Frontiers in Applied Mathematics and Statistics

Hepatitis B is the main public health problem of the whole world. In epidemiology, mathematical models perform a key role in understanding the dynamics of infectious diseases. This paper proposes Pade approximation (Pa) with Differential Evolution (DE) for obtaining solution of Hepatitis-B model which is nonlinear numerically. The proposed strategy transforms the nonlinear model into optimization problem by using Pade approximation. Initial conditions are converted into problem constraints and constraint problem become unconstraint by using penalty function. DE is obtained numerical solution of Hepatitis-B model by solving the established problem of optimization. There is no need to choose step lengths in proposed Pade-approximation based Differential Evolution (PaDE) technique and also PaDE converges to true steady state points. Finally, a convergence and error analysis evidence that the convergence speed of PaDE is greater than Non-Standard Finite Difference (NSFD) method for different time steps.

Pneumonia has been a major airborne transmitted disease and continues to pose a major public health burden in both developed and developing countries of the world. In this study, we constructed and analyzed a nonlinear deterministic compartmental mathematical model for assessing the community-level impacts of vaccination, other protection measures like practicing good hygiene, avoiding close contacts with sick people and limiting exposure to cigarette smoke, etc. and treatment on the transmission dynamics of pneumonia disease in a population of varying size. Our model exhibits two kinds of equilibrium points: pneumonia disease-free equilibrium point, and pneumonia endemic equilibrium point(s). Using center manifold criteria, we have verified that the pneumonia model exhibits backward bifurcations whenever its effective reproduction number [R]P < 1 and in the same region, the model shows the existence of more than one endemic equilibrium point where some of which are stable and others are unstable. Thus, for pneumonia infection, the necessity of the pneumonia effective reproduction number [R]P < 1, although essential, it might not be enough to completely eradicate the pneumonia infection from the considered community. Our examination of sensitivity analysis shows that the pneumonia infection transmission rate denoted by {beta} plays a crucial role to change the qualitative dynamics of pneumonia infection. By taking standard data from published literature, our numerical computations show that the numerical value of pneumonia infection model effective reproduction number is [R]P = 8.31 at {beta} = 4.21 it implies that the disease spreads throughout the community. Finally, our numerical simulations show that protection, vaccination, and treatment against pneumonia disease have the effect of decreasing pneumonia expansion.

The paper assumed that Nf2-Amot complex regulates the phosphorylation cascade so that each cell in the early mammalian embryo differentiates properly in silico. To confirm the validity of the assumption, it was necessary to verify whether Nf2-Amot complex has an impact on the resulting differentiation. The living embryo is unsuitable for the confirmation since the early mammalian embryo is too small to observe and too ethically sensitive to invade. In such cases, computational models can be used as experimental subjects for operations that cannot be applied to the living embryo. Previous models on the embryo, however, could not verify the assumption because they had not modeled Nf2-Amot complex, and they seldom modeled the Hippo signaling pathway. Therefore, the paper introduced a model of Nf2-Amot complex to the previous study that had modeled the Hippo signaling pathway. Testing the model under diverse conditions revealed that the existence of Nf2-Amot complex reproduces the ideal cell differentiation observed in the living embryo. In this sense, the validity of the model was confirmed. Furthermore, diverse cell-cell contacts that induce various concentrations of Nf2-Amot complex also resulted in ideal cell differentiation. These results suggested the correctness of the assumption in silico.

Capacitation is a complex maturation process mammalian sperm must undergo in the female genital tract to be able to fertilize an egg. This process involves, amongst others, physiological changes in flagellar beating pattern, membrane potential, intracellular ion concentrations and protein phosphorylation. Typically, in a capacitation medium, only a fraction of sperm achieve this state. The cause for this heterogeneous response is still not well understood and remains an open question. Here, one of our principal results is to develop a discrete regulatory network, with mostly deterministic dynamics in conjunction with some stochastic elements, for the main biochemical and biophysical processes involved in the early events of capacitation. The model criterion for capacitation requires the convergence of specific levels of a select set of nodes. Besides reproducing several experimental results and providing some insight on the network interrelations, the main contribution of the model is the suggestion that the degree of variability in the total amount and individual number of ion transporters among spermatozoa regulates the fraction of capacitated spermatozoa. This conclusion is consistent with recently reported experimental results. Based on this mathematical analysis, experimental clues are proposed for the control of capacitation levels. Furthermore, cooperative and interference traits that become apparent in the modelling among some components also call for future theoretical and experimental studies. Author summaryFertilization is one of the fundamental processes for the preservation of life. In mammals, sperm undergo a complex process during their passage through the female tract known as capacitation, which enables them for fertilization. At the present time, it is accepted from experimental observation, though not understood, that only a fraction of the sperm is capacitated. In this work, by means of a network mathematical model for regulatory sperm intracellular signaling processes involved in mice capacitation, we find that the variability in the distribution of the number of ion transporters intervenes in the regulation of the capacitation fraction. Experimental verification of this suggestion could open a line of research geared to the regulation of the degree of heterogeneity in the number of ion transporters as a fertility control. The model also uncovers, through in silico overactivation and loss of function of network nodes, synergetic traits which again call for experimental verification.

