Background: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptio...Background: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike 'assumed' distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration. Methods: It is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes. Results: The simulations explore the efficacy of the two-stage estimation procedure;these cover the estimation of the growth equation and mortality-recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms. Conclusions: The Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.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
By analyzing the susceptible-infected model, susceptible-infected-recovered-susceptible model and susceptible infected-recovered model, we get the improved Kermachk-Mckendrick model. And by applying the controlled thr...By analyzing the susceptible-infected model, susceptible-infected-recovered-susceptible model and susceptible infected-recovered model, we get the improved Kermachk-Mckendrick model. And by applying the controlled threshold value, we get the conditions of isolated rate for infectious disease eventually disappeared.展开更多
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
We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control...We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control.Approximate minimum principles are obtained usingEkeland’s vari ational principle.展开更多
基金partially supported by the USDA National Institute of Food and Agriculture,Mc Intire Stennis Project OKL0 3063the Division of Agricultural Sciences and Natural Resources at Oklahoma State Universityprovided by the USDA Forest Service,Research Joint Venture 17-JV-11242306045,Old-Growth Forest Dynamics and Structure,to Mark Ducey
文摘Background: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike 'assumed' distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration. Methods: It is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes. Results: The simulations explore the efficacy of the two-stage estimation procedure;these cover the estimation of the growth equation and mortality-recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms. Conclusions: The Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.
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.
文摘By analyzing the susceptible-infected model, susceptible-infected-recovered-susceptible model and susceptible infected-recovered model, we get the improved Kermachk-Mckendrick model. And by applying the controlled threshold value, we get the conditions of isolated rate for infectious disease eventually disappeared.
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.
基金This work was supported in part by a grant from the International Development Research Centre Ottawa,Canada
文摘We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control.Approximate minimum principles are obtained usingEkeland’s vari ational principle.
