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Pandemic is an unprecedented public health situation, especially for human beings with comorbidity. Vaccination and non-pharmaceutical interventions only remain extensive measures carrying a significant socioeconomic impact to defeating pandemic. Here, we formulate a mathematical model with comorbidity to study the transmission dynamics as well as an optimal control-based framework to diminish COVID-19. This encompasses modeling the dynamics of invaded population, parameter estimation of the model, study of qualitative dynamics, and optimal control problem for non-pharmaceutical interventions (NPIs) and vaccination events such that the cost of the combined measure is minimized. The investigation reveals that disease persists with the increase in exposed individuals having comorbidity in society. The extensive computational efforts show that mean fluctuations in the force of infection increase with corresponding entropy. This is a piece of evidence that the outbreak has reached a significant portion of the population. However, optimal control strategies with combined measures provide an assurance of effectively protecting our population from COVID-19 by minimizing social and economic costs.

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Deterministic chaos has been studied extensively in various fields. Some of the ideas emerging out of these studies have been put to novel applications. However, it is unknown whether natural ecological systems support chaotic dynamics. There is no concrete evidence which suggests that ecosystem evolution is chaotic in certain situations. This is very intriguing because ecosystems do possess all the necessary qualifications to be able to support such a dynamical behavior. The present paper attempts to answer the above question with the help of a few systems modeling different but very common ecological situations. A new methodology for the analysis of a class of model ecological systems is presented. Simulation experiments suggest that natural terrestrial systems are not suitable candidates where one should look for chaos. Additionally, our study also points out that the failure of attempts to observe chaos in natural populations might have resulted because biological interactions are not conducive for such a behavior to be supported. The cause of these failures may not be the poor data quality or demerits in the analysis techniques.

In this paper, a mathematical model is proposed and analysed to study the dynamics of one-prey two-predators system with ratio-dependent predators growth rate. Criteria for local stability, instability and global stability of the nonnegative equilibria are obtained. The permanent co-existence of the three species is also discussed. Finally, computer simulations are performed to investigate the dynamics of the system.

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