Assessing and understanding the Size of the human fetus provides essential information for the management of the health of the newborn and its mother, a quality that is commonly measured by Fetal Ultrasound. Many equations for making such calculations have been considered, none of which have been entirely satisfactory. As we have shown in the previous paper in this series, a consideration of the formation of the body, in units of numbers of cells, N, an approach we call Binary Cellular Analysis, provides new quantitative tools for estimating the Size and Age of the fetus. These tools include a new mathematical basis for creating useful expressions, Binary Cellular Estimated Fetal Weight Equations, for calculating human Fetal Weight from Ultrasound Measurements. As we show here, the most promising of these expressions, the Abdominal Circumference Binary Cellular Estimated Fetal Weight Equation, performs better than other currently used Fetal Weight Equations, while also yielding up new methods for improving ultrasound size assessment. Web-based calculators (https://kidzgrowth.com) make these mathematical manipulations available for obstetric care.

The cumulative records of COVID-19 are rapidly increasing day by day in India. The key question prevailing in minds of all is when will it get over? There have been several attempts in literature to address this question using time series, Machine learning, epidemiological and statistical models. However due to high level of uncertainty in the domain and lack of big historical data, the performance of these models suffer. In this work, we present an intuitive model that uses a combination of epidemiological model (SEIR) and mathematical curve fitting method to forecast spread of COVID-19 in India in future. By using the combination model, we get characteristics benefits of these models under limited knowledge and historical data about the novel Coronavirus. Instead of fixing parameters of the standard SEIR model before simulation, we propose to learn them from the real data set consisting of progression of Corona spread in India. The learning of model is carefully designed by understanding that available data set consist of records of cases under full, partial to zero lockdown phases in India. Hence, we make two separate predictions by our propose model. One under the situation of full lockdown in India and, other with partial to zero restrictions in India. With continued strict lockdown after May 03, 2020, our model predicted May 14, 2020 as the date of peak of Coronavirus in India. However, in current scenario of partial to zero lockdown phase in India, the peak of Coronavirus cases is predicted to be July 31, 2020. These two predictions presented in this work provide awareness among citizens of India on importance of control measures such as full, partial and zero lockdown and the spread of Corona disease infection rate. In addition to this, it is a beneficial study for the government of India to plan the things ahead.

In developing tissues, signal transduction from morphogen gradients conveys positional information to cells, resulting in cell specification and differentiation. One such morphogen is bone morphogenetic protein (BMP), of the TGF-{beta} superfamily, whose signaling network is highly conserved across many species. In Danio rerio (zebrafish), this signaling pathway directs dorsoventral axis formation during early embryogenesis. Many of the molecules that play a role in this network are well-understood; however, the mechanisms through which they achieve noise attenuation and gradient robustness have not been fully defined. Specifically, the heterodimer-heterotetramer complex has been shown to be required for signal transduction[1], but current understanding and modeling of the BMP membrane receptors at this stage has not given any insight into evolutionary drivers of the requirement. In this study, we develop a stochastic model of receptor oligomerization with the published reports of binding kinetics of BMP ligand-receptor interactions to mechanistically assess zebrafish phenotype variability related to the distributions of noise and stochasticity. We can also analyze time-dependent signaling and frequency metrics that are not available in traditional, deterministic modeling. Fast Fourier Transform and cumulative energy spectral density visualization show that the heterodimer-heterotetramer complex may function as part of a low-pass filter mechanism in the dorsal-ventral axis formation process, specifically tuned to the noise of the system. Under dynamic conditions such as the mid-blastula transition (MBT), wherein the morphogen gradient rapidly changes shape, established metrics of noise and information transduction, such as coefficient of variation and mutual information, overlook important temporal effects that may be particularly relevant during development. As the BMP signaling pathway is highly conserved and has been implicated in human bone growth and wound healing, its study in simpler systems stands to accelerate our comprehension of BMP network structure and molecular mechanisms with potential application in regenerative medical studies.

