Reaction-diffusion Model Research Articles

A reaction–diffusion epidemic model with virus mutation and media coverage: Theoretical analysis and numerical simulation

In this paper, a novel COVID-19 reaction–diffusion model with virus mutation and media coverage is investigated. First, the solution’s uniform boundedness for the system is established. Then, the basic reproduction numbers for ordinary and mutant viruses spread in heterogeneous environments are defined. Furthermore, the endemic equilibrium’s asymptotic distribution for the system is explored. In addition, when one diffusion coefficient tends to zero and the other diffusion coefficients are greater than zero and fixed, the solution of the system will asymptotically approach endemic equilibrium. Next, a theoretical analysis of how high-frequency media coverage affects the development of the COVID-19 epidemic is conducted. Theoretical research shows that high-frequency media coverage will lead to the disappearance of the disease. Meantime, global sensitivity analysis on the basic reproduction numbers R01 and R02 are performed. Finally, theoretical simulations and instance predictions are carried out. Because of the complexity of the Shanghai epidemic and changes in management and control, the infection rates β1(t),β2(t) are given in the form of a piecewise function with more practical significance, and they are used to predict the epidemic trend of COVID-19 in Shanghai. Through a series of numerical simulations and analysis, the key indicators of the Shanghai COVID-19 epidemic are as follows: (1) The basic reproduction numbers in the early, middle, and late stages of COVID-19 are R¯0(1:34)=0.9152, R¯0(35:49)=3.1476, and R¯0(50:140)=0.6547, respectively; (2) This epidemic round in Shanghai will peak at 3,270 new daily confirmed cases on the 49th day (April 15); (3) The final size of the epidemic will reach 63,470 confirmed cases; (4) This round of COVID-19 epidemic in Shanghai, China, is expected to be fully cleared in late June to early July. The above conclusions are basically consistent with the facts. Of course, with the rise in temperature and strict control, the epidemic situation in Shanghai, China, is expected to be cleared earlier. Our results provide new ideas for preventing and controlling the COVID-19 epidemic.

In-situ polymerized zinc phytate chelated Si-C-P geopolymer hybrid coating constructed by incorporating chitosan oligosaccharide and DOPO for flame-retardant plywood

Zinc phytate (ZnPA) chelated geopolymer and in-situ polymerized chitosan oligosaccharide (COS)/DOPO explored a challenging and exciting flame retardant organic–inorganic hybrid coating via sol–gel method to find halogenated high-efficiency bio-flame retardants are widely used as decorative materials for flame-retardant plywood. Herein, the results show that the appropriated ZnPA (0.5 wt%) enhances the flame retardancy of the geopolymer coating, the peak of heat release rate (p-HRR) decreases from 136.09 to 99.39 kW·m−2, the fire growth index (FGI) dropped from 0.47 to 0.28 kW m−2 s−1, the fire performance index (FPI) climbs from 1.00 to 2.52 s·m2·kW−1, and the flame retardancy index (FRI) climbs from 1.00 to 2.52. Meanwhile, through the microstructural analysis of the residual layer, a strong, compact and non-flammable resilient residues is obtained after combustion; Moreover, the pyrolysis kinetics show that the Z.-L.-T. three-dimensional diffusion reaction model governs the pyrolysis of hybrid coatings. Appropriated ZnPA increases the pyrolysis Eα from 138.95 kJ·mol−1 to 160.75 kJ·mol−1 at 741 ∼ 890 ℃ and improves the thermal stability of the coating. The Si-C-P structure improves the formaldehyde adsorption rate by 46.5%. Therefore, this study, by preparing COS-pretreated ZnPA and DOPO to co-construct alkali-excited cenospheres-based Si-C-P coating, reveals the sustainable development of modified bio-based geopolymer flame-retardant coatings in the construction industry.

