Impact of vaccination on the dynamics of COVID-19: A mathematical study using fractional derivatives

Arrival of a new disease marks a yearlong destruction of human lives and economy in general, and if the disease turns out to be a pandemic the loss is frightening. COVID-19 is one such pandemic that has claimed millions of lives till date. There is a suffering throughout the world due to various factors associated with the pandemic, be it loss of livelihoods because of sudden shutdown of companies and lockdown, or loss of lives due to lack of medical aid and inadequate vaccination supplies. In this study, we develop a six-compartmental epidemiological model incorporating vaccination. The motivation behind the study is to analyze the significance of higher vaccination efficacy and higher rate of population getting vaccinated in controlling the rise in infectives and thereby the untimely demise of various individuals. The work begins with an ordinary differential equation model followed by stability analysis of the same, after which a fractional-order derivative model of the same is formulated and the existence of uniformly stable solution for the system is proved. In addition to this, we present the stability of the equilibria in general for the fractional model framed. The sensitivity analysis of the basic reproduction number along with its correlation with various parameters is presented. In addition to this, sensitivity of certain state variables in the fractional model with respect to different fractional orders as well with respect to different infection rate is exhibited in this work. Factors related to lockdown and usage of face shields are incorporated in the entire study, and importance of these is highlighted in the study as well. The major takeaway from the study is that mere vaccination will not suffice in eradication of the virus. The vaccine efficacy plays a major role along with other intervention included in the model. The numerical simulations are carried out in MATLAB software using ode45 and fde12. [ FROM AUTHOR] Copyright of International Journal of Biomathematics is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

Comments on "Optimal control analysis of a COVID-19 and tuberculosis co-dynamics model".

Study of dynamics of COVID-19 and its co-infection with other diseases through mathematical models is the major focus of recent advancement in mathematical modeling of infectious diseases. There are numerous mathematical models on COVID-19 which describe its dynamics for different geographic regions. However, there are very few research papers dealing with co-infection of COVID-19 and TB. As both TB and COVID-19 are infectious diseases of same nature it becomes very difficult to predict the co-dynamics of these two diseases. The formulation of a correct mathematical model is very important in any kind of modeling and if the mathematical model is not proper then any prediction based on this may not be valid. This letter highlights the important limitations in the proposed mathematical model of co-dynamics of COVID-19 and TB by [1].

A co-infection model on TB - COVID-19 with optimal control and sensitivity analysis.

COVID-19 had been declared a public health emergency by the World Health Organization in the early 2020. Since then, this deadly virus has claimed millions of lives worldwide. Amidst its chaotic spread, several other diseases have faced negligence in terms of treatment and care, of which one such chronic disease is Tuberculosis. Due to huge rise in COVID-19 cases, there had been a drastic decrease in notification of TB cases which resulted in reversal of global TB target progress. Apart from these due to the earlier co-infections of TB with SARS and MERS-CoV viruses, the TB-COVID-19 co-infection posed a severe threat in the spread of the disease. All these factors backed to be major motivation factor in development of this model. Leading with this concern, a TB - COVID-19 co-infection model is developed in this study, considering possibility of waning immunity of both diseases. Considering different epidemiological traits, an epidemiological model with 11 compartments is developed and the co-dynamics is analysed. A detailed stability and bifurcation analysis is performed for the TB only sub-model, COVID-19 only sub-model and the complete TB - COVID-19 model. Impact of key parameters namely, infection rate, waning immunity, and face mask efficacy on disease prevalence is discussed in detail. Sensitivity analysis by means of normalized forward sensitivity index of the basic reproduction number and LHS-PRCC approach is carried to provide a thorough understanding of significance of various parameters in accelerating as well as controlling the disease spread. Optimal control analysis is presented extensively, incorporating controls related to timely and improved TB treatment, and enhanced COVID-19 tests and isolation facilities to curb the spread of these infectious diseases. The simulation results obtained from each of these analyses stress on the importance of different control measures in mitigation of the diseases and are illustrated accordingly. The study suggests that in the times of a pandemic, other disease treatment and care must not be neglected, and adequate care must be taken so that mortality due to co-infection and unavailability of timely treatment can be avoided.

Mathematical modeling of COVID-19 in India and Nepal with optimal control and sensitivity analysis.