Estimation of COVID-19 dynamics and its evolution is a multidisciplinary effort, which requires the unification of heterogeneous disciplines (scientific, mathematics, epidemiological, biological/bio-chemical, virologists and health disciplines to mention the most relevant) to work together in a better understanding of this pandemic. Time series analysis is of great importance to determine both the similarity in the behavior of COVID-19 in certain countries/states and the establishment of models that can analyze and predict the transmission process of this infectious disease. In this contribution, an analysis of the different states of the United States will be carried out to measure the similarity of COVID-19 time series, using dynamic time warping distance (DTW) as a distance metric. A parametric methodology is proposed to jointly analyze infected and deceased persons. This metric allows to compare time series that have a different time length, making it very appropriate for studying the United States, since the virus did not spread simultaneously in all the states/provinces. After a measure of the similarity between the time series of the states of United States was determined, a hierarchical cluster was created, which makes it possible to analyze the behavioral relationships of the pandemic between different states and to discover interesting patterns and correlations in the underlying data of COVID-19 in the United States. With the proposed methodology, nine different clusters were obtained, showing a different behavior in the eastern zone and western zone of the United States. Finally, to make a prediction of the evolution of COVID-19 in the states, Logistic, Gompertz and SIR model was computed. With these mathematical model it is possible to have a more precise knowledge of the evolution and forecast of the pandemic.

Several countries have witnessed multiple waves of the COVID-19 pandemic between 2020 and 21. The method in [8] is applied here to analyze the COVID-19 waves in India and the UK. For this, a birth-death model is fitted to the active and total cases data for 30 days periods called windows starting from 16th March 2020 up to 10th May 2021. Peculiarities of the parameters suggested a classification of the above windows into three categories: (i) whose fitted parameters predicted a rise in the number of active cases before a fall to zero, (ii) which predicted a decrease, without rising, in the active cases to zero and (iii) which predicted an increase in the active cases until the entire susceptible population gets infected. It follows that some of the type (iii) windows are of the same or lesser concern when compared to some type (i) windows. Further analysis of the type (iii) windows leads to the identification of those which could be indicators of the start of a new wave of the pandemic. The study thus proposes a method for using the present data for identifying pandemic waves in the near future.

Current research is an attempt to understand the CoVID-19 pandemic curve through statistical approach of probability density function with associated skewness and kurtosis measures, change point detection and polynomial fitting to estimate infected population along with 30 days projection. The pandemic curve has been explored for above average affected countries, six regions and global scale during 64 days of 22nd January to 24th March, 2020. The global cases infection as well as recovery rate curves remained in the ranged of 0 - 9.89 and 0 - 8.89%, respectively. The confirmed cases probability density curve is high positive skewed and leptokurtic with mean global infected daily population of 6620. The recovered cases showed bimodal positive skewed curve of leptokurtic type with daily recovery of 1708. The change point detection helped to understand the CoVID-19 curve in term of sudden change in term of mean or mean with variance. This pointed out disease curve is consist of three phases and last segment that varies in term of day lengths. The mean with variance based change detection is better in differentiating phases and associated segment length as compared to mean. Global infected population might rise in the range of 0.750 to 4.680 million by 24th April 2020, depending upon the pandemic curve progress beyond 24th March, 2020. Expected most affected countries will be USA, Italy, China, Spain, Germany, France, Switzerland, Iran and UK with at least infected population of over 0.100 million. Infected population polynomial projection errors remained in the range of -78.8 to 49.0%.

The outbreak of COVID-19 has an undeniable global impact, both socially and economically. March 11th, 2020, COVID-19 was declared as a pandemic worldwide. Many governments, worldwide, have imposed strict lockdown measures to minimize the spread of COVID-19. However, these measures cannot last forever; therefore, many countries are already considering relaxing the lockdown measures. This study, quantitatively, investigated the impact of this relaxation in the United States, Germany, the United Kingdom, Italy, Spain, and Canada. A modified version of the SIR model is used to model the reduction in lockdown based on the already available data. The results showed an inevitable second wave of COVID-19 infection following loosening the current measures. The study tries to reveal the predicted number of infected cases for different reopening dates. Additionally, the predicted number of infected cases for different reopening dates is reported.

