In this paper, we consider a forager–exploiter model with logistic growth and nonlinear resource consumption ...ut=Δu−χ1∇⋅(u∇w)+η1(u−u2),x∈Ω,t>0,vt=Δv−χ2∇⋅(v∇u)+η2(v−v2),x∈Ω,t>0,wt=Δw−uw(1+u+v)γ−vw(1+u+v)θ−μw+r(x,t),x∈Ω,t>0in a smooth bounded domain Ω⊂Rn(n≥1) with nonnegative initial data and null Neumann boundary condition, where χ1,χ2,η1,η2,μ>0 and r is a given nonnegative function. We prove that if γ>0andθ>(n−2)+n+2,then the classical solution exists globally and remains bounded. Moreover, the large time behavior is investigated when r satisfies certain conditions.
A logistic SIS epidemic reaction-diffusion model with the nonlinear incidence rate SqIp in a spatially heterogeneous environment is concerned. The L∞-estimates of solutions are established under ...p,q>0 to explore the long-time behaviors of the model. In a special case, asymptotic dynamics and bifurcation behaviors under p=1 are analyzed in terms of the basic reproduction number. Asymptotic profiles of the endemic equilibrium for small and large diffusion rates of the susceptible and infected populations are investigated. It is found that the infectious disease is hard to control even under a policy of mobility regulation when 0<p<1 and q>1.
Novel coronavirus respiratory disease COVID-19 has caused havoc in many countries across the globe. In order to contain infection of this highly contagious disease, most of the world population is ...constrained to live in a complete or partial lockdown for months together with a minimal human-to-human interaction having far reaching consequences on countries’ economy and mental well-being of their citizens. Hence, there is a need for a good predictive model for the health advisory bodies and decision makers for taking calculated proactive measures to contain the pandemic and maintain a healthy economy. This paper extends the mathematical theory of the classical Susceptible–Infected–Removed (SIR) epidemic model and proposes a Generalized SIR (GSIR) model that is an integrative model encompassing multiple waves of daily reported cases. Existing growth function models of epidemic have been shown as the special cases of the GSIR model. Dynamic modeling of the parameters reflect the impact of policy decisions, social awareness, and the availability of medication during the pandemic. GSIR framework can be utilized to find a good fit or predictive model for any pandemic. The study is performed on the COVID-19 data for various countries with detailed results for India, Brazil, United States of America (USA), and World. The peak infection, total expected number of COVID-19 cases and thereof deaths, time-varying reproduction number, and various other parameters are estimated from the available data using the proposed methodology. The proposed GSIR model advances the existing theory and yields promising results for continuous predictive monitoring of COVID-19 pandemic.
•Proposed a generalized SIR (GSIR) framework for pandemic modeling and prediction.•An integrative model that captures the pandemic data via distinct waves emerging at different times.•Explicit expressions are derived for the susceptible, infected and removed populations.•Dynamic modeling of system parameters that can capture the impact of policy decisions.•GSIR model is tested on COVID-19 data and can serve as a good alternative to existing methods.
To explore pesticide uptake from soil into a growing potato, a moving-boundary dynamic model is proposed on the basis of the radical diffusion process of a chemical to a sphere. This model, which ...considers the logistic growth of the potato tuber, describes two hypothetical processes of chemical diffusion within a growing tuber. The model was tested in an illustrative case study for an application of chlorpyrifos. Results indicate that the distribution of chlorpyrifos concentrations along the potato radius is significantly affected by the tuber development. In comparison of our results to results from a classic model using a fixed boundary, the proposed dynamic model yields a quick and big jump for both the average concentration and bioconcentration factor (BCF) of chlorpyrifos in the potato as a result of the sigmoid expansion boundary. Overall, the dynamic model predicts that chlorpyrifos BCFs in the potato at harvest are higher than those using the classical model. In comparison of model results to measured uptake of chlorpyrifos into potato at harvest, the dynamic model shows better performance than the classical model. Our results provide a new perspective on pesticide uptake into potatoes and inform human health risk assessment for pesticides applied at different tuber growth stages.
