We have proved that any 3-dimensional dynamical system of ordinary differentialequations(in short, 3D ODE)With time-independent invariants can be rewritten asHaniltonian systems with respect to generalized Poisson bra...We have proved that any 3-dimensional dynamical system of ordinary differentialequations(in short, 3D ODE)With time-independent invariants can be rewritten asHaniltonian systems with respect to generalized Poisson brackets and theHamiltonians are these invariants. As an example,we discuss the Kermack-Mckendrick modelfor epidemics in detail. The results we obtained are generalizatioof those obtained by Y. Nutku.展开更多
The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose...The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure <em>exact</em> (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the <em>exact</em> equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called <em>basic reproduction ratio</em>, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.展开更多
In this paper,based on the classic Kermack-McKendrick SIR model,we propose an ordinary differential equation model to re-examine the COVID-19 epidemics in Wuhan where this disease initially broke out.The focus is on t...In this paper,based on the classic Kermack-McKendrick SIR model,we propose an ordinary differential equation model to re-examine the COVID-19 epidemics in Wuhan where this disease initially broke out.The focus is on the impact of all those major nonpharmaceutical interventions(NPIs)implemented by the local public healthy authorities and government during the epidemics.We use the data publicly available and the nonlinear least-squares solver lsqnonlin built in MATLAB to estimate the model parameters.Then we explore the impact of those NPIs,particularly the timings of these interventions,on the epidemics.The results can help people review the responses to the outbreak of the COVID-19 inWuhan,while the proposed model also offers a framework for studying epidemics of COVID-19 and/or other similar diseases in other places,and accordingly helping people better prepare for possible future outbreaks of similar diseases.展开更多
文摘We have proved that any 3-dimensional dynamical system of ordinary differentialequations(in short, 3D ODE)With time-independent invariants can be rewritten asHaniltonian systems with respect to generalized Poisson brackets and theHamiltonians are these invariants. As an example,we discuss the Kermack-Mckendrick modelfor epidemics in detail. The results we obtained are generalizatioof those obtained by Y. Nutku.
文摘The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure <em>exact</em> (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the <em>exact</em> equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called <em>basic reproduction ratio</em>, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.
基金Research partially supported by NSERC of Canada(No.RGPIN-2016-04665)CP was supported by the”Short-term Study Abroad Program for PhD Students”of Northeast Normal University(China).
文摘In this paper,based on the classic Kermack-McKendrick SIR model,we propose an ordinary differential equation model to re-examine the COVID-19 epidemics in Wuhan where this disease initially broke out.The focus is on the impact of all those major nonpharmaceutical interventions(NPIs)implemented by the local public healthy authorities and government during the epidemics.We use the data publicly available and the nonlinear least-squares solver lsqnonlin built in MATLAB to estimate the model parameters.Then we explore the impact of those NPIs,particularly the timings of these interventions,on the epidemics.The results can help people review the responses to the outbreak of the COVID-19 inWuhan,while the proposed model also offers a framework for studying epidemics of COVID-19 and/or other similar diseases in other places,and accordingly helping people better prepare for possible future outbreaks of similar diseases.
