Nonautonomous Model Research Articles

Modelling malaria dynamics with partial immunity and protected travellers: optimal control and cost-effectiveness analysis

A mathematical model of malaria dynamics with naturally acquired transient immunity in the presence of protected travellers is presented. The qualitative analysis carried out on the autonomous model reveals the existence of backward bifurcation, where the locally asymptotically stable malaria-free and malaria-present equilibria coexist as the basic reproduction number crosses unity. The increased fraction of protected travellers is shown to reduce the basic reproduction number significantly. Particularly, optimal control theory is used to analyse the non-autonomous model, which incorporates four control variables. The existence result for the optimal control quadruple, which minimizes malaria infection and costs of implementation, is explicitly proved. Effects of combining at least any three of the control variables on the malaria dynamics are illustrated. Furthermore, the cost-effectiveness analysis is carried out to reveal the most cost-effective strategy that could be implemented to prevent and control the spread of malaria with limited resources.

Robustness of Attractors for Non-autonomous Kirchhoff Wave Models with Strong Nonlinear Damping

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: $$u_{tt}- (1+\epsilon \Vert \nabla u\Vert ^2)\Delta u-\sigma (\Vert \nabla u\Vert ^2) \Delta u_t+f(u)=g(x,t)$$, where $$\epsilon \in [0,1]$$ is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: $$1\le p<p^{**}\equiv \frac{N+4}{(N-4)^+}$$, (i) the related process has a pullback attractor $${\mathscr {A}}_\epsilon $$ in natural energy space $${\mathcal {H}}=(H^1_0\cap L^{p+1})\times L^2$$ for each $$\epsilon $$, and it is upper semicontinuous on the perturbation $$\epsilon $$; (ii) the related process has a partially strong pullback exponential attractor $${\mathcal {M}}_\epsilon $$ for each $$\epsilon $$, and it is Holder continuous on $$\epsilon \in [0,1]$$. These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case.

Time series analysis for multi-dimensional dynamical systems combining wavelet transformation and local principal component analysis

Abstract To analyze trajectories for systems of many degrees of freedom, we propose a new method called wavelet local principal component analysis (WlPCA) combining the wavelet transformation and local PCA in time. Our method enables us to reduce the dimensionality of time series both in degrees of freedom and frequency so that characteristic features of oscillatory behavior can be captured. To test the new method, we apply WlPCA to a non-autonomous model of multiple degrees of freedom, the Froeschlé maps of $N=2$ and $N=4$, which correspond to autonomous systems of three and five degrees of freedom, respectively. The eigenvalues and eigenvectors obtained by WlPCA reveal those times when frequency variation exhibits switching between relatively stationary features. Moreover, further analyses indicate which degrees of freedom and frequencies are involved in the switching. We confirm that the switching corresponds to the onset of transport in phase space. These findings imply that, even for systems of larger degrees of freedom, barriers can exist in phase space that block transport for a finite time, thereby dividing the phase space into multiple quasi-stationary regions. Thus, our method is promising for understanding dynamics in systems of many degrees of freedom, such as vibrational-energy redistribution in molecules.

Nonautonomous dynamics of acute cell injury.

Medical conditions due to acute cell injury, such as stroke and heart attack, are of tremendous impact and have attracted huge amounts of research effort. The biomedical research that seeks cures for these conditions has been dominated by a qualitative, inductive mind-set. Although the inductive approach has not been effective in developing medical treatments, it has amassed enough information to allow construction of quantitative, deductive models of acute cell injury. In this work we develop a modeling approach by extending an autonomous nonlinear dynamic theory of acute cell injury that offered new ways to conceptualize cell injury but possessed limitations that decrease its effectiveness. Here we study the global dynamics of the cell injury theory using a nonautonomous formulation. Different from the standard scenario in nonlinear dynamics that is determined by the steady state and fixed points of the model equations, in this nonautonomous model with a trivial fixed point, the system property is dominated by the transient states and the corresponding dynamic processes. The model gives rise to four qualitative types of dynamical patterns that can be mapped to the behavior of cells after clinical acute injuries. The nonautonomous theory predicts the existence of a latent stress response capacity (LSRC) possessed by injured cells. The LSRC provides a theoretical explanation of how therapies, such as hypothermia, can prevent cell death after lethal injuries. The nonautonomous theory of acute cell injury provides an improved quantitative framework for understanding cell death and recovery and lays a foundation for developing effective therapeutics for acute injury.