A diffusion-reaction computational study to reveal the depolymerization mechanisms of epoxy composites for recycling

It has been recently discovered that the thermosetting matrix of engineering composites can be fully depolymerized in organic solvents through covalent bond exchange reactions (BERs) between the polymer network and solvent molecules. This breakthrough enables the eco-friendly and sustainable recovery of valuable fiber reinforcements using mild processing conditions. However, current investigations have been limited to proof-of-concept experimental demonstrations, leaving unanswered questions regarding the influence of temperature, solvent choice, and fiber arrangement on composite depolymerization performance. These factors are crucial for the commercialization and widespread industrial implementation of this technique. To address this significant knowledge gap, this study aims to establish the relationship between composite depolymerization speed and various material and processing conditions. A multiscale diffusion-reaction computational model is defined based on the finite element method, which links the microscale BER rate to the continuum-level composite depolymerization kinetics. Specifically, it reveals how the processing temperature, solvent diffusivity, fiber content, and fiber arrangement affect the overall composite depolymerization speed. The study enhances our understanding of the underlying mechanisms of composite recycling using organic solvents. As a result, it provides valuable insights for industrial stakeholders, allowing them to optimize depolymerization conditions, make informed material selections, and develop suitable business models for waste management.

A peridynamic model for advection–reaction–diffusion problems

In this paper, a bond-based peridynamic (PD) advection–reaction–diffusion model is formulated to simulate processes involving chemical reactions, flows, and diffusion. In the formulation, the reaction rate, represented by a body force-like local term, depends on the local concentrations of the reactants, and the dependence varies according to the reaction type. Both linear and nonlinear dependencies are considered. A system of coupled PD equations for each substance is solved numerically. We verify and validate the PD advection–reaction–diffusion model by comparing results with those from a corresponding classical model and from experiments in both 1D and 2D, for a bimolecular reaction (A+B→AB) in a flowing aqueous environment, similar to purifying polluted water flowing in a channel, via chemical reactions. The model is also used to simulate bimolecular advection–reaction–diffusion in heterogeneous media with impermeable inclusions, and to simulate polymer degradation through hydrolysis and diffusion, similar to how biodegradable polymer implants dissolve.

Source-sink dynamics on networks: Persistence and extinction

Dynamics of populations with logistic growth leads to global persistence on arbitrary networks, whereas bistable dynamics is always associated with local extinction. In this paper we study a reaction-diffusion model on networks with a combination of both logistic and bistable reactions. We analyze a system of source-sink dynamics on networks with migration survival probabilities and derive conditions for strong persistence or local extinction. We show how the persistence or extinction of small populations depend on the diffusion strength, migration survival probability, network structure, and the sum and distribution of per capita growth rates.

Nonlinear stability analysis of a chemical reaction–diffusion system

A reaction–diffusion model, known as the Sel’kov–Schnakenberg model, is considered. The nonlinear stability of the constant steady state is studied by using a special Liapunov functional and a maximum principle for regular solutions.

Stability and bifurcation in a reaction–diffusion model with nonlinear boundary conditions

In this paper, we investigate the existence, stability, local and global bifurcation of the steady state solutions for a diffusive logistic model with nonlinear boundary conditions. Supercritical and subcritical bifurcation of steady state solutions are obtained by virtue of the Crandall–Rabinowitz theorem and the Lyapunov–Schmidt reduction. The local stability and the possibility of obtaining an Allee effect are also analyzed. Furthermore, the result on global bifurcation is obtained as well by employing the Rabinowitz theorem.

Global stability of a PDE-ODE model for acid-mediated tumor invasion

In this paper, we study the global dynamics of a general reaction-diffusion model based on acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition of healthy cells and tumor cells on tumor progression in the long term, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. A key feature of this model is the density-limited tumor diffusion term for tumor cells, which might give rise to the degeneracy of the parabolic equation. To overcome this difficulty, we combine the construction of suitable Lyapunov functionals and upper/lower solutions. This paper continues and improves the work begun in [15].

Free energy dissipation enhances spatial accuracy and robustness of self-positioned Turing pattern in small biochemical systems.

Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Turing proposed a general mechanism for pattern formation exemplified by a reaction-diffusion model with two chemical species in a large system. However, in small biological systems such as a cell, the existence of multiple Turing patterns and strong noise can lower the spatial order. Recently, a modified reaction-diffusion model with an additional chemical species is shown to stabilize the Turing pattern. Here, we study non-equilibrium thermodynamics of this three-species reaction-diffusion model to understand the relationship between energy cost and the performance of self-positioning. By using computational and analytical approaches, we show that beyond the onset of pattern formation the positioning error decreases as energy dissipation increases. In a finite system, we find that a specific Turing pattern exists only within a finite range of total molecule number. Energy dissipation broadens this range, which enhances the robustness of Turing pattern against molecule number fluctuations in living cells. The generality of these results is verified in a realistic model of the Muk system underlying DNA segregation in Escherichia coli, and testable predictions are made for the dependence of the accuracy and robustness of the spatial pattern on the ATP/ADP ratio.

LPF: a framework for exploring the wing color pattern formation of ladybird beetles in Python.

Biological pattern formation is one of the complex system phenomena in nature, requiring theoretical analysis based on mathematical modeling and computer simulations for in-depth understanding. We propose a Python framework named LPF to systematically explore the highly diverse wing color patterns of ladybirds using reaction-diffusion models. LPF supports GPU-accelerated array computing for numerical analysis of partial differential equation (PDE) models, concise visualization of ladybird morphs, and evolutionary algorithms for searching mathematical models with deep learning models for computer vision. LPF is is available on GitHub at https://github.com/cxinsys/lpf. Supplementary data are available at Bioinformatics online.

Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases

We study some reaction-diffusion systems for infectious diseases and investigate how the dynamics is impacted by the movement of populations and spatial heterogeneity of the environment. General conditions for the existence, uniqueness and stability of the coexistence steady states are established. Our analysis revealed two mechanisms for the coexistence of strains: (i) when the ratio of the transmission probability of two strains falls into some intermediate ranges; (ii) when the diffusion rate of two strains falls into proper ranges. Interestingly, when there is no coexistence of strains, it is possible for the “weak” strain to be dominant for some intermediate range of diffusion rates, in strong contrast to small and large diffusion cases where the “weak” strain always goes extinct.

Temporal control by cofactors prevents kinetic trapping in retroviral Gag lattice assembly

For retroviruses like HIV to proliferate, they must form virions shaped by the self-assembly of Gag polyproteins into a rigid lattice. This immature Gag lattice has been structurally characterized and reconstituted in vitro, revealing the sensitivity of lattice assembly to multiple cofactors. Due to this sensitivity, the energetic criterion for forming stable lattices is unknown, as are their corresponding rates. Here, we use a reaction-diffusion model designed from the cryo-ET structure of the immature Gag lattice to map a phase diagram of assembly outcomes controlled by experimentally constrained rates and free energies, over experimentally relevant timescales. We find that productive assembly of complete lattices in bulk solution is extraordinarily difficult due to the large size of this ∼3700 monomer complex. Multiple Gag lattices nucleate before growth can complete, resulting in loss of free monomers and frequent kinetic trapping. We therefore derive a time-dependent protocol to titrate or “activate” the Gag monomers slowly within the solution volume, mimicking the biological roles of cofactors. This general strategy works remarkably well, yielding productive growth of self-assembled lattices for multiple interaction strengths and binding rates. By comparing to the in vitro assembly kinetics, we can estimate bounds on rates of Gag binding to Gag and the cellular cofactor IP6. Our results show that Gag binding to IP6 can provide the additional time delay necessary to support smooth growth of the immature lattice with relatively fast assembly kinetics, mostly avoiding kinetic traps. Our work provides a foundation for predicting and disrupting formation of the immature Gag lattice via targeting specific protein-protein binding interactions.

