Role of immigration and emigration on the spread of COVID-19 in a multipatch environment: a case study of India.

Human mobility has played a critical role in the spread of COVID-19. The understanding of mobility helps in getting information on the acceleration or control of the spread of disease. The COVID-19 virus has been spreading among several locations despite all the best efforts related to its isolation. To comprehend this, a multi-patch mathematical model of COVID-19 is proposed and analysed in this work, where-in limited medical resources, quarantining, and inhibitory behaviour of healthy individuals are incorporated into the model. Furthermore, as an example, the impact of mobility in a three-patch model is studied considering the three worst-hit states of India, i.e. Kerala, Maharashtra and Tamil Nadu, as three patches. Key parameters and the basic reproduction number are estimated from the available data. Through results and analyses, it is seen that Kerala has a higher effective contact rate and has the highest prevalence. Moreover, if Kerala is isolated from Maharashtra or Tamil Nadu, the number of active cases will increase in Kerala but reduce in the other two states. Our findings indicate that the number of active cases will decrease in the high prevalence state and increase in the lower prevalence states if the emigration rate is higher than the immigration rate in the high prevalence state. Overall, proper travel restrictions are to be implemented to reduce or control the spread of disease from the high-prevalence state to other states with lower prevalence rates.

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 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.

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.

Modeling and optimal control analysis of COVID-19: Case studies from Italy and Spain.

Coronavirus disease 2019 (COVID-19) is a viral disease which is declared as a pandemic by WHO. This disease is posing a global threat, and almost every country in the world is now affected by this disease. Currently, there is no vaccine for this disease, and because of this, containing COVID-19 is not an easy task. It is noticed that elderly people got severely affected by this disease specially in Europe. In the present paper, we propose and analyze a mathematical model for COVID-19 virus transmission by dividing whole population in old and young groups. We find disease-free equilibrium and the basic reproduction number (R 0). We estimate the parameter corresponding to rate of transmission and rate of detection of COVID-19 using real data from Italy and Spain by least square method. We also perform sensitivity analysis to identify the key parameters which influence the basic reproduction number and hence regulate the transmission dynamics of COVID-19. Finally, we extend our proposed model to optimal control problem to explore the best cost-effective and time-dependent control strategies that can reduce the number of infectives in a specified interval of time.

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.

A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic.

In this paper, we propose a mathematical model to assess the impacts of using face masks, hospitalization of symptomatic individuals and quarantine of asymptomatic individuals in combating the COVID-19 pandemic in India. We calibrate the proposed model to fit the four data sets, viz. data for the states of Maharashtra, Delhi, Tamil Nadu and overall India, and estimate the rate of infection of susceptible with symptomatic population and recovery rate of quarantined individuals. We also estimate basic reproduction number to illustrate the epidemiological status of the regions under study. Our simulations infer that the infective population will be on increasing curve for Maharashtra and India, and settling for Tamil Nadu and Delhi. Sophisticated techniques of sensitivity analysis are employed to determine the impacts of model parameters on basic reproduction number and symptomatic infected individuals. Our results reveal that to curtail the disease burden in India, specific control strategies should be implemented effectively so that the basic reproduction number is decreased below unity. The three control strategies are shown to be important preventive measures to lower disease transmission rate. The model is further extended to its stochastic counterpart to encapsulate the variation or uncertainty observed in the disease transmissibility. We observe the variability in the infective population and found their distribution at certain fixed time, which shows that for small populations, the stochasticity will play an important role.

The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection: a mathematical study.

In this paper, we present a deterministic non-linear mathematical model for the transmission dynamics of HIV and TB co-infection and analyze it in the presence of screening and treatment. The equilibria of the model are computed and stability of these equilibria is discussed. The basic reproduction numbers corresponding to both HIV and TB are found and we show that the disease-free equilibrium is stable only when the basic reproduction numbers for both the diseases are less than one. When both the reproduction numbers are greater than one, the co-infection equilibrium point may exist. The co-infection equilibrium is found to be locally stable whenever it exists. The TB-only and HIV-only equilibria are locally asymptotically stable under some restriction on parameters. We present numerical simulation results to support the analytical findings. We observe that screening with proper counseling of HIV infectives results in a significant reduction of the number of individuals progressing to HIV. Additionally, the screening of TB reduces the infection prevalence of TB disease. The results reported in this paper clearly indicate that proper screening and counseling can check the spread of HIV and TB diseases and effective control strategies can be formulated around 'screening with proper counseling'.

Temporal variation of Ixodes ricinus intensity on the rodent host Apodemus flavicollis in relation to local climate and host dynamics.

The risk to humans of contracting tick-borne zoonotic diseases depends on the risk of a bite from an infected tick, which can be broken down into its component parts as the number of host-seeking ticks in the environment, in particular nymphs, and the prevalence of tick-borne pathogens they are carrying. In turn, the prevalence of tick-borne pathogens is dependent upon tick biting intensity on hosts that support transmission between ticks; namely rodents. These ticks once fed moult into the next life stage and search for the next blood meal, thus posing a zoonotic risk. Here, we analyse tick biting intensity on rodents in a known tick-borne encephalitis (TBE) focus in Trentino (northern Italy). We examine patterns of tick demography and the influence of host densities and climate on ticks' generation time, development rates, tick density and intensity. During the period 2000-2004, a population of the yellow-necked mouse, Apodemus flavicollis, the most important TBE transmission host, was intensively monitored. Ticks feeding on individual rodents were counted, distinguishing between the larval and nymph life-stages. Local temperature and relative humidity was calculated using both data-loggers in the field site and regional weather stations. We investigated which factors had a predictive value both on feeding tick intensity and on the overall density of larvae or nymphs feeding on rodents in a year. We observed a negative effect of rodent density on tick intensity, while temperature influenced positively both larvae and nymph intensity. Overall larval density was higher in the years and trapping grids where rodent density was higher, while for nymphs no such effect was observed. The best explanatory variable for nymph density was the larval density in the previous year, confirming the discrete nature of tick demography. This provides important information in terms of monitoring the risk to humans of acquiring pathogen-infected ticks.

Seasonal population dynamics of ticks, and its influence on infection transmission: a semi-discrete approach.

In this paper, a simple semi-discrete (ticks' feeding is assumed to occur only during the summers of each year) model for tick population dynamics is presented. Conditions for existence, uniqueness, and stability of a positive equilibrium are found; the system is then studied numerically using parameter estimates calibrated for the tick Ixodes ricinus in Trentino, Italy, and the sensitivity to parameters is examined. Then, this model is extended to consider a tick-transmitted infection of one species of hosts, while other hosts are incompetent to the infection. Assuming, for simplicity, that the infection is not affecting the total number either of hosts or ticks, a threshold condition for infection persistence is obtained. The dependence of the equilibrium infection prevalence on parameters is studied numerically; in particular, we considered how infection prevalence depends on host densities. This analysis reveals that a 'dilution effect' occurs both for competent and for incompetent hosts; this means that, besides a lower threshold for host densities for infection to persist, there exists also an upper threshold: if host densities were higher than the upper threshold, the infection would go to extinction. Numerically, it is found that, for realistic parameter values, the upper threshold is not much higher than observed densities.