future. Methods: We fitted the data on the most relevant SARS-CoV-2 variants recorded by the World Health Organization (WHO). The function exploited for the fit is related to the total number of infected subjects in the world since the start of the epidemic. Results: We found that the number of relevant SARS-CoV-2 variants up to November 2021 was about 44. Moreover, the number of new relevant variants per ten million cases turned out to be 1.64 in November 2021, slightly decreased in comparison to the value of 2.29 in March 2020. Conclusions: Our simple mathematical model can evaluate the number of relevant SARS-CoV-2 variants as the cumulative number of cases increase worldwide and may represent a useful tool in planning strategies to effectively contrast the pandemic.

MULTI-WAVE COVID-19 PANDEMIC DYNAMICS IN ICELAND IN TERMS OF DOUBLE SIGMOIDAL BOLTZMANN EQUATION (DSBE)

The world is facing multi-wave transmission of COVID-19 pandemics, and investigations are rigorously carried out on modeling the dynamics of the pandemic. Multi-wave transmission during infectious disease epidemics is a big challenge to public health. Here we introduce a simple mathematical model, the double sigmoidal-Boltzmann equation (DSBE), for analyzing the multi-wave Covid-19 spread in Iceland in terms of the number of cumulative cases. Simulation results and the main parameters that characterize multi waves are derived, yielding important information about the behavior of the multi-wave pandemics over time. The result of the current examination reveals the effectiveness and efficacy of DSBE for exploring the Covid 19 dynamics in Iceland and can be employed to examine the pandemic situation in different countries undergoing multi-waves.

The quest for a unified theory on biomechanical palm risk assessment through theoretical analysis and observation

Several methodologies related to the biomechanical risk assessment and the uprooting and breaking potential of palms are reviewed and evaluated in this study. Also a simple mathematical model was designed, to simulate the results of critical wind speed predictions for a tall coconut palm by using classic beam theory and Brazier buckling. First, the review presents arguments that assess the applicability of some influential claims and tree and palm risk assessment methods that have been amply marketed in the last 20 years. Then, the analysis goes beyond the classical procedures and theories that have influenced the arboricultural industry and related press so far. And afterwards, rationale behind several postulated ideas are presented, that are hoped to be fruitful in the path towards a new biomechanical theory for the biomechanical risk assessment of palms. The postulated model envisages the palm stem as a viscoelastic and hollow cylinder that is not only prone to buckling, ovalization and kinking, but also fatigue, shear, splitting and crack propagation. This envisaging was also the main reason why simple Brazier buckling formulation was experimentally applied to simulate the breaking risk of a cocostem. This study also enables a better understanding of the wide range of factors that may influence the mechanical behaviour of trees and palms under (wind) loading.

A simple mathematical model for the evaluation of the long first wave of the COVID-19 pandemic in Brazil

We propose herein a mathematical model to predict the COVID-19 evolution and evaluate the impact of governmental decisions on this evolution, attempting to explain the long duration of the pandemic in the 26 Brazilian states and their capitals well as in the Federative Unit. The prediction was performed based on the growth rate of new cases in a stable period, and the graphics plotted with the significant governmental decisions to evaluate the impact on the epidemic curve in each Brazilian state and city. Analysis of the predicted new cases was correlated with the total number of hospitalizations and deaths related to COVID-19. Because Brazil is a vast country, with high heterogeneity and complexity of the regional/local characteristics and governmental authorities among Brazilian states and cities, we individually predicted the epidemic curve based on a specific stable period with reduced or minimal interference on the growth rate of new cases. We found good accuracy, mainly in a short period (weeks). The most critical governmental decisions had a significant temporal impact on pandemic curve growth. A good relationship was found between the predicted number of new cases and the total number of inpatients and deaths related to COVID-19. In summary, we demonstrated that interventional and preventive measures directly and significantly impact the COVID-19 pandemic using a simple mathematical model. This model can easily be applied, helping, and directing health and governmental authorities to make further decisions to combat the pandemic.

Two Mathematical Approaches to Inferring the Internal Structure of Proteins from their Shape

sing a simple mathematical model, we propose two approaches to externally infer how the amino-acid sequence is folded in a protein. One is the previously proposed differential geometric approach. The other is a new category theoretical approach proposed in this paper. As an example, we consider detecting the presence of internal singularities from the outside. Knowledge of Category theory is not required. Proteins are represented as a loop of triangles. In both approaches, the outer contour of the loop is examined to detect the presence of singular triangles (such as isolated triangles) inside. By considering the interaction between loops, the new approach allows us to detect more singular triangles than the previous approach. We hope that this research will provide a new perspective on protein structure analysis and promote further collaboration between mathematics and biology.