Mathematical Analysis for Transmission of COVID-19

In this chapter, we discuss the basics of compartmental models in epidemiology and requisite analysis.

Coronavirus Disease 2019 (COVID-19) is declared as pandemic on 11 March 2020 by World Health Organization (WHO). There are apparent dissimilarities in incidence and mortality of COVID-19 cases in different parts of world. Developing countries in Asia and Africa with fragile health system are showing lower incidence and mortality compared to developed countries with superior health system in Europe and America. Most countries in Asia and Africa have national Bacillus Calmette-Guerin (BCG) vaccination programme, while Europe and America do not have such programme or have ceased it. At present, there is no known Food and Drug Administration (FDA)-approved treatment available for COVID-19 disease. There is no vaccine available currently to prevent COVID-19 disease. As mathematical modelling is ideal for predicting the rate of disease transmission as well as evaluating efficacy of possible public health prevention measures, we have created a mathematical model with seven compartments to understand nationwide BCG vaccine recommendation on COVID-19 transmission, severity and mortality. We have computed two basic reproduction numbers, one at vaccine-free equilibrium point and other at non-vaccine-free equilibrium point, and carried out local stability, sensitivity and numerical analysis. Our result showed that individuals with BCG vaccinations have lower risk of getting COVID-19 infection, shorter hospital stays and increased rate of recovery. Furthermore, countries with long-standing universal BCG vaccination policies have reduced incidence, mortality and severity of COVID-19. Further research will focus on exploring the immediate benefits of vaccination to healthcare workers and patients as well as benefits of BCG re-vaccination.

The present world is in dire straits due to the deadly SARS coronavirus-2 (CoV-2) outbreak, and the experts are trying heart and soul to discover any prevention and/or remedy. The people from all walks of life in the universe are fighting to defeat this novel coronavirus. In this case, doctors are in the front line fighters who have put themselves at a risk. In this paper, we have formulated a non-linear system of five differential equations of COVID-19 based on the tendency of doctors to be infected. The target of this study is to take a look at the transmission of COVID-19 from asymptomatic populations to the doctors. The model is analyzed with the determination of the basic reproduction number, equilibrium, and related stability analysis at both equilibrium points. The graph of the basic reproductive ratio for different parameters has been drawn to show the disease behavior. Finally, we have simulated our model numerically for visualizing the analytical findings. Our study shows that the asymptomatic population increases as the disease (COVID-19) transmission rate increases. The number of infected population increases with the infection rate. These increasing asymptomatic and infected populations lead the doctors to get infected by contacting with them. Thus, the whole medical service system is getting down over time.

The novel corona virus or Covid-19 spread had its inception in November of 2019, and in March 2020, it was declared as a pandemic. Since its initial stage, it has now already infected over 5 million people, leading to the lockdown of countries around the world, and a halt on global as well as national travel across the globe. Based on this, the research proposes a mathematical Covid-19 model to study the outcome of these classified zones under different control strategies. In the nonlinear mathematical model, the total population has been divided into seven compartments, namely Susceptible, Exposed, Red zone, Orange zone, Green zone, Hospitalized, and Recovered. The spectral radius is calculated to analyze dynamics of the Covid-19. To control the spread of the virus, the parameters of controls are Medical Intervention, Partial Lockdown, and Strict Lockdown. This model has been validated with numerical data. The conclusion validates the implementation of lockdown in curbing Covid-19 cases.

The treatment of corona virus disease is not possible without any vaccine. However, spreading of the deadly virus can be controlled by various measures being imposed by Government like lockdown, quarantine, isolation, contact tracing, social distancing and putting face mask on mandatory basis. As per information from the Department of Medical Health and Family Welfare of Rajasthan on 19 September 2020, corona virus COVID-19 severely affected the state of Rajasthan, resulting in cumulative positive cases 113,124, cumulative recovered 93,805 and cumulative deaths 1322. Without any appropriate treatment, it may further spread globally as it is highly communicable and because potentially affecting the human body respiratory system, which could be fatal to mankind. Therefore, to reduce the spread of infection, authors are motivated to construct a predictive mathematical model with sustainable conditions as per the ongoing scenario in the state of Rajasthan. Mathematica software has been used for numerical evaluation and graphical representation for variation of infection, recovery, exposed, susceptibles and mortality versus time. Moreover, comparative analysis of results obtained by predictive mathematical model has been done with the exact data plotting by curve fitting as obtained from Rajasthan government website. As a part of analysis and result, it is noted that due to the variation of transmission rate from person to person corresponding rate of infection goes on increasing monthly and mortality rate found high as shown and discussed numerically. Further, we can predict that the situation will become worse in the winter months especially in month of December due to unavailability of proper vaccine. This model may become more efficient when the researchers, experts from medical sciences and technologist work together.

