Search

In the present paper, we propose and study a fractional-order prey-predator type ecological model with the functional response of Holling type II and the effect of harvesting. In addition, we have also considered the influence of a super-predator on the conventional predator. The complex dynamical behaviour of the proposed model system including existence and uniqueness criteria, finiteness and nonnegativity of the solutions have studied rigorously. In addition, we have determined existence criteria of several equilibria and analyzed the asymptotic nature of those equilibria. We have enriched our analysis with the inclusion of fractional Hopf bifurcation and the criteria of global stability of the equilibrium points. Finally, some numerical simulation works have been incorporated to validate the analytical analysis.

COVID-19 has become one of the most severe health issues of the twenty-first century. More than 30 million people have been affected by this deadly disease and more than nine lakh people have been died due to this disease worldwide. Government authorities are forced to impose lockdown in their respective countries. In a word, COVID-19 has stopped the natural flow of life in worldwide. In this situation, scientists and researchers of different fields are trying to find out appropriate mitigating measures for COVID-19. However, there is no approved pharmaceutical measures like treatment or vaccination available to the health workers. So, the governments are strengthening on the non-pharmaceutical measures like maintaining social distance, use of face mask and sanitizer and on lockdown policy. In this scenario, determining the dynamics of the disease can help governments, health workers, scientists to decide future action to be taken in order to tackle the disease. In this chapter, we have proposed and analyzed an SEAIR type fractional order epidemic model in order to investigate the transmission mechanisms of the COVID-19. We have studied the non-negativity and boundedness of the solutions, and the existence and uniqueness of the solutions of the system. The explicit expression of the most important parameter, namely the basic reproduction number is derived and the further investigations of the system are performed in terms of the basic reproduction number. Apart from studying the local dynamics of the fractional order system, we have considered the environmental fluctuations in the deterministic system. We have studied the stochastic stability of the infected equilibrium points. Finally, important results are illustrated through numerical simulation works.

In the present study, we develop and analyze an SEIR epidemic model to assess the consequences of media awareness program and treatment control. The disease transmission rate as well as the treatment function are taken in the saturated form. Different equilibrium points and their stability are discussed. The threshold parameter, basic reproduction number is obtained and it is seen that the system may posses a backward bifurcation. An optimal control problem is formulated with treatment and media awareness parameters as control parameters and solved it analytically. The cost-effectiveness analysis is performed to find out the best strategy to be applied to control the transmission of diseases. In addition to our analytical results, several numerical simulations are also illustrated. Finally, a brief discussion is given regarding the role of treatment and media awareness program.

The coronavirus disease 2019 (COVID-19) outbreak, due to SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), originated in Wuhan, China and is now a global pandemic. The unavailability of vaccines, delays in diagnosis of the disease, and lack of proper treatment resources are the leading causes of the rapid spread of COVID-19. The world is now facing a rapid loss of human lives and socioeconomic status. As a mathematical model can provide some real pictures of the disease spread, enabling better prevention measures. In this study, we propose and analyze a mathematical model to describe the COVID-19 pandemic. We have derived the threshold parameter basic reproduction number, and a detailed sensitivity analysis of this most crucial threshold parameter has been performed to determine the most sensitive indices. Finally, the model is applied to describe COVID-19 scenarios in India, the second-largest populated country in the world, and some of its vulnerable states. We also have short-term forecasting of COVID-19, and we have observed that controlling only one model parameter can significantly reduce the disease's vulnerability.

In this paper, we proposed and analyzed a susceptible-infected-recovered (SIR) type epidemic model to investigate the effect of transport-related infectious diseases namely tuberculosis, measles, rubella, influenza, sexually transmitted diseases, etc. The existence and stability criteria of both the diseases include free equilibrium point and endemic equilibrium point which are established and the threshold parametric condition for which the system passes through a transcritical bifurcation is also obtained. Optimal control strategy for control parameters is formulated and solved both theoretically and numerically. Lastly, we not only illustrate our theoretical results through graphical illustrations but also computer simulation is used to show that our model would be a good model to study the SARS epidemic in 2003.

In this paper, we propose and analyze a fractional-order SIS epidemic model with the saturated treatment and disease transmission. The existence and uniqueness, nonnegativity and finiteness of solutions for our suggested model have been studied. Different equilibria of the model are found and their local and global stability analyses are also examined. Furthermore, the conditions for fractional backward and fractional Hopf bifurcation are also analyzed in both the commensurate and incommensurate fractional-order model. We study how the control parameter and the order of the fractional derivative play role in local as well as global stability of equilibrium points and Hopf bifurcation. We have demonstrated the analytical results of our proposed model system through several numerical simulations.

Highlights • A mathematical model has been proposed to analyse the pandemic COVID-19. • The model has been analysed both theoretically and numerically. • The procedure to control the basic reproduction number R 0 has been provided. • We have formed an optimal control problem where governmental policy is the control. • The model is used for short term prediction of COVID-19 in three states of India., As there is no vaccination and proper medicine for treatment, the recent pandemic caused by COVID-19 has drawn attention to the strategies of quarantine and other governmental measures, like lockdown, media coverage on social isolation, and improvement of public hygiene, etc to control the disease. The mathematical model can help when these intervention measures are the best strategies for disease control as well as how they might affect the disease dynamics. Motivated by this, in this article, we have formulated a mathematical model introducing a quarantine class and governmental intervention measures to mitigate disease transmission. We study a thorough dynamical behavior of the model in terms of the basic reproduction number. Further, we perform the sensitivity analysis of the essential reproduction number and found that reducing the contact of exposed and susceptible humans is the most critical factor in achieving disease control. To lessen the infected individuals as well as to minimize the cost of implementing government control measures, we formulate an optimal control problem, and optimal control is determined. Finally, we forecast a short-term trend of COVID-19 for the three highly affected states, Maharashtra, Delhi, and Tamil Nadu, in India, and it suggests that the first two states need further monitoring of control measures to reduce the contact of exposed and susceptible humans.

In this paper, we have proposed and formulated an epidemic model with two types of diseases-one is comparative weaker and the other is comparatively stronger with the assumption that both the diseases are active simultaneously in the system. The dynamical behavior of the model; equilibrium analyses with their existence criteria and local stability criteria have been discussed rigorously. With the help of second generation matrix method, we evaluate basic reproduction number of the proposed model. We propose an optimal control problem considering treatment as control parameter and solve it in order to minimize the compound loss due to the presence of infection. All the theoretical results are verified with some appropriate computer simulation works.

This paper formulates and analyses a susceptible-exposed-infected-recovered (SEIR) type epidemic model with the effect of transport-related infection between two cities in the presence of treatment control. The dispersal of populations from one city to another city has an important impact on the dynamics of disease evolution. The basic reproduction number is calculated for all the different cases of the proposed model system. It is found out that the disease-free equilibrium is disease-free if the basic reproduction is less than unity, otherwise the disease may remain in the system. In addition, the optimal control problem is constructed and solved analytically and numerically by considering the treatment control as a control variable. Further, we present a numerical simulation to confirm the analytical results. Finally, we show a comparison of the result of our predicted model with the real data of severe acute respiratory syndrome (SARS) outbreak in 2003 in Hong Kong.