Cancer is characterized by the uncontrolled growth of cells with the ability of invading local organs and/or tissues and of spreading to other sites. Several kinds of mathematical models have been proposed in the literature, involving different levels of refinement, for the evolution of tumors and their interactions with chemotherapy drugs. In this article, we present the solution of a state estimation problem for tumor size evolution. A system of nonlinear ordinary differential equations is used as the state evolution model, which involves as state variables the numbers of tumor, normal and angiogenic cells, as well as the masses of the chemotherapy and anti-angiogenic drugs in the body. Measurements of the numbers of tumor and normal cells are considered available for the inverse analysis. Parameters appearing in the formulation of the state evolution model are treated as Gaussian random variables and their uncertainties are taken into account in the estimation of the state variables, by using an algorithm based on the auxiliary sampling importance resampling particle filter. Test cases are examined in the article dealing with a chemotherapy protocol for pancreatic cancer.
In this paper, we combine algorithm of Liu & West for the Particle Filter (PF) with SIRU-type epidemic model to monitor and forecast cases of Covid-19 in Brazil from February up to September. We filter the number of cumulative reported cases and estimate model parameters and more importantly unreported infectious cases (asymptomatic and symptomatic infectious individuals). The parameters under study are related to the attenuation factor of the transmission rate and the fraction of asymptomatic infectious becoming reported as symptomatic infectious. Initially, the problem is analysed through Particle Swarm Optimization (PSO) based simulations to provide initial guesses, which are then refined by means of PF simulations. Subsequently, two additional steps are performed to verify the capability of the adjusted model to predict and forecast new cases. According to the results, the pandemic peak is expected to take place in mid-June 2020 with about 25,000 news cases per day. As medical and hospital resources are limited, this result shows that public health interventions are essential and should not be relaxed prematurely, so that the coronavirus pandemic is controlled and conditions are available for the treatment of the most severe cases.
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