The paper introduces the data-driven optimal control problem combined with a class of nonlinear SEIRS epidemic model. By extending the classical SEIR model framework, a nonlinear SEIRS epidemic model is formulated by considering the immune loss rate of recovered population and a nonlinear incidence rate with saturation effect. In the SEIRS epidemic model, the number of population in each compartment is unknown and the parameters are time-varying. Combined with the ODE system derived from the SEIRS epidemic model, we leverage real-time data to define the loss function and obtain a data-driven optimal control problem. Employing the generalized Pontryagin’s maximum principle, we state the necessary conditions for optimal solution to the data-driven optimal control problem. Furthermore, we meticulously devise an algorithmic framework, inclusive of detailed steps, to address this complex optimization task. We conduct numerical experiments using reported COVID-19 data, which enable us to estimate unknown population numbers and obtain time-varying parameters within the SEIRS epidemic model. The results of numerical experiments validate the effectiveness and rationality of our algorithm. Ultimately, by controlling the growth of the number of infected population and dead population over the subsequent 30 days, we obtain the temporal evolution trend of key parameters. Notably, our findings emphasize the strategic importance of reducing both the infection rate and the immune loss rate as effective means to control propagation of epidemics. This conclusion, rooted in our data-driven approach, offers a fresh perspective on epidemic control strategies.

This paper is concerned with a mathematical model of competition for resource where species consume noninteracting resources. This system of differential equations is formally obtained by renormalizing the MacArthur's competition model at equilibrium, and agrees with the trait-continuous model studied by Mirrahimi S, Perthame B, Wakano JY [J. Math. Biol. 64(7): 1189-1223, 2012]. As a dynamical system, self-organized generation of distinct species occurs. The necessary conditions for survival are given. We prove the existence of the evolutionary stable distribution (ESD) through an optimization problem and present an independent algorithm to compute the ESD directly. Under certain structural conditions, solutions of the system are shown to approach the discrete ESD as time evolves. The time discretization of the system is proven to satisfy two desired properties: positivity and energy dissipation. Numerical examples are given to illustrate certain interesting biological phenomena.

It is an increasingly challenging task to explore the risk measurement for multidimensional portfolios with nonlinear correlative assets. A risk measurement scheme based on the mixed copula theory is proposed in this paper, where the mixed copula is constructed by the linear combination of three single Archimedean copulas, embodying greater flexibility than single copula in connecting different types of marginal distributions. In the scenario, ARMA-EGARCH model with t innovation is employed to fit marginal distributions, and the parameter values of the mixed copulas are inferred by maximum likelihood estimation (MLE) method, and interior point algorithm is used to calculate the extreme values of the MLE, VaR and CVaR, corresponding to the optimal portfolio with the minimum risk. Finally, an empirical study on five international stock market indexes in Europe is performed to verify the feasibility and effectiveness of the scheme.

Correlation and risk measurement are important for reliability and safety evaluation of many practical systems. Clayton copula and 180° rotated Clayton copula are suitable for measuring the lower-lower tail correlation and the upper-upper tail correlation reflecting positive correlation, respectively. The 90° rotated Clayton copula and 270° rotated Clayton copula are severally appropriate for measuring the lower-upper tail correlation and the upper-lower tail correlation reflecting negative correlation. In this paper, considering the possibility of variable correlation state transition under the condition of unfixed component copula function weight, a mixed Clayton copula function of Markov transformation is constructed by using the above four kinds of Clayton copula, and the marginal distributions of variables are modeled by combining ARMA(p,q)−GARCH(m,n) model, then the correlation and its corresponding risk measurement models are constructed based on mixed Clayton copula. Finally, the empirical results based on crude oil futures price and Chinese CSI 300 stock index futures price show that, compared with time-varying Normal copula, time-varying T copula and Markov-switching GRG copula, Markov-switching mixed-Clayton copula model can achieve better effect of parameter estimation. The calculation on the correlation and risk measurement further verifies the validity and reliability of the models.

