In this study, we develop a mathematical model incorporating age-specific transmission dynamics of COVID-19 to evaluate the role of vaccination and treatment strategies in reducing the size of COVID-19 burden. Initially, we establish the positivity and boundedness of the solutions of the non controlled model and calculate the basic reproduction number and do the stability analysis. We then formulate an optimal control problem with vaccination and treatment as control variables and study the same. Pontryagin’s Minimum Principle is used to obtain the optimal vaccination and treatment rates. Optimal vaccination and treatment policies are analysed for different values of the weight constant associated with the cost of vaccination and different efficacy levels of vaccine. Findings from these suggested that the combined strategies (vaccination and treatment) worked best in minimizing the infection and disease induced mortality. In order to reduce COVID-19 infection and COVID-19 induced deaths to maximum, it was observed that optimal control strategy should be prioritized to the population with age greater than 40 years. Varying the cost of vaccination it was found that sufficient implementation of vaccines (more than 77 %) reduces the size of COVID-19 infections and number of deaths. The infection curves varying the efficacies of the vaccines against infection were also analysed and it was found that higher efficacy of the vaccine resulted in lesser number of infections and COVID induced deaths. The findings would help policymakers to plan effective strategies to contain the size of the COVID-19 pandemic.
COVID -19 pandemic has resulted in more than 257 million infections and 5.15 million deaths worldwide. Several drug interventions targeting multiple stages of the pathogenesis of COVID -19 can significantly reduce induced infection and thus mortality. In this study, we first develop SIV model at within-host level by incorporating the intercellular time delay and analyzing the stability of equilibrium points. The model dynamics admits a disease-free equilibrium and an infected equilibrium with their stability based on the value of the basic reproduction number R
0. We then formulate an optimal control problem with antiviral drugs and second-line drugs as control measures and study their roles in reducing the number of infected cells and viral load. The comparative study conducted in the optimal control problem suggests that if the first-line antiviral drugs show adverse effects, considering these drugs in reduced amounts along with the second-line drugs would be very effective in reducing the number of infected cells and viral load in a COVID-19 infected patient. Later, we formulate a time-optimal control problem with the goal of driving the system from any initial state to the desired infection-free equilibrium state in finite minimal time. Using Pontryagin’s Minimum Principle, it is shown that the optimal control strategy is of the bang-bang type, with the possibility of switching between two extreme values of the optimal controls. Numerically, it is shown that the desired infection-free state is achieved in a shorter time when the higher values of the optimal controls. The results of this study may be very helpful to researchers, epidemiologists, clinicians and physicians working in this field.
Introduction and importance:
Kartagener’s syndrome is a rare, ciliopathic autosomal recessive genetic disorder that comprises a triad of situs inversus, chronic sinusitis, and bronchiectasis leading to recurrent respiratory infections due to ciliary dyskinesia and thereby progressive deterioration of lung function. Additional clinical features of infertility, otitis media, and rhinitis are also seen in patients.
Case presentation:
The authors hereby present a case of Kartagener’s syndrome in a 40-year-old male with a repeated respiratory infection and bronchial asthma. He was received at the emergency room with symptoms of hemoptysis, shortness of breath, and chest pain. Diagnosis of cystic bronchiectasis with superadded infection was made based on clinical examinations and radiological assessments. He was treated in high-dependency unit. After 5 days of relieving therapeutic interventions in the hospital, he was discharged without further complication.
Clinical discussions:
Early diagnosis of Kartagener’s syndrome is likely to be beneficial as it helps delay deterioration of lung function to prevent complications and improve the quality of life of patients but the diagnosis of this syndrome is usually delayed as it is a rare disease, especially in countries with lack of complex diagnostic facilities. So, assessment for this syndrome has to be done in patients presenting with chronic and recurrent respiratory infections for correct timely diagnosis to have a good patient-centric healthcare facility.
COVID-19 pandemic has caused the most severe health problems to adults over 60 years of age, with particularly fatal consequences for those over 80. In this case, age-structured mathematical modeling could be useful to determine the spread of the disease and to develop a better control strategy for different age groups. In this study, we first propose an age-structured model considering two different age groups, the first group with population age below 30 years and the second with population age above 30 years, and discuss the stability of the equilibrium points and the sensitivity of the model parameters. In the second part of the study, we propose an optimal control problem to understand the age-specific role of treatment in controlling the spread of COVID -19 infection. From the stability analysis of the equilibrium points, it was found that the infection-free equilibrium point remains locally asymptotically stable when R 0 < 1 , and when R 0 is greater than one, the infected equilibrium point remains locally asymptotically stable. The results of the optimal control study show that infection decreases with the implementation of an optimal treatment strategy, and that a combined treatment strategy considering treatment for both age groups is effective in keeping cumulative infection low in severe epidemics. Cumulative infection was found to increase with increasing saturation in medical treatment.
Leprosy (Hansen’s disease) is an infectious, neglected tropical disease caused by the Mycobacterium Leprae (M. Leprae). About 2,02,189 new cases are diagnosed worldwide each year. Lepra reactions are an off shoot of leprosy infection causing major nerve damage leading to disability. Early detection of lepra reactions through the study of biomarkers can prevent subsequent disabilities. Motivated by these observations, in this study, we have proposed and analyzed a three-dimensional mathematical model to capture the dynamics of susceptible schwann cells, infected schwann cells, and the bacterial load based on the pathogenesis of leprosy. We did the stability analysis, numerical simulations, and also performed the sensitivity analysis using Spearman’s rank correlation coefficient, partial rank correlation coefficient, and Sobol’s index methods. We later performed the optimal control studies with both multi-drug therapy and steroid interventions as control variables. Finally, we did the comparative and effectiveness study of these different control interventions.
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