Abstract Background The rising number of novel pathogens threatening the human population has motivated the application of mathematical modeling for forecasting the trajectory and size of epidemics. ...Materials and methods We summarize the real-time forecasting results of the logistic equation during the 2015 Ebola challenge focused on predicting synthetic data derived from a detailed individual-based model of Ebola transmission dynamics and control. We also carry out a post-challenge comparison of two simple phenomenological models. In particular, we systematically compare the logistic growth model and a recently introduced generalized Richards model (GRM) that captures a range of early epidemic growth profiles ranging from sub-exponential to exponential growth. Specifically, we assess the performance of each model for estimating the reproduction number, generate short-term forecasts of the epidemic trajectory, and predict the final epidemic size. Results During the challenge the logistic equation consistently underestimated the final epidemic size, peak timing and the number of cases at peak timing with an average mean absolute percentage error (MAPE) of 0.49, 0.36 and 0.40, respectively. Post-challenge, the GRM which has the flexibility to reproduce a range of epidemic growth profiles ranging from early sub-exponential to exponential growth dynamics outperformed the logistic growth model in ascertaining the final epidemic size as more incidence data was made available, while the logistic model underestimated the final epidemic even with an increasing amount of data of the evolving epidemic. Incidence forecasts provided by the generalized Richards model performed better across all scenarios and time points than the logistic growth model with mean RMS decreasing from 78.00 (logistic) to 60.80 (GRM). Both models provided reasonable predictions of the effective reproduction number, but the GRM slightly outperformed the logistic growth model with a MAPE of 0.08 compared to 0.10, averaged across all scenarios and time points. Conclusions Our findings further support the consideration of transmission models that incorporate flexible early epidemic growth profiles in the forecasting toolkit. Such models are particularly useful for quickly evaluating a developing infectious disease outbreak using only case incidence time series of the early phase of an infectious disease outbreak.
We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. It is a 2×2 reaction-diffusion system describing the ...interaction of cells and a chemical signal. We study Cauchy problem for the original system and its transformed system, which is one of hyperbolic-parabolic balance laws. Our initial data are generic perturbations of a constant ground state, i.e. the initial mass of perturbation is non-zero. In the case of non-diffusive chemical, we obtain optimal L2 time decay rates for the solution with finite initial data. In the case of diffusive chemical, optimal L2 rates are also obtained with additional assumption on the smallness of the initial amplitude but still allowing large oscillation.
Solar power production in Europe has raised from about 130 MW to 110 GW of installed capacity (corresponding to 90 GWh to 120 TWh in annual electricity generation) during the present century. ...Together with wind power, it constitutes the largest growth within renewable energy sources in the last decades. At present however, clear signs of saturation can be observed in the leading areas of solar power in Europe. Here, the development of solar power in Europe is analysed and, for the three leading countries (Germany, Italy, Spain) a logistic growth path at the national level and a proportionality between saturation level of the growth curve of each country and its gross domestic product (GDP) is found. The sum of the next three countries (France, UK, Belgium) is well described by a logistic path with a time offset relative to the first group of three, and the sum of the two logistic paths, representing in total about 85% of European solar power production, describes the growth pattern in the corresponding area very well. Based on this, an estimate of a future saturation level for solar power in Europe is obtained by extrapolation. Finally, a model based on logistic growth patterns and learning curves that links solar power production data to investment data, is proposed. The proposed model is validated and calibrated on historical European data and extrapolated into the future. In a future scenario where European investments continue to decrease, a saturation level that is fully in line with our GDP based 200 TWh/y estimate is found and the application of the findings is discussed in a global context.
•Solar power production in the leading European areas show clear signs of saturation.•Proportionality between national saturation levels and gross domestic products is observed.•A model linking investment and solar power production is proposed and validated.•With current investment trends, solar power is not likely to fill the foreseen role.
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•Early life was an open chemical system of reversible reactions with branched autocatalytic cycles (ACs).•Individual ACs show logistic growth like a biological population.•Pairs of ...ACs can interact just like biological species, including competition, predation, etc.•Simple chemical ecosystems can exhibit successional dynamics and historical contingence, implying evolution.•A chemical ecosystem framework explains how adaptive evolution can precede genetic encoding.
It is becoming widely accepted that very early in life’s origin, even before the emergence of genetic encoding, reaction networks of diverse small chemicals might have manifested key properties of life, namely self-propagation and adaptive evolution. To explore this possibility, we formalize the dynamics of chemical reaction networks within the framework of chemical ecosystem ecology. To capture the idea that life-like chemical systems are maintained out of equilibrium by fluxes of energy-rich food chemicals, we model chemical ecosystems in well-mixed compartments that are subject to constant dilution by a solution with a fixed concentration of input chemicals. Modelling all chemical reactions as fully reversible, we show that seeding an autocatalytic cycle with tiny amounts of one or more of its member chemicals results in logistic growth of all member chemicals in the cycle. This finding justifies drawing an instructive analogy between an autocatalytic cycle and a biological species. We extend this finding to show that pairs of autocatalytic cycles can exhibit competitive, predator-prey, or mutualistic associations just like biological species. Furthermore, when there is stochasticity in the environment, particularly in the seeding of autocatalytic cycles, chemical ecosystems can show complex dynamics that can resemble evolution. The evolutionary character is especially clear when the network architecture results in ecological precedence, which makes a system’s trajectory historically contingent on the order in which cycles are seeded. For all its simplicity, the framework developed here helps explain the onset of adaptive evolution in prebiotic chemical reaction networks, and can shed light on the origin of key biological attributes such as thermodynamic irreversibility and genetic encoding.
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