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The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula
Yuxin Zhou,
Jun Zhang,
Yufeng Diao
Issue:
Volume 11, Issue 5, October 2022
Pages:
116-122
Received:
7 September 2022
Accepted:
5 October 2022
Published:
11 October 2022
DOI:
10.11648/j.acm.20221105.11
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Abstract: In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.
Abstract: In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentia...
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Computational Modelling of Two Strain Meningitis Disease Outbreak
Timothy Kiprono Yano,
Jacob Bitok
Issue:
Volume 11, Issue 5, October 2022
Pages:
123-129
Received:
11 September 2022
Accepted:
26 September 2022
Published:
11 October 2022
DOI:
10.11648/j.acm.20221105.12
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Abstract: Meningococcal meningitis is a significant contributor to increased deaths globally, particularly the vulnerable children aged between 0-5 years. This paper formulates a robust two-strain epidemic model for the transmission dynamics of bacterial meningitis by incorporating interventions such as treatment and vaccination. The aim of the article is to formulate a meningitis epidemic model and study the time dependent dynamics of meningitis in the presence of antibiotic resistance to treatment threats while assessing the impact of vaccination proportion. The study uses the 4th order Runge Kutta numerical approach to solve the problem and Maple mathematical tool to undertake simulations. The meningitis model qualitative study reveals existence of disease-free state when infection dies out and endemic state when disease persist in the community. The disease-free case is found to be stable only if effective reproduction number Re < 1 and the community enjoys disease free scenario. Meningitis disease-free state reveals a locally asymptotically stable (LAS) transmission dynamics. The endemic equilibrium state i.e., Re > 1 exists and persistence occurs in the community. The impact of parameter control measures on the spread of meningitis disease through sensitivity study of the key parameter, i.e., Re, which revealed the key target parameters that can wipe out meningitis disease. We perform numerical solution of the considered model equations to display the qualitative findings and describe the asymptotical transmission dynamics of the disease. The effects of meningitis disease prevention and control approaches are analyzed. Key findings are shown using graphs and tables. We obtain a threshold vaccination proportion value beyond which the meningitis disease will be perfectly wiped out of the community and below which the disease acquires endemic state.
Abstract: Meningococcal meningitis is a significant contributor to increased deaths globally, particularly the vulnerable children aged between 0-5 years. This paper formulates a robust two-strain epidemic model for the transmission dynamics of bacterial meningitis by incorporating interventions such as treatment and vaccination. The aim of the article is to...
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Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis
Timothy Kiprono Yano,
Jacob Bitok
Issue:
Volume 11, Issue 5, October 2022
Pages:
130-139
Received:
24 September 2022
Accepted:
9 October 2022
Published:
17 October 2022
DOI:
10.11648/j.acm.20221105.13
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Abstract: A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.
Abstract: A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antib...
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Optimal Control Analysis of Meningococcal Meningitis Disease with Varying Population Size
Timothy Kiprono Yano,
Jacob Bitok,
Rael Jerop
Issue:
Volume 11, Issue 5, October 2022
Pages:
140-149
Received:
20 September 2022
Accepted:
4 October 2022
Published:
17 October 2022
DOI:
10.11648/j.acm.20221105.14
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Abstract: Meningococcal meningitis is a fatal and scary highly infectious disease especially in the African meningitis disease belt and globally, its every community’s desire to wipe out meningitis disease by considering its prevention and control mechanisms. The paper formulates and analyzes a Meningococcal meningitis epidemic model that describes the spreading mechanisms of meningitis in a community with varying population. The stability analysis approach of non-linear systems is used to distinguish the properties of an epidemic deterministic compartmental model. The effective threshold reproductive value is determined by Jacobian approach and the stability study for the zero disease and endemic states are determined. Sensitivity indices analysis of the effective reproductive number to the crucial parameter values are established and rated accordingly. Using Pontryagin's approach to an optimal problem, the model was extended to include the following four control intervention measures: effort to prevent a disease infection by providing education needed, efforts to treat that minimizes sensitive and resistant strains and immunity control effort. The optimal control study of the applied control intervention efforts reveals that the use of prevention techniques and treatment efforts leads to a larger decrease of infections, thus becoming are the best intervention control strategy to eliminate the meningitis disease. Numerical analysis study was done for a combination of other strategies and main results are displayed using graphs.
Abstract: Meningococcal meningitis is a fatal and scary highly infectious disease especially in the African meningitis disease belt and globally, its every community’s desire to wipe out meningitis disease by considering its prevention and control mechanisms. The paper formulates and analyzes a Meningococcal meningitis epidemic model that describes the sprea...
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Numerical Investigation of Unsteady Hydromagnetic Stokes Free-Convective Fluid Flow Past an Infinite Vertical Porous Plate with Variable Suction in a Rotating System
Mayaka Augustine Ayanga,
Mathew Ngugi Kinyanjui,
Jeconia Okelo Abonyo,
Johana Kibet Sigey
Issue:
Volume 11, Issue 5, October 2022
Pages:
150-159
Received:
30 August 2022
Accepted:
12 October 2022
Published:
24 October 2022
DOI:
10.11648/j.acm.20221105.15
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Abstract: In this paper, Stokes first problem for an unsteady hydromagnetic free convective flow of a viscous incompressible fluid past an infinite vertical porous plate subjected to a variable suction in a rotating system has been studied. The specific equations governing the flow are nondimensionalized to obtain the dimensionless forms of the governing equations. The resulting dimensionless governing partial differential equations are solved numerically by the finite difference method based on the forward-time central-space scheme. The resulting numerical schemes are simulated in MATLAB software to obtain the profiles of the flow variables such as velocity, temperature, species concentration and magnetic induction. The main findings of this study are that an increase in the joule heating parameter results in a uniform increase in the velocity and temperature profiles near the plate but remain constantly distributed away from the plate. This observation implies that the flow is influenced substantially by the strength of joule heating near the plate and in the bulk of the fluid. The results are useful in industrial water treatment systems which rely on physical forces to aid in the removal of pollutants. Moreover, the results are applicable in the separation of isotopes contained in a mixture of very light molecular-weight gases such as hydrogen and helium and medium molecular-weight gases like nitrogen and air.
Abstract: In this paper, Stokes first problem for an unsteady hydromagnetic free convective flow of a viscous incompressible fluid past an infinite vertical porous plate subjected to a variable suction in a rotating system has been studied. The specific equations governing the flow are nondimensionalized to obtain the dimensionless forms of the governing equ...
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