Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

Shakourifar, Mohammad; Enright, W. H.

doi:10.1137/100793098

Cited by 19 publications

(3 citation statements)

References 30 publications

(59 reference statements)

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“…-VIDEs (with a time dependent delay) ( [9]): -Solving Problems which depend on parameters and parameter determination.…”

Section: Sdc Rk Methods For Other Classes Of Odesmentioning

confidence: 99%

Reducing the Uncertainty When Approximating the Solution of ODEs

Enright

2012

IFIP Advances in Information and Communication Technology

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Abstract. One can reduce the uncertainty in the quality of an approximate solution of an ordinary differential equation (ODE) by implementing methods which have a more rigorous error control strategy and which deliver an approximate solution that is much more likely to satisfy the expectations of the user. We have developed such a class of ODE methods as well as a collection of software tools that will deliver a piecewise polynomial as the approximate solution and facilitate the investigation of various aspects of the problem that are often of as much interest as the approximate solution itself. We will introduce measures that can be used to quantify the reliability of an approximate solution and discuss how one can implement methods that, at some extra cost, can produce very reliable approximate solutions and therefore significantly reduce the uncertainty in the computed results.

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“…-VIDEs (with a time dependent delay) ( [9]): -Solving Problems which depend on parameters and parameter determination.…”

Section: Sdc Rk Methods For Other Classes Of Odesmentioning

confidence: 99%

Reducing the Uncertainty When Approximating the Solution of ODEs

Enright

2012

IFIP Advances in Information and Communication Technology

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“…Our model system can be studied in the framework of a system of delay integro-differential equations (DIDEs) abstracted like in (3.1). We are in a good position to know that this type of system has been under previous investigations (Koto, 2002; Zhang and Vandewalle, 2004; Khasawneh and Mann, 2011; Shakourifar and Enright, 2011), even from the field of population dynamics (Kuang, 1993). A practical method for solving (3.1) is the direct adaptation of the (one-step) Runge–Kutta method, cf.…”

Section: Numerical Implementationsmentioning

confidence: 96%

A COVID-19 epidemic model integrating direct and fomite transmission as well as household structure

Wijaya¹,

Ganegoda

Jayatunga³

et al. 2020

Preprint

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This paper stresses its base contribution on a new SIR-type model for COVID-19 including direct and fomite transmission as well as the effect of distinct household structures. To what extent increasing the physical-distancing-related contact radius and enhancing mass control (public curfew, lockdown, workplace clearance, and school closure) reduce the number of predicted active cases is studied via parameter estimation.

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“…Our model system can be studied in the framework of a system of delay integro-differential equations (DIDEs) abstracted like in ( 14 ). We are in a good position to know that this type of system has been under previous investigations [ 50 – 53 ], even from the field of population dynamics [ 54 ]. A practical method for solving ( 14 ) is the direct adaptation of the (one-step) Runge–Kutta method, cf.…”

Section: Appendix A: Model Derivationmentioning

confidence: 99%

An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19

et al. 2021

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This paper stresses its base contribution on a new SIR-type model including direct and fomite transmission as well as the effect of distinct household structures. The model derivation is modulated by several mechanistic processes inherent from typical airborne diseases. The notion of minimum contact radius is included in the direct transmission, facilitating the arguments on physical distancing. As fomite transmission heavily relates to former-trace of sneezes, the vector field of the system naturally contains an integral kernel with time delay indicating the contribution of undetected and non-quarantined asymptomatic cases in accumulating the historical contamination of surfaces. We then increase the complexity by including the different transmission routines within and between households. For airborne diseases, within-household interactions play a significant role in the propagation of the disease rendering countrywide effect. Two steps were taken to include the effect of household structure. The first step subdivides the entire compartments (susceptible, exposed, asymptomatic, symptomatic, recovered, death) into the household level and different infection rates for the direct transmission within and between households were distinguished. Under predefined conditions and assumptions, the governing system on household level can be raised to the community level. The second step then raises the governing system to the country level, where the final state variables estimate the total individuals from all compartments in the country. Two key attributes related to the household structure (number of local households and number of household members) effectively classify countries to be of low or high risk in terms of effective disease propagation. The basic reproductive number is calculated and its biological meaning is invoked properly. The numerical methods for solving the DIDE-system and the parameter estimation problem were mentioned. Our optimal model solutions are in quite good agreement with datasets of COVID-19 active cases and related deaths from Germany and Sri Lanka in early infection, allowing us to hypothesize several unobservable situations in the two countries. Focusing on extending minimum contact radius and reducing the intensity of individual activities, we were able to synthesize the key parameters telling what to practice.

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Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

Cited by 19 publications

References 30 publications

Reducing the Uncertainty When Approximating the Solution of ODEs

Reducing the Uncertainty When Approximating the Solution of ODEs

A COVID-19 epidemic model integrating direct and fomite transmission as well as household structure

An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19

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