Mathematical modeling of the spread of the coronavirus under strict social restrictions.

We formulate a simple susceptible-infectious-recovery (SIR) model to describe the spread of the coronavirus under strict social restrictions. The transmission rate in this model is exponentially decreasing with time. We find a formula for basic reproduction function and estimate the maximum number of daily infected individuals. We fit the model to induced death data in Italy, United States, Germany, France, India, Spain, and China over the period from the first reported death to August 7, 2020. We notice that the model has excellent fit to the disease death data in these countries. We estimate the model's parameters in each of these countries with 95% confidence intervals. We order the strength of social restrictions in these countries using the exponential rate. We estimate the time needed to reduce the basic reproduction function to one unit and use it to order the quality of social restrictions in these countries. The social restriction in China was the strictest and the most effective and in India was the weakest and the least effective. Policy-makers may apply the Chinese successful social restriction experiment and avoid the Indian unsuccessful one.

Computational modeling of human papillomavirus with impulsive vaccination.

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.

COMPARING HEALTH EFFECTS OF RADON IN MANITOBA AND PRINCE EDWARD ISLAND.

We compare lung cancer (LC) rates, smoking percentages and radon levels in Manitoba (MN) and Prince Edward Island (PE). Smoking does not explain high LC in MN, but indoor radon does. We show that population attributable risk in PE is only 37% of that in MN. Radon above 200 Bq m -3 increases LC in PE by only 4 cases, but in MN by 195 cases.

New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations.

A two-step fifth and a multi-step 5+3r order iterative method are derived, r≥1 for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite's measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.

Estimating the burden of lung cancer and the efficiency of home radon mitigation systems in some Canadian provinces.

Lung cancer (LC) is the leading cause of death of cancer in Canada in both men and women, and indoor radon is the second leading cause of LC after tobacco smoking. The Population Attributable Risk (PAR) is used to assess radon exposure risk. In this work we estimate the burden of LC in some Canadian provinces. We use the PAR to identify the radon levels responsible for most LC cases. Finally, we use the PAR function of the two variables, radon action and target levels, to search for a possible optimal mitigation program. METHODS: The LC burden for Ontario, Alberta, Manitoba, Quebec and British Columbia was estimated using provincial radon and mortality data. Then the PAR and LC cases for these provinces were estimated over the period 2006-2009 at different given indoor radon exposure levels. Finally, the PAR function when radon action levels and radon target levels are variables was analyzed. RESULTS: The highest burden of LC in 2006-2009 was in Ontario and Quebec. During the period 2006-2009, 6% of houses in Ontario, 9% of houses in Alberta, 19% of houses in Manitoba, 7% of houses in Quebec, and 5% of houses in British Columbia had radon levels higher than 200 Bq/m3 and were responsible about 913, 211, 260, 972, and 258 lives, respectively. Radon mitigation programs could have prevented these LC cases. The BEIR VI assumption for the United States (US) population, 95% of LC deaths in men and 90% of LC deaths in women are Ever-Smokers (ES), can be applied to the Canadian population. The PAR is a linear function in the target radon value with an estimated slope of 0.0001 for Ontario, Alberta, Quebec and British Columbia, and 0.0004 for Manitoba. The PAR is almost a square root function in the radon action level. The PAR is sensitive to changes in the radon mitigation program and as such, any improvement is a worthwhile investment.

Population attributable risk associated with lung cancer induced by residential radon in Canada: Sensitivity to relative risk model and radon probability density function choices: In memory of Professor Jan M. Zielinski.

Indoor radon has been identified as the second leading cause of lung cancer after tobacco smoking. The Population Attributable Risk (PAR) estimates the proportion of lung cancer cases associated with indoor radon exposure. Different relative risk (RR) models have been used in the literature to calculate PAR. The aim of this study is to assess how sensitive PAR is to the relative risk model and radon probability distribution functions choices. METHODS: Using Canadian observed first floor radon data collected by Health Canada during the period October 2010 to March 2011, seven common PAR radon models used for North American miners and dwelling scenarios were applied. The death rates used for this study were from the period 2006-2009. Smoking data (Ever Smoking ES and Never Smoking NS) collected in 2009 was also used in this study. The original discrete radon data for Canada overall and for each of its provinces are estimated using log-normal and Gaussian kernel density estimator distributions. PAR was then calculated for Canada and its provinces using the empirical, log-normal, and Gaussian kernel estimates distributions. Finally, cancer death cases attributable to radon are reported for the constant relative risk model for the three distributions and the reduction in the cases when the action level 200Bq/m3 is applied. RESULTS: PAR for the Canadian data is sensitive to the model choice, and it varies with a range of 10% for ES and 32% for NS, respectively. There is little difference in results between miners' models and dwelling models. PAR values for ES females are greater than those for ES males, except in Saskatchewan, Northwest Territories, Nunavut, and Yukon. The male-female range overlaps. Gaussian kernel estimator produces PAR estimates similar to the commonly used log-normal distribution. CONCLUSION: Many lung cancer cases could be prevented in Canada by reducing indoor radon. PAR is sensitive to the choice of RR model. Miners' models can be used for residential radon. Empirical, log-normal, and Gaussian kernel density estimation with support [0,∞) can all be applied to radon data.

Controlling malaria with indoor residual spraying in spatially heterogenous environments.

Indoor residual spraying-spraying insecticide inside houses to kill mosquitoes-has been one of the most effective methods of disease control ever devised, being responsible for the near-eradication of malaria from the world in the third quarter of the twentieth century and saving tens of millions of lives. However, with malaria resurgence currently underway, it has received relatively little attention, been applied only in select physical locations and not always at regular intervals. We extend a time-dependent model of malaria spraying to include spatial heterogeneity and address the following research questions: 1. What are the effects of spraying in different geographical areas? 2. How do the results depend upon the regularity of spraying? 3. Can we alter our control strategies to account for asymmetric phenomena such as wind? We use impulsive partial differential equation models to derive thresholds for malaria control when spraying occurs uniformly, within an interior disc or under asymmetric advection effects. Spatial heterogeneity results in an increase in the necessary frequency of spraying, but control is still achievable.