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In this paper, a mathematical ordinary differential tumor-immune model is proposed based on an immune checkpoint inhibitor, which is an innovative method for tumor immunotherapies. Two important factors in tumor-immune response are the programmed cell death protein 1 (PD-1) and its ligand PD-L1. The model consists of three populations: tumor cells, activated T cells and anti-PD-1. By analyzing the dynamics of the model, it is found that there is always a unique tumor-free equilibrium and at most two tumor interior equilibria. The nonexistence of nontrivial positive periodic orbits is established by using the new Dulac function, and then a global dynamics of the model is obtained. The conclusions of our analysis show that increasing the possibility of T cells killing tumor cells (p), early detection of tumor cells, or the use of PD-1 inhibitors to activate T cells are effective in eliminating tumor cells. [ABSTRACT FROM AUTHOR]

Changes in cellular Ca 2 + concentration control a variety of physiological activities including hormone and neurotransmitter release, muscular contraction, synaptic plasticity, ionic channel permeability, apoptosis, enzyme activity, gene transcription and reproduction process. Spatial–temporal Ca 2 + dynamics due to Ca 2 + release, buffering and re-uptaking plays a central role in studying the Ca 2 + regulation in T lymphocytes. In most cases, Ca 2 + has its major signaling function when it is elevated in the cytosolic compartment. In this paper, a two-dimensional mathematical model to study spatiotemporal variations of intracellular Ca 2 + concentration in T lymphocyte cell is proposed and investigated. The cell is assumed to be a circular shaped geometrical domain for the representation of properties of Ca 2 + dynamics within the cell including important parameters. Ca 2 + binding proteins for the dynamics of Ca 2 + are itself buffer and other physiological parameters located in Ca 2 + stores. The model incorporates the important biophysical processes like diffusion, reaction, voltage-gated Ca 2 + channel, leak from endoplasmic reticulum (ER), efflux from cytosol to ER via sarco–ER Ca 2 + adenosine triphosphate (SERCA) pumps, buffers and Na + / Ca 2 + exchanger. The proposed mathematical model is solved using a finite difference method and the finite element method. Appropriate initial and boundary conditions are incorporated in the model based on biophysical conditions of the problem. Computer simulations in MATLAB R2019b are employed to investigate mathematical models of reaction–diffusion equation. The effect of source, buffer, Na + /Ca 2 + exchanger, etc. on spatial and temporal patterns of Ca 2 + in T lymphocyte has been studied with the help of numerical results. From the obtained results, it is observed that, the coordinated combination of the incorporated parameters plays a significant role in Ca 2 + regulation in T lymphocytes. ER leak and voltage-gated Ca 2 + channel provides the necessary Ca 2 + to the cell when required for its proper functioning, while on the other side buffers, SERCA pump and Na + /Ca 2 + exchanger makes balance in the Ca 2 + concentration, so as to prevent the cell from death as higher concentration for longer time is harmful for the cell and can cause cell death. [ABSTRACT FROM AUTHOR]