Impact of weather seasonality and sexual transmission on the spread of Zika fever

We establish a compartmental model to study the transmission of Zika virus disease including spread through sexual contacts and the role of asymptomatic carriers. To incorporate the impact of the seasonality of weather on the spread of Zika, we apply a nonautonomous model with time-dependent mosquito birth rate and biting rate, which allows us to explain the differing outcome of the epidemic in different countries of South America: using Latin Hypercube Sampling for fitting, we were able to reproduce the different outcomes of the disease in various countries. Sensitivity analysis shows that, although the most important factors in Zika transmission are the birth rate of mosquitoes and the transmission rate from mosquitoes to humans, spread through sexual contacts also highly contributes to the transmission of Zika virus: our study suggests that the practice of safe sex among those who have possibly contracted the disease, can significantly reduce the number of Zika cases.

An Effective Disturbance-Observer-Based Nonlinear Controller for a Pump-Controlled Hydraulic System

Researches for improving the control performance of hydraulic systems in both control accuracy and energy efficiency have never stopped in aerospace and industrial applications. The existence of nonlinearities, uncertainties, and unknown terms in the system dynamics, however, significantly limits the desired performance. To realize improvements by dealing with these problems, an advanced position controller incorporated with effective disturbance observers (DOs) applied for a pump-controlled hydraulic system is proposed in this article. Here, uncertainties are considered as certainties (nominal terms) and their deviations. To eliminate certain nonlinearities in the system dynamics, the proposed controller is designed based on a simplified robust sliding-mode-backstepping scheme. The lumped unknown terms, which mainly degrade the performance of the controller, in pressure dynamics and force dynamics are expanded by using equivalent nonautonomous models. To effectively approximate the terms and to ensure usability of the estimated results inside the control framework, two different high-order DOs are developed. Asymptotic convergences of these observers are achieved by adopting nonlinear combinations of the estimation errors. Effectiveness and feasibility of the designed observers and the closed-loop system for an asymptotically tracking performance in the presence of bounded time-varying disturbances are then confirmed by Lyapunov-based proofs and extensive experiments.

Families of halo orbits in the elliptic restricted three-body problem for a solar sail with reflectivity control devices

Solar sail halo orbits designed in the Sun-Earth circular restricted three-body problem (CR3BP) provide inefficient reference orbits for station-keeping since the disturbance due to the eccentricity of the Earth’s orbit has to be compensated for. This paper presents a strategy to compute families of halo orbits around the collinear artificial equilibrium points in the Sun-Earth elliptic restricted three-body problem (ER3BP) for a solar sail with reflectivity control devices (RCDs). In this non-autonomous model, periodic halo orbits only exist when their periods are equal to integer multiples of one year. Here multi-revolution halo orbits with periods equal to integer multiples of one year are constructed in the CR3BP and then used as seeds to numerically continue the halo orbits in the ER3BP. The linear stability of the orbits is analyzed which shows that the in-plane motion is unstable while the out-of-plane motion is neutrally stable and a bifurcation is identified. Finally, station-keeping is performed which shows that a reference orbit designed in the ER3BP is significantly more efficient than that designed in the CR3BP, while the addition of RCDs improve station-keeping performance and robustness to uncertainty in the sail lightness number.

A non-autonomous impulsive food-chain model with delays

A non-autonomous almost periodic prey–predator system with impulsive effects and multiple delays is proposed in this paper, Holling’s type-IV systems and ratio-dependent functional responses are also involved in the model. By applying absolute inequalities, integral inequalities, differential inequalities and the mean-value theorem and other mathematical analysis techniques, we obtain some sufficient conditions which guarantee the permanence of the system. Moreover, we obtain the existence and the uniqueness of the almost periodic solution which is uniformly asymptotically stable by constructing a series of Lyapunov functionals. Finally, we present several numerical examples to verify the theoretical results and present some discussions of pest management in the agricultural ecological system.