India is witnessing the second wave of the COVID-19 disease from the first half of February 2021. The method in [5] is applied here to analyze the second wave in India. We start with fitting a birth-death model to the active and total cases data for the period from 13th to 28th February 2021. This initial dataset is expanded step by step by adding the two future weeks data to it until 14th May 2021. This resulted in six models in total. The efficacy of each model is tested in terms of predictions made for the next two weeks. The infectivity rates are found to be ever-increasing in the case of the five initial models. The infectivity rate for the sixth model, which is based on the data from 13th February to 14th May 2021, shows a decreasing nature with an increase in time. This indicates a decline in the second wave, which may start from 4th June 2021 according to the fitted parameters.

The main purpose of the study is to introduce the wavelength models developed to measure the size of outbreaks based on the COVID-19 example. In this way, the wavelengths of the outbreaks can be calculated, ensuring that the outbreaks are valid, reliable and easy to follow at the national and international level. Wavelength models consist of approved case, death, recovered case and net wavelength models. Thus, the size of the outbreak can be measured both individually and as a whole. COVID-19 cases of 181 countries were used to demonstrate the application of the models. The prominent findings in the applied wavelength models are as follows: the countries with the highest case wavelength are USA, Italy, Spain and Germany, respectively. However, Italy ranks first in the death wavelength, followed by Spain, the USA and France. On the other hand, China has taken the first place in the recovered case wavelength. This country was followed by Spain and Germany and Italy, respectively. Based on all these wavelength models mentioned, net wavelength lengths are calculated. According to the findings of net wavelengths obtained, Canada ranked first, followed by United Kingdom, USA and Italy, respectively.

Estimation of statistical quantities plays a cardinal role in handling of convoluted situations such as COVID-19 pandemic and forecasting the number of affected people and fatalities is a major component for such estimations. Past researches have shown that simplistic numerical models fare much better than the complex stochastic and regression-based models when predicting for countries such as India, United States and Brazil where there is no indication of a peak anytime soon. In this research work, we present two models which give most accurate results when compared with other forecasting techniques. We performed both short-term and long-term forecasting based on these models and present the results for two discrete durations.

In this manuscript, we model and visualize the region-wise trends of the evolution to COVID-19 infections employing a SIR epidemiological model. The SIR dynamics are expressed using stochastic differential equations. We first optimize the parameters of the model using RMSE as loss function on the available data using L-BFGS-B gradient descent optimisation to minimise this loss function. This helps to gain better approximation of the models parameter for specific country or region. The derived parameters are aggregated to project future trends for the Indian subcontinent for next 180 days, which is currently at an early stage within the infection cycle. The projections are meant to function a suggestion for strategies for the socio-political counter measures to mitigate COVID-19. This study considers the current data for India from various open sources. The SIR models prediction is found following the actual trends till date. The inflection point analysis is important to find out which countries have reached their inflection point of the number of infection. We found that if current restrictions like lockdown in India continues with same control, then India will observe it[s] peak in active patients count on 22 May 2020, it will take 28 August 2020 for 90% of the peak active infections to end. Inspired from the study of DDI Lab at Singapore university of technology and design (SUTD), this study additionally tries to model and quantify the variations in the count of active patients in the country which might occur due to effect of waiver in restrictions. It should be noted that these results were predicted using COVID-19 data of India till 03 May 2020.

The COVID-19 is declared as a public health emergency of global concern by World Health Organisation (WHO) affecting a total of 201 countries across the globe during the period December 2019 to January 2021. As of January 25, 2021, it has caused a pandemic outbreak with more than 99 million confirmed cases and more than 2 million deaths worldwide. The crisp of this paper is to estimate the global risk in terms of CFR of the COVID-19 pandemic for seventy deeply affected countries. An optimal regression tree algorithm under machine learning technique is applied which identified four significant features like diabetes prevalence, total number of deaths in thousands, total number of confirmed cases in thousands, and hospital beds per 1000 out of fifteen input features. This real-time estimation will provide deep insights into the early detection of CFR for the countries under study. CFR[Formula]as suggested by (Boldog et al., 2020, Chakraborty et al. 2019, Russell et al., 2020) Diabetes Prevalenceproportion of a population who have diabetes in a given period of time. Stringency Indexit provides a computable parameter to evaluate the effectiveness of the nationwide lock down in a particular country. GDP Per Capitait is a metric that breaks down a countrys economic output per person and is calculated by [Formula] Population Densityit is a measurement of population per unit area. It refers to the number of people living in an area per square kilometre. HDIit is a statistic composite index of life expectancy, education (literacy rate, gross enrolment ratio at different levels and net attendance ratio) and per capita income indicators which are used to rank countries into four tiers of human development.

