<abstract><p>In this paper, the recent trends of CoVID-19 infection spread have been studied to explore the advantages of leaky vaccination dynamics in SEVR (Susceptible Effected Vaccinated Recovered) compartmental model with the help of <italic>Caputo-Fabrizio</italic> (CF) and <italic>Atangana-Baleanu derivative in the Caputo sense</italic> (ABC) non-singular kernel fractional derivative operators with memory effect within the model to show possible long–term approaches of the infection along with limited defensive vaccine efficacy that can be designed numerically over the closed interval ranging from 0 to 1. One of the main goals is to provide a stepping information about the usefulness of the aforementioned non-singular kernel fractional approaches for a lenient case as well as a critical case in CoVID-19 infection spread. Another is to investigate the effect of death rate on state variables. The estimation of death rate for state variables with suitable vaccine efficacy has a significant role in the stability of state variables in terms of basic reproduction number that is derived using next generation matrix method, and order of the fractional derivative. For non-integral orders the pandemic modeling sense viz, CF and ABC, has been compared thoroughly. Graphical presentations together with numerical results have proposed that the methodology is powerful and accurate which can provide new speculations for CoVID-19 dynamical systems.</p></abstract>

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In this article, we have described a mathematical model for tumor-immune system with impact of small molecules drug, and solved by multi step differential transform method. The mathematical model has four compartments; population of tumor cell, CD8 T-killer cell, CD4 T-helper cell and amount of drug in the blood stream. Calculations are presented to verify the theoretical results so obtained for tumor free equilibrium point. The model is fit for removing tumor with passage of time irrespective of the initial population or size of the tumor.

In this study, we discussed the fractional order model of tumor-immune system based on Liu‘s model. We examine the dynamic behaviour of tumor growth and investigate the conditions of tumor removal mathematically. We discussed qualitative analysis on the mathematical model and defined the existence and uniqueness conditions. Local stability is also checked for tumor-free equilibrium point. We give facts about that tumor growth rate, source rate of immune cells, and death rate of immune cells play vital role in tumor dynamics. Numerical simulations are demonstrated to reveal the analytical results.

Monoclonal antibody (mAb) drugs are used to kill malignant tumors by making them able to see into the immune system. In this article, we developed a mathematical model with the help of tumor‐immune system with antibodies effect and efficacy of mAb drug. We defined the existence of solutions of the model, and by the Lyapunov method, we showed that the solutions are bounded; thereafter, we illustrated the local and global stability phenomenon, which is described by the Lyapunov method and localization of compact invariant sets (LCIS) method for disease‐free equilibrium points including the biological significance. In actual fact, this model suggested some ideas about the malignant tumors reciprocate to mAb drug dosage and its efficacy, eventually to overcome malignancy. It is likely to control the steadiness of the tumor, which is very important for any kind of treatment. The numerical calculations for the proposed model have been carried out by multi‐step differential transformation method (Ms‐DTM). The simulation of the model is depicted through Figures 1–5, which shows the effect of interaction rate of tumor cells and antibodies and dose of the drug for various values of efficacy.