How vector feeding preference through an infectious host relates to the seasonal transmission rates in a mathematical vector‐host model

Vector‐borne diseases, such as leishmaniasis, dengue, malaria, and yellow fever, transmitted by microparasites show periodic fluctuations in their prevalence. The novelty of this research is to assess the relationship between the vector feeding preference for an infectious host and the annual seasonal transmission through a vector‐host mathematical model. For the first time, numerical simulations illustrate that by increasing the vector feeding preference value in the transmission dynamics, periodic fluctuations accentuate and the endemic equilibrium average increases in vector and host populations. Moreover, increasing the vector feeding preference value, the amplitude strengthens for the infectious host and vector populations. This periodic behavior shows a similar pattern with the Peruvian incidence data from 2000 to 2016 for Andean cutaneous leishmaniasis provided by the Ministry of Health of Peru (MINSA). In addition, using the Floquet theory, the time average method and the linear operator method provides for the first time that the basic reproduction number for a nonautonomous system depends explicitly on the vector feeding preference for the infectious host. The nonautonomous model system shows that is a threshold parameter for the local stability of the disease‐free periodic solution. Therefore, the vector feeding preference is an important factor that should be considered and attended to for future research. Public and veterinary health in Peru and other countries should consider the vector feeding preference for specific host to vector control.

Convergence of solutions of nonautonomous Nicholson's Blowflies model with impulses

This paper deals with a nonautonomous Nicholson's blowflies model with impulses. It is shown that under the proper conditions every positive solution of the model approaches to a constant as t tends to infinity.

Stability Analysis of Single-Phase Low-Voltage AC Microgrids With Constant Power Terminals

This express brief presents the stability analysis of single-phase microgrids (SP-MG) operating under master–slave connection with constant power terminals. The SP-MG is composed of linear elements, nonlinear loads, and distributed generators modeled as PQ constant terminals interconnected through power electronic converters. Lyapunov’s direct method through a Hamiltonian representation of the grid is used to demonstrate stability. The non-autonomous model of the SP-MG is transformed into an autonomous equivalent model based on the dynamics of the error. The proposed analysis shows that if there is an admissible trajectory ${x} ^{\boldsymbol {\star }}$ solution of the power flow equations, then the SP-MG is stable in the sense of Lyapunov.

Optimization of LNG terminal parameters for a wide range of gas tanker sizes: the case of the port of Świnoujście

LNG terminals are built to handle tankers of specific size, with cargo capacities within a narrow scope. This is related to the differences in cargo, fender and mooring systems used for LNG tanker handling of various sizes. The research problem solved in the article is the development of the method for optimal design of a universal LNG cargo handling facility that enables safe operations of LNG tankers in a wide range of cargo capacity that covers almost entire spectrum of global fleet tanker sizes. The article presents a methodology of optimizing the parameters of LNG cargo terminals to accommodate both small bunker ships with cargo capacity of 500 m3 (50 metres in length) to Q-flex type tankers capable of carrying up to 220 000 m3 (320 m in length). The authors have determined conditions for the safe operation of these tankers in sea LNG terminals and described differences in the construction of cargo, fendering and mooring systems. The optimization of both location and terminal parameters for a wide range of gas tanker sizes as well as approach channels leading to the LNG berths was performed using a specially designed two-stage simulation method of optimization. In the first stage the best location of a universal LNG terminal and its berths in the existing port basin is determined. The second stage defines optimal parameters of approach waterways to the berths of a universal LNG terminal. The optimization criterion at both stages was the minimization of the costs to build and to operate a universal LNG terminal. The developed optimization methodology was actually used in the design of the universal LNG terminal in the outer port of Świnoujście. The tests made use of real time simulation (RTS) and non-autonomous models of ships, in which ship movement is controlled by a human (pilot, captain). Simulation tests were performed on a multi-bridge ship handling Polaris simulator with a 3D projection, from Kongsberg Maritime AS. This full-mission bridge simulator (FMBS) is located at the Marine Traffic Engineering Centre, Maritime University of Szczecin. Two simulation ship movement models were built and verified for testing the manoeuvres of port entry and berthing. These are: Q-flex type tanker (length: 320 m) and an LNG bunker ship, 6,000 m3 capacity, 104 m in length. The test results were used in the design of the universal LNG terminal in the outer port of Świnoujście and approach waterways leading to the berths (now this investment project is in progress).