Following power law, Farrs law and IDEA model, we analyze the data of COVID-19 pandemic for India up to 2 May, 2020 and for Germany, France, Italy, the USA, Singapore, China and Denmark up to 26 April, 2020. The cumulative total number of infected persons as a function of elapsed time has been fitted with power law to find the scaling exponent ({gamma}). The reduction in{gamma} in different countries signals the reduction in the growth of infection, possibly, due to long-term Government intervention. The extent of infection and reproduction rate R0 of the same are also examined using Farrs law and IDEA model. The new cases per day with time assume Gaussian bell shaped curve, obeying the rule that faster rise follows faster decay. In India and Singapore, the peak of the bell shaped curve is still elusive. It is found that, till date, countries such as Denmark and India implementing sooner lockdown have underwent lower number of new cases of infection. Daily variation shows, R0 of all the countries is reducing, ushering in fresh hopes to combat COVID-19. Finally, we try to make a prediction as to the date on which the different countries will come down to daily cases of infection as low as one hundred (100).

In this paper, we estimate the reproductive number R0 of COVID-19 based on Wallinga and Lipsitch framework [11] and a novel statistical time delay dynamic system. We use the observed data reported in CCDCs paper to estimate distribution of the generation interval of the infection and apply the simulation results from the time delay dynamic system as well as released data from CCDC to fit the growth rate. The conclusion is: Based our Fudan-CCDC model, the growth rate r of COVID-19 is almost in [0.30, 0.32] which is larger than the growth rate 0.1 estimated by CCDC [9], and the reproductive number R0 of COVID-19 is estimated by 3.25 [&le;] R0 [&le;] 3.4 if we simply use R = 1 + r * Tc with Tc = 7.5, which is bigger than that of SARS. Some evolutions and predictions are listed.

Modeling the evolution of Covid-19 incidence rate is critical to deciding and assessing non-medical intervention strategies that can lead to successful containment of the pandemic. This research presents a mathematical model to empirically assess measures related to various pandemic containment strategies, their similarities and a probabilistic estimate on the evolution of Covid-19 incidence rates. The model is built on the principle that, the exponential rise and decay of the number of confirmed Covid-19 infections can be construed as a set of concurrent non-linear waves. These waves can be characterized by a linear combination of Gaussian and Cauchy Lorentz functions collectively termed as Gaussian-Lorentzian Composite (GLC) function. The GLC function is used for non-linear approximation of officially confirmed Covid-19 incidence rates in each country. Results of fitting GLC based models to incidence rate trends of 20 different countries proves that the models can empirically explain the growth and decay trajectory Covid-19 infections in a given population.

A relevant problem in the study of the Covid-19 pandemic is the study of its temporal evolution. Such evolution depends on a number of factors, among which the average rate of contacts between susceptible and infected individuals, the duration of infectiousness and the transmissibility, that is the probability of infection after a contact between susceptible and infected individuals. In a previous study, we analyzed the potentiality of a number of distributions to describe the evolution of the pandemic and the potentiality of each distribution to mathematically predict the evolution of the pandemic in Italy. Since the number of daily tests was changing and increasing with time, we used the ratio of the new daily cases per swab. We considered distributions of the type of Gauss (normal), Gamma, Beta, Weibull, Lognormal and in addition of the type of the Planck blackbody radiation law. The Planck law, describing the amount of energy of the electromagnetic radiation emitted by a black body at each wavelength or at each frequency, marked in 1900 the beginning of Quantum Mechanics. The result of our analysis was that, among the considered distributions, the Planck law has the best potentiality to mathematically predict the evolution of the pandemic and the best fitting capability. In this paper, we analyze the time evolution of the second wave of the Covid-19 pandemic in Italy and in particular we predict the ratio of the new daily cases per swab at Christmas 2020 using the data in the interval from 17 Oct to 21 Nov. According to Figure 4 and Figure 8, the prediction for such a ratio around Christmas is approximately within 6% and 7%. In this study there is also an attempt to account for the effects of the governmental containment measures.