Italy faced the COVID-19 crisis in the early stages of the pandemic. In the present study, a SEIR compartment mathematical model has been proposed. The model considers four stages of infection: susceptible(S), exposed (E), infected (I) and recovered (R). Basic reproduction number \(R_0\) which estimates the transmission potential of a disease has been calculated by the next-generation matrix technique. We have estimated the model parameters using real data for the Coronavirus transmission. To get a dipper insight into the transmission dynamics, we have also studied four of the most pandemic affected regions of Italy. Basic reproduction number stood differently for different regions of Italy i.e. Lombardia (2.1382), Veneto (1.7512), Emilia Romagna (1.6331), Piemonte (1.9099) and for Italy at 2.0683. The sensitivity of \(R_0\) corresponding to various disease transmission parameters has also been demonstrated via numerical simulations. Besides, it has been demonstrated with the help of simulations that earlier lockdown and rapid isolation of infective individuals would have been helpful in a dual way; by substantially reducing the number of susceptible people on one hand and preponing the end of the pandemic on the other. This paper also includes complete theoretical analysis of the proposed model including the epidemic feasibility of the model and existence of endemic equilibrium point. We have also derived the conditions under which the disease became endemic. Since the existence of an endemic equilibrium point refers to the possibility of backward bifurcation, we have given a detailed analysis regarding the same. All the theoretical analysis is supported by detailed numerical simulations to understand the transmission dynamics of COVID-19 While analyzing different regions of Italy it was found that Lombardia was the hardest hit and had the highest number of infectives. We have also forecasted the future scenario of the pandemic in Italy. The model predicts that the COVID-19 epidemic shall die out from the worst affected Lombardia region by approximately by November 2020.

COVID-19 is a major pandemic threat of 2019–2020 which originated in Wuhan. As of now, no specific anti-viral medication is available. Therefore, many countries in the world are fighting to control the spread by various means. In this chapter, we model COVID-19 scenario by considering compartmental model. The set of dynamical system of nonlinear differential equation is formulated. Basic reproduction number \(R_{0}\) is computed for this dynamical system. Endemic equilibrium point is calculated and local stability for this point is established using Routh-Hurwitz criterion. As COVID-19 has affected more than 180 countries in several ways like medically, economy, etc. It necessitates the effect of control strategies applied by various government worldwide to be analysed. For this, we introduce different types of time dependent controls (which are government rules or social, medical interventions) in-order to control the exposure of COVID-19 and to increase recovery rate of the disease. By using Pontryagins maximum principle, we derive necessary optimal conditions which depicts the importance of these controls applied by the government during this epidemic.

The widespread of the novel coronavirus (2019-nCoV) has adversely affected the world and is treated as a Public Health Emergency of International Concern by the World Health Organization. Assessment of the basic reproduction number with the help of mathematical modeling can evaluate the dynamics of virus spread and facilitate critical information for effective medical interventions. In India, the disease control strategies and interventions have been applied at the district level by categorizing the districts as per the infected cases. In this study, an attempt has been made to estimate the basic reproduction number R₀ based on publically available data at the district level in India. The susceptible-exposed-infected-critically infected-hospitalization-recovered (SEICHR) compartmental model is constructed to understand the COVID-19 transmission among different districts. The model relies on the twelve kinematic parameters fitted on the data for the outbreak in India up to May 15, 2020. The expression of basic reproduction number R₀ using the next-generating matrix is derived and estimated. The study also employs three time-dependent control strategies to control and minimize the infection transmission from one district to another. The results suggest an unstable situation of the pandemic that can be minimized with the suggested control strategies.

In this chapter, a fractional SEIR model and its robust first-order unconditionally convergent numerical method is proposed. Initial conditions based on Namibian data as of 4 July 2020 were used to simulate two scenarios. In the first scenario, it is assumed that the proper control mechanisms for kerbing the spread of COVID-19 are in place. In the second scenario, a worst-case scenario is presented. The worst case is characterised by ineffective COVID-19 control mechanisms. Results indicate that if proper control mechanisms are followed, Namibia can contain the spread of COVID-19 in the country to a lowest level of 1, 800 positive cases by October 2020. However, if no proper control mechanisms are followed, Namibia can hit a potentially unmanageable level of over 14, 000 positive cases by October 2020. From a mathematical point of view, results indicate that the fractional SEIR model and the proposed method are appropriate for modelling the chaotic nature observed in the spread of COVID-19. Results herein are fundamentally important to policy and decision-makers in devising appropriate control and management strategies for curbing further spread of COVID-19 in Namibia.