Computational modeling of pitting corrosion

Abstract Pitting corrosion damage is a major problem affecting material strength and may result in difficult to predict catastrophic failure of metallic material systems and structures. Computational models have been developed to study and predict the evolution of pitting corrosion with the goal of, in conjunction with experiments, providing insight into pitting processes and their consequences in terms of material reliability. This paper presents a critical review of the computational models for pitting corrosion. Based on the anodic reaction (dissolution) kinetics at the corrosion front, transport kinetics of ions in the electrolyte inside the pits, and time evolution of the damage (pit growth), these models can be classified into two categories: (1) non-autonomous models that solve a classical transport equation and, separately, solve for the evolution of the pit boundary; and (2) autonomous models like cellular automata, peridynamics, and phase-field models which address the transport, dissolution, and autonomous pit growth in a unified framework. We compare these models with one another and comment on the advantages and disadvantages of each of them. We especially focus on peridynamic and phase-filed models of pitting corrosion. We conclude the paper with a discussion of open areas for future developments.

A Mathematical Model for the Population Dynamics of Malaria with a Temperature Dependent Control

In this work, we develop a mathematical model for malaria transmission that investigates the impact of temperature on the efficacy of ITNs. The disease free state of the autonomous version of the formulated model is seen to be globally asymptotically stable in the absence of disease induced mortality when the associated reproduction number is less than unity. Also, when the associated reproduction number of the non-autonomous model is greater than unity, the disease persist in the population. Sensitivity and uncertainty analysis of the autonomous model, using the associated reproduction number, number of infected humans and vectors as response functions, show that the probability of transmission from human to vectors ( $$\beta _{hv}$$ ), maximum biting rate of vectors ( $$b_{max}$$ ), vector carrying capacity ( $$k_v$$ ) as well as ITN efficacy and coverage ( $$\varepsilon _b$$ and $$b_o$$ ) have the most impact on all three response functions. In the non-autonomous model, for temperature values in the range $$[18\pm 4-32\pm 4]\,^\circ $$ C and using the number of infected humans and vectors as response functions, it can be seen that the bed-net coverage lost its significance for temperature values in the range $$[22\pm 4-30\pm 4]\,^\circ $$ C and $$[22\pm 4-26\pm 4] \,^\circ $$ C respectively. Numerical simulations of the time-averaged reproduction number for the case where the efficacy of the bed-net is temperature-dependent showed that temperature values in the range [22–27] $$\,^\circ $$ C effects the efficacy of the bed-nets most. Numerical simulations of the non-autonomous model for the case where ITN efficacy is temperature-dependent, we determine temperature ranges that make for increase in the effectiveness of the bed-nets in the fight against malaria.

Application of Optimal Control Theory to Newcastle Disease Dynamics in Village Chicken by Considering Wild Birds as Reservoir of Disease Virus

In this study, an optimal control theory was applied to a nonautonomous model for Newcastle disease transmission in the village chicken population. A notable feature of this model is the inclusion of environment contamination and wild birds, which act as reservoirs of the disease virus. Vaccination, culling, and environmental hygiene and sanitation time dependent control strategies were adopted in the proposed model. This study proved the existence of an optimal control solution, and the necessary conditions for optimality were determined using Pontryagin’s Maximum Principle. The numerical simulations of the optimal control problem were performed using the forward–backward sweep method. The results showed that the use of only the environmental hygiene and sanitation control strategy has no significant effect on the transmission dynamics of the Newcastle disease. Additionally, the combination of vaccination and environmental hygiene and sanitation strategies reduces more number of infected chickens and the concentration of the Newcastle disease virus in the environment than any other combination of control strategies. Furthermore, a cost-effective analysis was performed using the incremental cost-effectiveness ratio method, and the results showed that the use of vaccination alone as the control measure is less costly compared to other control strategies. Hence, the most effective way to minimize the transmission rate of the Newcastle disease and the operational costs is concluded to be the timely vaccination of the entire population of the village chicken, improvement in the sanitation of facilities, and the maintenance of a hygienically clean environment.