The pandemic started in the late 2019 and is still waving in claiming millions of lives with virus being mutated to deadlier form. This pandemic has caught attention toward interventions like improved detection of the infected, better quarantine facilities and adequate medical facilities in terms of hospital beds and other medical aid. In this study, we developed a 7-compartment epidemiological model, with inclusion of identified and unidentified infected population along with media factor associated with the aware identified infected population. This is included by using Holling function in the nonlinear incidence, that is responsible for reduction in infection rate via identified infected. The model is fitted to the observed active COVID-19 cases data, collected for a period of 11 months between July 2020 to May 2021 of Nepal and India, and the infection rate as well as the basic reproduction number is obtained for the first wave and second wave of the pandemic in both countries. A comparative analysis on the effect of different parameters on the disease prevalence for both the countries is presented in this work. Sensitivity analysis, time series behavior and optimal control analysis with control parameters equating with reduced infection rate, enhanced detection rate, improved quarantine and hospitalization rate are presented in detail. By means of PRCC, sensitivity analysis is performed and the key parameters influencing the disease prevalence are identified. A detailed study on impact of several parameters in the COVID-19 prevalence, thereby suggesting the interventions to be implemented is discussed in the work. Predictions till June 30, 2021, are obtained using the second wave data for both the countries, and a declining trend is observed for both the countries for the next 30 to 40 days. The estimated values of the infection rates and the hospitalization rates obtained are higher for India compared to Nepal. An optimal control analysis for both the countries is described in detail providing the difference in infectives and recoveries with and without any controls or interventions. The study suggests that improved treatment facilities, testing drives and other non-pharmaceutical interventions would bring down the infected cases to a major extent.

Mathematical modeling of COVID-19 in India and its states with optimal control.

A pandemic is an epidemic spread over a huge geographical area. COVID-19 is 5 th such pandemic documented after 1918 flu pandemic. In this work, we frame a mathematical epidemic model taking inspiration from the classic SIR model and develop a compartmental model with ten compartments to study the coronavirus dynamics in India and three of its most affected states, namely, Maharashtra, Karnataka, and Tamil Nadu, with inclusion of factors related to face mask efficacy, contact tracing, and testing along with quarantine and isolation. We fit the developed model and estimate optimum values of disease transmission rate, detection rate of undetected asymptomatic, and the same of undetected symptomatic. A sensitivity analysis is presented stressing on the importance of higher face mask usage, rapid testing, and contact tracing for curbing the disease spread. An optimal control analysis is performed with two control parameters to study the increase and decrease of the infected population with and without control. This study suggests that improved and rapid testing will help in identifying more infectives, thereby contributing in the decline of disease transmission rate. Optimal control analysis results on stressing on the importance of abiding by strict usage of face mask and social distancing for drastic decrease in number of infections. Time series behaviour of the symptomatic, asymptomatic, and hospitalized population is studied for a range of parameters, resulting in thorough understanding of significance of different parameters.

Modeling of COVID-19 with limited public health resources: a comparative study of three most affected countries.

COVID-19 has become a deadly pandemic in the recent times claiming millions of lives worldwide in a grievous manner. Most of the countries in the world have limited number of medical resources (hospitals, beds, ventilators, etc.), and in the case of large outbreak, it becomes very difficult to provide treatment to every infected individual. In this study, we propound a mathematical model where we classify the infected into two subcategories-asymptomatic and symptomatic. This model further accounts for the effect of limited medical resource for infected people and using face masks in combating the pandemic. Focusing on these aspects, we analyze the model and exploit the available data for assessing the pattern in three most affected countries, namely USA, India and Brazil. The developed model is calibrated to fit data for these three countries and estimate the transmission rate of symptomatic, asymptomatic individuals. The rate at which the individuals who are quarantined recover is estimated as well. Along with these estimations, a comparative study based on the basic reproduction number estimated for the three countries is presented. Standard methods of sensitivity analysis are performed to analyze the ways in which basic reproduction number is impacted upon due to changes in different parameters of the model. Further, we obtain disease-free equilibrium and endemic equilibrium of the model. It is observed that backward bifurcation occurs if the capacity of treatment is small and bistable equilibria are shown that makes the system more sensitive to the initial conditions. Sufficient conditions for the local asymptomatic stability of the endemic equilibrium and disease-free equilibrium of the system are obtained. The results of this study imply that to curb the severity of the increasing cases of the disease in these countries, effective strategies to control disease spread should be implemented so that the basic reproduction number can be decreased below the threshold value which is certainly less than unity. The use of protective masks in public is shown to be an important preventive measure to lower disease transmission rate. Also, the quantity of medical resources should increase so that every infected person can get better treatment.