The infectious coronavirus disease is spreading at an alarming rate, not only in India but also globally too. The impact of coronavirus disease (COVID- 19) outbreak needs to be analyzed statistically and modelled to know its behaviour so as to predict the same for future. An exhaustive statistical analysis of the data available for the spread of this infection, specifically on the number of positive cases, active cases, death cases and recovered cases, and connection between them could probably suggest some key factors. This has been achieved in this paper by analyzing these four dominant cases. This helped to know the relationship between the current and the past cases. Hence, in this paper, an approach of statistical analysis of COVID-19 data specific to metropolitan cities of India is done. A regression model has been developed for prediction of active cases with 10 lag days in four metropolitan cities of India. The data used for developing the model is considered from 26th April to 31st July (97 days), tested for the month of August. Further, an Artificial Neural Network (ANN) model using back propagation algorithm for active cases for all India and Bangalore has been developed to see the comparison between the two models. This is different from the other existing ANN models as it uses the lag relationships to predict the future scenario. In this case, data is divided into training, validation and testing sets. Model is developed on the training sets and is checked on the validation set, tested on the remaining, and then, it is implemented for prediction.

The ensuing paper expounds a new mathematical model for a pandemic instigated by novel coronavirus (COVID-19) with influence of quarantine on transmission of COVID-19, using Caputo fractional-order derivative for various fractional order. Basic reproduction number for the SEQAIR model has been calculated in the study and additionally proving the existence and uniqueness of the solution using the fixed-point theorem. Furthermore, numerical solution is revealed using the Adams–Bashforth–Moulton method, and its application for real-world data is deliberated.

Within a very short period, the corona infection virus (COVID-19) has created a global emergency situation by spreading worldwide. This virus has dissimilar effects in different geographical regions. In the beginning of the spread, the number of new cases of active corona virus has shown exponential growth across the globe. At present, for such infection, there is no vaccination or anti-viral medicine specific to the recent corona virus infection. Mathematical formulation of infection models is exceptionally successful to comprehend epidemiological models of ailments, just as it causes us to take vital proportions of general wellbeing interruptions to control the disease transmission and the spread. This work based on a new mathematical model analyses the dynamic behaviour of novel corona virus (COVID-19) using Caputo–Fabrizio fractional derivative. A new modified SEIRQ compartment model is developed to discuss various dynamics. The COVID-19 transmission is studied by varying reproduction number. The basic number of reproduction \(R_{0}\) is determined by applying the next generation matrix. The equilibrium points for disease-free and endemic states are computed with the help of basic reproduction number \(R_{0}\) to check the stability property. The Picard approximation and Banach’s fixed point theorem based on iterative Laplace transform are useful in establishing the existence and stability behaviour of the fractional-order system. Finally, numerical computations of the COVID-19 fractional-order system are presented to analyse the dynamical behaviour of the solutions of the model. Also, a fractional-order SEIRQ COVID-19 model with vaccinated people has also been formulated and its dynamics with impact on the propagation behaviour is studied.

Epidemic diseases are well known to be fatal and cause great loss worldwide—economically, socially and mentally. Even after around nine months, since the Coronavirus Disease 2019 (COVID-19) began to spread, people are getting infected all over the world. This is one of the areas where human medical advancements fail because by the time the disease is identified and its treatment is figured out, most of the population is already exposed to it. In such cases, it becomes easier to take steps if the dynamics of the disease and its sensitivity to various factors is known. This chapter deals with developing a mathematical model for the spread of Coronavirus disease, by employing a number of parameters that affect its spread. A compartmental modelling approach using ordinary differential equation has been used to formulate the set of equations that describe the model. We have used the next generation matrix method to find the basic reproduction number of the system and proved that the system is locally asymptotically stable at the disease-free equilibrium for \(R_0<1\). Stability and existence of endemic equilibrium have been discussed, followed by sensitivity of infective classes to parameters like proportion of vaccinated individuals and precautionary measures like social distancing. It is expected that after the vaccine is developed and is available to use, as the proportion of vaccinated individuals will increase, the infection will decrease in the population which can gradually lead to herd immunity. Since, the vaccine is still under development, non-intervention measures play a major role in coping with the disease. The disease generally transmits when the water droplets from an infected individuals’ mouth or nose are inhaled by a healthy individual. The best measures that should be adopted are social distancing, washing one’s hands frequently, and covering one’s mouth with mask, quarantine and lockdowns. Thus, as more and more precautionary measures are taken, it would gradually reduce the infection which has also been proved numerically by the sensitivity analysis of ‘w’ in our dynamical analysis.