Sodium lignosulfonate improves the compatibility of melamine polyphosphate with graphite-containing alkali-activated cenospheres-based coating for flame-retarding plywood

Cenospheres, a common industrial solid waste generated by coal-fired power plants, have become a significant environmental concern due to their emissions. This has created a need for practical solutions. In response, a novel organic-inorganic hybrid coating, the melamine polyphosphate@sodium lignosulfonate@graphite-containing alkali-activated cenospheres-based coating (MPP@LS@FG-ACC) has been synthesized using the sol-gel method for flame-retarding plywood. The research demonstrates that an appropriate LS (1.00 wt%) pretreated MPP improves its compatibility with the inorganic siliceous coating, enables more phosphorus-containing groups within MPP to participate in the as-formed Si-P-C flame retardant, which transforms into interpenetrating, robust, and nonflammable resilient residues during burning. Consequently, the flame retardancy is enhanced, as evidenced by a reduction in the peak heat release rate (from 98.65 to 63.73 kW·m−2), an increase in the flame retardancy index (from 1 to 2.58), and a decrease in smoke production by approximately 49 %. Additionally, the Z.-L.-T. Three-dimensional diffusion-reaction model governs the composite coatings' pyrolysis. Kinetics calculations clarify that an appropriate LS makes its Eα rise from 94.13 to 122.89 kJ·mol−1 at 970–771 ℃, corresponding to enhanced thermal stability for blocking heat and fire. Hence, this research presents an effective strategy for enhancing the compatibility of MPP by LS pretreatment and proposes an efficient approach for reusing cenospheres, improving the flame retardancy of cenospheres-derived building materials.

Spatio-temporal dynamics of the metabolome of climacteric fruit during ripening and postharvest storage.

Fruit quality traits are determined to a large extent by their metabolome. The metabolite content of climacteric fruit changes drastically during ripening and postharvest storage and has been investigated extensively. However, the spatial distribution of metabolites and how it changes in time has received much less attention as fruit are usually considered as homogenous plant organs. Yet, spatio-temporal changes of starch, which is hydrolysed during ripening, has been used for ages as a ripening index. As vascular transport of water, and, hence, convective transport of metabolites slows down in mature fruit and even stalls after detachment, spatio-temporal changes in their concentration are likely affected by diffusive transport of gaseous molecules that act as substrate (O2), inhibitor (CO2) or regulator (ethylene, NO) of the metabolic pathways that are active during climacteric ripening. In this review we discuss such spatio-temporal changes of the metabolome and how they are affected by transport of metabolic gases and gaseous hormones. As there are currently no measurement techniques available to measure the metabolite distribution repeatedly by nondestructive means, we introduce reaction-diffusion models as an in silico tool to compute it. We show how the different components of such a model can be integrated and used to better understand the role of spatio-temporal changes of the metabolome in ripening and postharvest storage of climacteric fruit that is detached from the plant and discuss future research needs.

Effect of the location of a protection zone in a reaction–diffusion model

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$ , and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$ ). We show the existence of two critical values $0< L_*\leq L^*$ , and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$ ; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$ ; only spreading happens when the length of protection zone is larger than $L^*$ . Based on the properties of $L_*$ , we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$ ), the protection zone should start from somewhere near $0$ ; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$ ), then the protection zone should start from somewhere away from $0$ , and as far away from $0$ as possible.

Generalized result on the global existence of positive solutions for a parabolic reaction diffusion model with a full diffusion matrix

"In this paper, we study the global existence in time of solutions for a parabolic reaction di usion model with a full matrix of di usion coe cients on a bounded domain. The technique used is based on compact semigroup methods and some estimates. Our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of nonlinearities."

