A fractional-order model for optimizing combination therapy in heterogeneous lung cancer: integrating immunotherapy and targeted therapy to minimize side effects.

This research presents a novel approach to address the complexities of heterogeneous lung cancer dynamics through the development of a Fractional-Order Model. Focusing on the optimization of combination therapy, the model integrates immunotherapy and targeted therapy with the specific aim of minimizing side effects. Notably, our approach incorporates a clever fusion of Proportional-Integral-Derivative (PID) feedback controls alongside the optimization process. Unlike previous studies, our model incorporates essential equations accounting for the interaction between regular and mutated cancer cells, delineates the dynamics between immune cells and mutated cancer cells, enhances immune cell cytotoxic activity, and elucidates the influence of genetic mutations on the spread of cancer cells. This refined model offers a comprehensive understanding of lung cancer progression, providing a valuable tool for the development of personalized and effective treatment strategies. the findings underscore the potential of the optimized treatment strategy in achieving key therapeutic goals, including primary tumor control, metastasis limitation, immune response enhancement, and controlled genetic mutations. The dynamic and adaptive nature of the treatment approach, coupled with economic considerations and memory effects, positions the research at the forefront of advancing precision and personalized cancer therapeutics.

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels.

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods.

A mathematical system of COVID-19 disease model: Existence, uniqueness, numerical and sensitivity analysis.

A compartmental mathematical model of spreading COVID-19 disease in Wuhan, China is applied to investigate the pandemic behaviour in Iran. This model is a system of seven ordinary differential equations including individual behavioural reactions, governmental actions, holiday extensions, travel restrictions, hospitalizations, and quarantine. We fit the Chinese model to the Covid-19 outbreak in Iran and estimate the values of parameters by trial-error approach. We use the Adams-Bashforth predictor-corrector method based on Lagrange polynomials to solve the system of ordinary differential equations. To prove the existence and uniqueness of solutions of the model we use Banach fixed point theorem and Picard iterative method. Also, we evaluate the equilibrium points and the stability of the system. With estimating the basic reproduction number R 0 , we assess the trend of new infected cases in Iran. In addition, the sensitivity analysis of the model is assessed by allocating different parameters to the system. Numerical simulations are depicted by adopting initial conditions and various values of some parameters of the system.