A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes

The research work in this paper attempts to describe the outbreak of Coronavirus Disease 2019 (COVID-19) with the help of a mathematical model using both the Ordinary Differential Equation (ODE) and Fractional Differential Equation. The spread of the disease has been on the increase across the globe for some time with no end in sight. The research used the data of COVID-19 cases in Nigeria for the numerical simulation which has been fitted to the model. We brought in the consideration of both asymptomatic and symptomatic infected individuals with the fact that an exposed individual is either sent to quarantine first or move to one of the infected classes with the possibility that susceptible individual can also move to quarantined class directly. It was found that the proposed model has two equilibrium points; the disease-free equilibrium point ( DFE ) and the endemic equilibrium point ( E 1 ) . Stability analysis of the equilibrium points shows ( E 0 ) is locally asymptotically stable whenever the basic reproduction number,R 0 < 1 and ( E 1 ) is globally asymptotically stable wheneverR 0 > 1 . Sensitivity analysis of the parameters in theR 0 was conducted and the profile of each state variable was also depicted using the fitted values of the parameters showing the spread of the disease. The most sensitive parameters in theR 0 are the contact rate between susceptible individuals and the rate of transfer of individuals from exposed class to symptomatically infected class. Moreover, the basic reproduction number for the data is calculated asR 0 ≈ 1.7031 . Existence and uniqueness of solution established via the technique of fixed point theorem. Also, using the least square curve fitting method together with the fminsearch function in the MATLAB optimization toolbox, we obtain the best values for some of the unknown biological parameters involved in the proposed model. Furthermore, we solved the fractional model numerically using the Atangana-Toufik numerical scheme and presenting different forms of graphical results that can be useful in minimizing the infection.

Respiratory dysfunction following cardiopulmonary bypass: verification of a non-invasive technique to measure shunt fraction

Respiratory dysfunction is a well recognized complication of cardiopulmonary bypass. The size of the pulmonary shunt fraction is the best indicator of respiratory dysfunction but its measurement conventionally requires use of a pulmonary artery catheter to measure mixed venous oxygen content. We compared pulmonary shunt fraction, based on a non-invasive technique using a previously described mathematical model, with shunt fraction measured invasively using a pulmonary artery catheter in 22 patients undergoing elective coronary artery surgery. The mean shunt fraction measured by the invasive technique was 19.6 +/- 2.0 (18.8-20.4)% of cardiac output at 24 h (+/- 1 SD and 90% confidence intervals) and 20.9 +/- 2.9 (19.8-22.0)% of cardiac output at 44 h post-surgery. There was good agreement between the two methods of measurement. The mean difference was 0.21 percentage points with 95% confidence interval -0.01 to 0.43. The limits of agreement (-1.17 to 1.59) are small enough to be confident that the non-invasive method can be used to give the same result as that obtained using a pulmonary artery catheter. The values for shunt fractions obtained by the non-invasive technique were 19.7 +/- 2.3 (18.8-20.6)% of cardiac output at 24 h and 20.7 +/- 2.5 (19.7-21.6)% of cardiac output at 44 h. The non-invasive measurement of the shunt fraction provided us with a simple, practical method for following a further ten patients over an extended period of time where prolonged catheterization is impractical.