Dynamical behaviour of an intraguild predator–prey model with prey refuge and hunting cooperation

ABSTRACT An intraguild predator–prey model including prey refuge and hunting cooperation is investigated in this paper. First, for the corresponding ordinary differential equation model, the existence and stability of all equilibria are given, and the existence of Hopf bifurcation, direction and stability of bifurcating periodic solutions are investigated. Then, for partial differential equation model, the diffusion-driven Turing instability is obtained. What is more, the existence and non-existence of the non-constant positive steady state of the reaction–diffusion model are established by the Leray–Schauder degree theory and some priori estimates. Next, some numerical simulations are performed to support analytical results. The results showed that prey refuge can change the stability of model and even have a stabilizing effect on this model, meanwhile the hunting cooperation can make such model without diffusion unstable, but make such model with diffusion stable. Lastly, a brief conclusion is concluded in the last section.

Epidemic dynamics and spatial segregation driven by cognitive diffusion and nonlinear incidence

AbstractWe propose susceptible‐infected‐susceptible epidemic reaction–diffusion models with cognitive movement and nonlinear incidence in a spatially heterogeneous environment. The cognitive dispersal term takes either random diffusion or symmetric diffusion. Building upon the ‐estimates of positive solutions under , we state the asymptotic dynamics for , . The numerical results reveal spatial segregation of susceptible and infected populations: (a) the heterogeneous random diffusion can segregate the population and reduce the infection fraction significantly; (b) the segregation phenomenon disappears as the ratio approaches one from below; (c) the disease‐free region strengthens the segregation induced by heterogeneous random diffusion; (d) the segregation governed by random diffusion is more sensitive to the incidence mechanism; (e) the distribution of steady states driven by symmetric diffusion is always similar to that by homogeneous diffusion.

In-situ polymerized zinc ions chelated Si-C-N cenospheres-based geopolymeric coating constructed by incorporating β-cyclodextrin and dopamine for flame-retarding plywood

Preparing halogen-free and efficient bio-flame retardants by recycling industrial solid waste is crucial for creating an environmentally friendly human settlement. Herein, β-cyclodextrin(β-CD) and dopamine(DA) are employed to chelate zinc ions (β-CD@Zn@DA) involved in cenospheres-based geopolymeric coating for further flame-retarding plywood. The results show that an appropriate dosage of β-CD (1.2 g, 1.00 wt%) improves the flame retardancy of the geopolymeric coating, and the peak heat release rate decreases from 107.17 kW·m−2 to 66.04 kW·m−2, the flame retardancy index increases from 1.00 to 1.41. Meanwhile, due to hydrogen bonding cross-linking and chelation of zinc ions, the coating is transformed into a dense and non-flammable ceramic-like blocking layer during firing to prevent heat or mass transfer. Furthermore, the pyrolysis kinetics identify that the Z.-L.-T. three-dimensional diffusion–reaction model governs the coatings’ pyrolysis, and an appropriate β-CD makes the pyrolysis Eα climb from 186.08 to 237.41 kJ·mol−1 at 980 ∼ 740 °C, corresponding to the resilient residues for blocking heat and fire. Therefore, in-situ polymerized zinc ions, chelated β-CD, and DA for facilely designing Si-C-N hybrid flame-retarding coating is explored, promoting the recycling of industrial solid waste and the development of more sustainable building materials.

Results for a Control Problem for a SIS Epidemic Reaction–Diffusion Model

This article is focused on investigating the mathematical model calibration of a reaction–diffusion system arising in the mathematical model of the spread of an epidemic in a society. We consider that the total population is divided into two classes of individuals, called susceptible and infectious, where a susceptible individual can become infectious, and that upon recovery, an infected individual can become susceptible again. We consider that the population lives in a spatially heterogeneous environment, and that the spread of the dynamics is governed by a reaction–diffusion system consisting of two equations, where the variables of the model are the densities of susceptible and infected individuals. In the reaction term, the coefficients are the rates of disease transmission and the rate of infective recovery. The main contribution of this study is the identification of the reaction coefficients by assuming that the infective and susceptible densities at the end time of the process and on overall spatial domain are observed. We apply the optimal control methodology to prove the main findings: the existence of positive solutions for the state system, the existence of at least one solution for the identification problem, the introduction of first-order necessary conditions, and the local uniqueness of optimal solutions.