Coronavirus disease is an infectious disease which is caused by a virus called coronavirus. The people who are infected with this disease will experience respiratory illness. This disease has been declared pandemic by ‘World Health Organization (WHO)’. There are many sectors that have been affected due to the lockdown practiced in the entire country, among agricultural sector, manufacturing sector, service sector, education sector, business sector, etc. In our research, we examined the impact of COVID-19 in India vis-a-vis different sectors. For the purposes of this research, we shortlisted two particular sectors, i.e. education and service sectors. These two sectors form the backbone of our country. While the impact on education sector has led to many young minds and vulnerable school kids being affected in an adverse way and has left them to cope with new practices such as online classes during the lockdown period, and on the other hand, in the service sector, employees are working from home which in some case has had an impact on the effectiveness and efficiency of their work. In this paper, we assessed the impact on these two sectors on Indian economy by analyzing the responses given by our respondents through the questionnaires (Google form), and we then combined the data points to study how students and employees are being affected during the present lockdown period, imposed due to COVID-19. This chapter will help the readers to get to know more about the thinking of the students and employees in lockdown, and how much they are affected by this pandemic. Towards the trailing part of our research, we have discussed about the possible steps that can be adopted in future, by the employers and educational institutions, in order to limit the damage to the sector and to make recovery in the future.

During the pandemic due to coronavirus disease-19 (COVID-19), technology is regarded as a boon as well as a curse to human life which has a great impact on surroundings, people and the society. One of the innovative, however, perilous (if misused) inventions of humans is the smartphone which is becoming more and more alarmingly common yet an urgent question to be addressed. A wide application of smartphone technology is observed during this pandemic. It has both positive as well as negative impact on the prominent areas which include education, business, health, social life and furthermore. Moreover, the impact of such an addiction is observed not only among youngsters but has influenced all age groups. This scenario is modelled in this research through non-linear ordinary differential equations, where individuals susceptible to smartphone use will be either positively or negatively infected/addicted, may suffer from health issues procuring medication. Threshold is calculated using the next generation matrix method. Stability analysis is done using graph theory, and for the validation of data, numerical simulation is carried out. This study gives results explaining positive and negative issues on health due to excessive use of smartphone.

The ongoing COVID-19 pandemic emerged as one of the biggest challenges of recent times. Efforts have been made from different corners of the research community to understand different dimensions of the disease. Some theoretical works have reported that disease becomes severe in the presence of environmental pollution. In this work, we propose a nonlinear mathematical model to study the influence of air pollution on the dynamics of the disease. The basic reproduction number plays a vital role in predicting the future of an epidemic. Therefore, we obtain the expression of the basic reproduction number and performed a detailed sensitivity and uncertainty analysis. The values of partial rank correlation coefficients (PRCC) have been calculated corresponding to six critical parameters. The positive values of PRCC for pollution-related parameters depicts that pollution enhances the chances of a rapid spread of COVID-19.

Ever since the transmission of novel coronavirus through human-to-human hit the world. As this disease is spreading every day, hospitalisation of individuals increased. Consequence of this, there is a sudden surge of millions of gloves, masks, hand sanitizers and the other essential equipment in each month. Disposal of these commodities is a big challenge for hospitals and COVID-centre, as they may became the reason of creating pollution and infect the surroundings. Increasing hospitalisation cases of COVID-19 results in raising bio-waste which creates pollution. Observing the scenario, a mathematical model with four compartments is constructed in this article. The threshold value indicates the intensity of pollution that emerged from bio-waste. Stability of the equilibrium point gave the necessary condition. Optimal control theory is outlined to achieve the purpose of this chapter by reducing pollution. Outcomes are analytically proven and also numerically simulated.

Enlightened by the Coronavirus, the present paper deals with a mathematical model of COVID-19 to investigate the impact of S-I-R-M model on the pregnant women and the newly borns due to the influence of availability of suitable conditions. The rates of infection, rate of recovery, rate of mortality for pregnant women before and after delivery and for newly born babies due to the transmission rate have been discussed for the present observed data. The numerical illustrations have been carried out for the parameters, functions and represented graphically by MATHEMATICA Software. Moreover some comparisons have been shown in the figure to estimate the impact of susceptible conditions and represent the particular cases of S-I-R-M model.

Owing to the pandemic issue of the coronavirus disease 2019 (COVID-19), it is imperative to keep up more than 1-m of social distancing and 37.5 °C temperature to stop the transmission of COVID-19 from human to human. Therefore, it is utmost requirement to make the smart belt installed with ultrasonic and LM35 sensors for distance and temperature measurements to reduce the transmission of COVID-19, respectively. The embedded sensors with NodeMCU show that once anything come in the proximity of 1-m near to the smart belt or helmet fixed to human body, it automatically makes an alarm for distance contact as well as temperature of incoming/outgoing body and sends an email to the controller with the help of Blynk application through Internet of things (IoT). These data can be stored in the cloud for the future purpose. However, the distance sensor has detected the movement of a person from 3 cm up to around 240 cm. The LM35 temperature sensor measures the actual temperature of the host body, i.e., 35.4 °C with time. With the help of this research, it is possible to interface a camera module which can detect the suspects. It could be interfaced with global positioning system (GPS) which can give location-wise data and help us to obtain the probability of suspects at a particular region. It is cost effective, i.e., $14/belt which can help to control the transmission of coronavirus from human to human.