GLOBAL DYNAMICS OF A SEASONAL MATHEMATICAL MODEL OF SCHISTOSOMIASIS TRANSMISSION WITH GENERAL INCIDENCE FUNCTION

In this paper, we investigate a nonautonomous and an autonomous model of schistosomiasis transmission with a general incidence function. Firstly, we formulate the nonautonomous model by taking into account the effect of climate change on the transmission. Through rigorous analysis via theories and methods of dynamical systems, we show that the nonautonomous model has a globally asymptotically stable disease-free periodic equilibrium when the associated basic reproduction ratio [Formula: see text] is less than unity. Otherwise, the system admits at least one positive periodic solutions if [Formula: see text] is greater than unity. Secondly, using the average of periodic functions, we further derive the autonomous model associated with the nonautonomous model. Therefore, we show that the disease-free equilibrium of the autonomous model is locally and globally asymptotically stable when the associated reproduction ratio [Formula: see text] is less than unity. When [Formula: see text] is greater than unity, the existence and global asymptotic stability of the endemic equilibrium is established under certain conditions. Finally, using linear and nonlinear specific incidence function, we perform some numerical simulations to illustrate our theoretical results.

Nontrivial periodic solution of a stochastic non-autonomous model with biodegradation of microcystins

In this paper, a stochastic non-autonomous model with biodegradation of microcystins is considered. Firstly, the threshold of the model that determines whether the microcystin-degrading bacteria extinct is obtained. Then, we investigate the persistence of species: it shows that there exists a unique nontrivial positive periodic solution. Finally, we conclude our study by a short discussion.

Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions

There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle–node bifurcations. In particular, the term “tipping”, or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle–node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter p(τ) and its corresponding “dynamic” (time-dependent) saddle–node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking timeτˆ at which the corresponding autonomous system with a time-independent parameter pa=p(τˆ) undergoes a bifurcation. A dimensionless parameter α=λp03V−2 is introduced, in which λ is the curvature of the autonomous saddle–node bifurcation according to parameter p(τ), which has an initial value of p0 and a constant rate of change V. We find that the breaking time τˆ is always less than the actual point of no return τ∗ after which the critical transition is irreversible; specifically, the relation τ∗−τˆ≃2.338(λV)−13 is analytically obtained. For a system with a small λV, there exists a significant window of opportunity (τˆ,τ∗) during which rapid reversal of the environment can save the system from catastrophe.

Bilberry vs. cowberry in a Scots pine boreal forest: Exclusion or coexistence in a post-fire succession?

Although the famous competition exclusion principle was thoroughly elaborated in the framework of multispecies population models, its illustrations with field data were facing certain obstacles. The data we deal with were collected by 6 consecutive examinations of bilberry (Vaccinium myrtillus) and cowberry (V. vitis-idaea) presence/absence in 20 × 20 cm quadrats located along permanent transects in a Scots pine boreal forest. These data feature a great sample size (2000 quadrats×6 examinations) and a 5-year interval for each pair of examination/re-examination, and the data have enabled us to construct a Markov chain model for the fine-scale dynamics of these dominant species at the later stages of post-fire succession. The discrete-time nonautonomous model consists of the five time-dependent, one-step transition matrices that describe changes occurring in the quadrat status for the 5 years in the following 4 terms: species-free quadrat; bilberry alone, cowberry alone, both species. Calibrated on the data, each one-step transition matrix transforms the current distribution of quadrats among their statuses exactly to the next one observed and yields a steady-state distribution as its (time-specific) positive eigenvector. Step-by-step variations in the elements of transition matrices do affect the components of the corresponding eigenvectors, but do not change the dominance relationships among them.Calculated as the 5-th power matrix root from the product of those five transition matrices (in the chronological order within their product), the geometric average, treated again as a transition matrix, provides for the spatial and temporal characteristics of the observed dynamics summarizing the whole observation period. We reveal the expected terminal stable outcome of species dynamics to be a distribution where 37% of quadrats are occupied by the bilberry alone, 11% by cowberry alone, 38% by the both species, and 14% be species-free. It is shown for the first time that the share of cowberry-alone quadrats decreases notably in the course of succession, whereas the share of species coexistence quadrats increase.The averaged characteristics of cyclicity and status duration in the fine-scale dynamics enable us to estimate the total time of the secondary post-fire succession as 46–50 years (till reaching a distribution close to the terminal one). Of the four statuses considered, the cowberry-alone quadrats have the least mean duration time (9 years), while the bilberry-and-cowberry quadrats have the longest time (18 years), which are explainable by different life spans of the shoots in proper species.

Dynamics analysis of stochastic epidemic models with standard incidence

In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.