Abstract.A new mathematical model is presented to study the early inflammatory effects in bone healing. It consists of a system of nonlinear ordinary differential equations that represents the interactions among macrophages, mesenchymal stem cells, and osteoblasts. A qualitative analysis of the model is performed to determine the equilibria and their corresponding stability properties. A set of numerical simulations is performed to support the theoretical results. The model is also used to numerically monitor the evolution of a broken bone for different types of fractures and to explore possible treatments to accelerate bone healing by administrating anti-inflammatory drugs.

A new mathematical model is presented to study the effects of macrophages on the bone fracture healing process. The model consists of a system of nonlinear ordinary differential equations that represents the interactions among classically and alternatively activated macrophages, mesenchymal stem cells, osteoblasts, and pro- and anti-inflammatory cytokines. A qualitative analysis of the model is performed to determine the equilibria and their corresponding stability properties. Numerical simulations are also presented to support the theoretical results, and to monitor the evolution of a broken bone for different types of fractures under various medical interventions. The model can be used to guide clinical experiments and to explore possible medical treatments that accelerate the bone fracture healing process, either by surgical interventions or drug administrations.

ABSTRACTCell degeneration, including that resulting in retinal diseases, is linked to metabolic issues. In the retina, photoreceptor degeneration can result from imbalance in lactate production and consumption as well as disturbances to pyruvate and glucose levels. To identify the key mechanisms in metabolism that may be culprits of this degeneration, we use a nonlinear system of differential equations to mathematically model the metabolic pathway of aerobic glycolysis in a healthy cone photoreceptor. This model allows us to analyze the levels of lactate, glucose, and pyruvate within a single cone cell. We perform numerical simulations, use available metabolic data to estimate parameters and fit the model to this data, and conduct a sensitivity analysis using two different methods (LHS/PRCC and eFAST) to identify pathways that have the largest impact on the system. Using bifurcation techniques, we find that the system has a bistable regime, biologically corresponding to a healthy versus a pathological state. The system exhibits a saddle node bifurcation and hysteresis. This work confirms the necessity for the external glucose concentration to sustain the cell even at low initial internal glucose levels. It also validates the role of β-oxidation of fatty acids which fuel oxidative phosphorylation under glucose- and lactate-depleted conditions, by showing that the rate of β-oxidation of ingested outer segment fatty acids in a healthy cone cell must be low. Model simulations reveal the modulating effect of external lactate in bringing the system to steady state; the bigger the difference between external lactate and initial internal lactate concentrations, the longer the system takes to achieve steady state. Parameter estimation for metabolic data demonstrates the importance of rerouting glucose and other intermediate metabolites to produce glycerol 3-phosphate (G3P), thus increasing lipid synthesis (a precursor to fatty acid production) to support their high growth rate. While a number of parameters are found to be significant by one or both of the methods for sensitivity analysis, the rate of β-oxidation of ingested outer segment fatty acids is shown to consistently play an important role in the concentration of glucose, G3P, and pyruvate, whereas the extracellular lactate level is shown to consistently play an important role in the concentration of lactate and acetyl coenzyme A. The ability of these mechanisms to affect key metabolites’ variability and levels (as revealed in our analyses) signifies the importance of inter-dependent and inter-connected feedback processes modulated by and affecting both the RPE’s and cone’s metabolism.

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Fitting Susceptible-Infected-Recovered (SIR) models to incidence data is problematic when not all infected individuals are reported. Assuming an underlying SIR model with general but known distribution for the time to recovery, this paper derives the implied differential-integral equations for observed incidence data when a fixed fraction of newly infected individuals are not observed. The parameters of the resulting system of differential equations are identifiable. Using these differential equations, we develop a stochastic model for the conditional distribution of current disease incidence given the entire past history of reported cases. We estimate the model parameters using Bayesian Markov Chain Monte-Carlo sampling of the posterior distribution. We use our model to estimate the transmission rate and fraction of asymptomatic individuals for the current Coronavirus 2019 outbreak in eight American Countries: the United States of America, Brazil, Mexico, Argentina, Chile, Colombia, Peru, and Panama, from January 2020 to May 2021. Our analysis reveals that the fraction of reported cases varies across all countries. For example, the reported incidence fraction for the United States of America varies from 0.3 to 0.6, while for Brazil it varies from 0.2 to 0.4.