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

In this paper, we design, analyze and numerically validate energy dissipating finite volume schemes for a competition-mutation equation with a gradient flow structure. The model describes the evolution of a population structured with respect to a continuous trait. Both semi-discrete and fully discrete schemes are demonstrated to satisfy the two desired properties: positivity of numerical solutions and energy dissipation. These ensure that the positive steady state is asymptotically stable. Moreover, the discrete steady state is proven to be the same as the minimizer of a discrete energy function. As a comparison, the positive steady state can also be produced by a nonlinear programming solver. Finally, a series of numerical tests is provided to demonstrate both accuracy and the energy dissipation property of the numerical schemes. The numerical solutions of the model with small mutation are shown to be close to those of the corresponding model with linear competition.

This paper considers numerical solutions to magneto hydrodynamics convective heat transfer over a permeable stretching wedge with thermal radiation and ohmic heating. Both the viscosity and thermal conductivity are assumed to vary as a linear function of the temperature, and dynamic viscosity is considered in a new form. Using an appropriate transformation, the governing partial differential equations are transformed into ordinary differential equations. Numerical solutions to these equations subject to corresponding boundary conditions are obtained by efficient numerical shooting technique coupled with Runge–Kutta–Fehlberg scheme. The effects of pertinent parameters are shown through tables and graphs, and meanwhile the associated transfer characteristics are analyzed in detail. The results show that: with the increase of viscosity variation parameter or the thermal conductivity, the local skin-friction coefficient increases but the local Nusselt number decreases. The increasing in viscosity variation parameter is to increase the velocity boundary layer thickness and decrease in the magnitude of the velocity gradient. Furthermore, thermal conductivity parameter has the same effect on the thermal boundary layer thickness and the temperature gradient.

According to rational expectation hypothesis, the government will take into account the future capital stock in the process of investment decision. By introducing anticipated capital stock into an economic model with investment delay, we construct a mixed functional differential system including delay and advanced variables. The system is converted to the one containing only delay by variable substitution. The equilibrium point of the system is obtained and its dynamical characteristics such as stability, Hopf bifurcation and its stability and direction are investigated by using the related theories of nonlinear dynamics. We carry out some numerical simulations to confirm these theoretical conclusions. The results indicate that both capital stock's anticipation and investment lag are the certain factors leading to the occurrence of cyclical fluctuations in the macroeconomic system. Moreover, the level of economic fluctuation can be dampened to some extent if investment decisions are made by the reasonable short-term forecast on capital stock. (C) 2015 Elsevier B.V. All rights reserved.

This paper is concerned with the discrete dynamics of an integro-differential model that describes the evolution of a population structured with respect to a continuous trait. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model.

In this paper, we present entropy satisfying schemes for solving an integro-differential 6 equation that describes the evolution of a population structured with respect to a continuous trait. 7 In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493–517] solutions are shown to converge 8 toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative 9 entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming 10 problem and can be computed by any well-established nonlinear programing algorithm. The schemes 11 are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive 12 and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is 13 presented to capture the global ESD for nonnegative initial data, which is made possible due to the 14 mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm 15 both accuracy and the entropy satisfying property and to underline the efficiency of capturing the 16 large time asymptotic behavior of numerical solutions in various settings. 17

In this paper, we present entropy satisfying schemes for solving an integro-differential equation that describes the evolution of a population structured with respect to a continuous trait. In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493-517] solutions are shown to converge toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming problem and can be computed by any well-established nonlinear programing algorithm. The schemes are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm both accuracy and the entropy satisfying property and to underline the efficiency of capturing the large time asymptotic behavior of numerical solutions in various settings.

This paper investigates the effect of thermal radiation on unsteady convection flow and heat transfer over a vertical permeable stretching surface in porous medium, where the effects of temperature dependent viscosity and thermal conductivity are also considered. By using a similarity transformation, the governing time-dependent boundary layer equations for momentum and thermal energy are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by the numerical shooting technique with fourth-fifth order Runge-Kutta scheme. Numerical results show that as viscosity variation parameter increases both the absolute value of the surface friction coefficient and the absolute value of the surface temperature gradient increase whereas the temperature decreases slightly. With the increase of viscosity variation parameter, the velocity decreases near the sheet surface but increases far away from the surface of the sheet in the boundary layer. The increase in permeability parameter leads to the decrease in both the temperature and the absolute value of the surface friction coefficient, and the increase in both the velocity and the absolute value of the surface temperature gradient.