In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, ...including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.
We consider a 2×2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, ...logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x→±∞, reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.
Assuming that the prey refuge is proportional to the prey density if its population size is below a critical threshold, or constant if its size is above the threshold, this paper proposes, and ...qualitatively analyzes, a Leslie–Gower predator–prey model assuming alternative feeding and harvesting in predators, and a Holling II function as the predator functional response. From the results of the mathematical analysis to the predator–prey models with proportional or constant prey refuge, the proposed model retains the same bifurcation cases obtained for each model analyzed. However, appropriate alterations of the parameters representing the critical threshold of prey population size and harvest in predators allows the formation of at least one limit cycle, stable or unstable, that lives in both vector fields of the proposed model.
In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier–Stokes equations. This paper introduces an exchange of oxygen between the drop ...and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by{nt+u⋅∇n=Δn−∇⋅(n∇c)+n−n2,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut=Δu+u⋅∇u+∇P−n∇φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 in conjunction with the initial data (n,c,u)(⋅,0)=(n0,c0,u0) and the boundary conditions∂c∂ν=1−c,∂n∂ν=n∂c∂ν,u=0,x∈∂Ω,t>0. Here, the fluid drop is described by Ω⊂RN being a bounded convex domain with smooth boundary. Moreover, φ is a given smooth gravitational potential.
Requiring sufficiently smooth initial data, the existence of a unique global classical solution for N=2 is proved, where ‖n‖Lp(Ω) is bounded in time for all p<∞, as well as the existence of a global weak solution for N=3.
Le comportement d'une suspension bactérienne dans une goutte de liquide incompressible est décrit par les équations de chemotaxis-Navier–Stokes. Cet article introduit un échange d'oxygène entre la goutte et son environnement et une croissance logistique de la population bactérienne. Le système généralise le prototype{nt+u⋅∇n=Δn−∇⋅(n∇c)+n−n2,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut=Δu+u⋅∇u+∇P−n∇φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 associé à la donnée initiale (n,c,u)(⋅,0)=(n0,c0,u0) et aux conditions du bord∂c∂ν=1−c,∂n∂ν=n∂c∂ν,u=0,x∈∂Ω,t>0 d'où Ω⊂RN soit un domaine borné et convexe avec un bord lisse. En outre, φ soit un potentiel lisse gravitationnel. En supposant que la donnée initiale soit suffisamment régulière, on démontre l'existence d'une solution classique unique pour N=2 telle que ‖n‖Lp(Ω) est borné pour p<∞ et l'existence d'une solution faible globale pour N=3.
This paper sets up a rumor spreading model in online social networks based on the European fox rabies SIR model. The model considers the impact of changing number of online social network users, ...combines the transmission dynamics to set up a population dynamics of rumor spreading model in online social networks. Simulation is carried out on online social network, and results show that the new rumor spreading model is in accordance with the real propagation characteristics in online social networks.
•We established a rumor spreading model in online social network with varying total population size.•We considered the effects of new registered user growth rate per month on the spread of rumors.•We considered the effects of inactive user rate per month on the spread of rumors.•We first introduced the concept of environmental intake capacity in the online social network.•We simulated an online social network with a change in the total number of users.
We consider the nonnegative solutions of the following chemotaxis-growth system{ut=Δu−∇⋅(u∇v1)+μ(u−uα),x∈Ω,t>0,v1,t=Δv1−v1+v2,x∈Ω,t>0,v2,t=Δv2−v2+v3,x∈Ω,t>0,⋅⋅⋅vk,t=Δvk−vk+u,x∈Ω,t>0, in a smooth ...bounded domain Ω⊂Rn(n⩾2) with nonnegative initial data and null Neumann boundary condition, where μ>0,α>0. This model can be regard as a chemotaxis type model with general indirect signal. We reveal the influence of space dimension n and the number of equations k≥2 by showing that the solutions are globally bounded whenever α>min{2,max{1,n+22k}}.
A stochastic HIV viral model with both logistic target cell growth and nonlinear immune response function is formulated to investigate the effect of white noise on each population. The existence of ...the global solution is verified. By employing a novel combination of Lyapunov functions, we obtain the existence of the unique stationary distribution for small white noises. We also derive the extinction of the virus for large white noises. Numerical simulations are performed to highlight the effect of white noises on model dynamic behaviour under the realistic parameters. It is found that the small intensities of white noises can keep the irregular blips of HIV virus and CTL immune response, while the larger ones force the virus infection and immune response to lose efficacy.
•A stochastic logistic HIV model with CTL immune response is formulated.•The existence of ergodic stationary distribution for small noises is established.•The sufficient conditions for virus extinction are obtained.•The influence of white noises is investigated under the realistic parameters.
In this paper, we investigate Cauchy problem for the two-dimensional incompressible chemotaxis-Navier-Stokes equations with the lower fractional ...diffusion{∂tn+u⋅∇n−Δn=−∇⋅(n∇c)+λn−μn2,∂tc+u⋅∇c−Δc=−cn,∂tu+u⋅∇u+Λ2αu+∇P=−n∇ϕ, where Λ:=(−Δ)12 and α∈12,1. We obtain the global-in-time existence and uniqueness of weak solution to the equations for a class of large initial data by making use of the coupled structure of system and damping effect of the logistic source, and developing the L43(R2) estimate for vorticity.
This work aims to study the effect of environmental noises like white noise and telegraph noise modeled by Markov switching on the dynamical properties of a modified logistic-growth-type epidemic ...model. Applying the Khasminskii method via a suitable construction of the Lyapunov function, we prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the epidemic model which is the stability of stochastic system in the weak sense. It is noteworthy that regime switching can induce state transitions of diseases between extinction and persistence. Further, the pulse control scheme is developed to drive the disease to extinction when χ<0. Several numerical simulations are provided to validate our analytical findings.
•The long-time dynamic behavior and pulse control strategy of a regime-switching stochastic SIR mode with logistic growth are investigated.•The sufficient conditions for extinction and existence of a unique ergodic stationary distribution are deduced.•The pulse control scheme is developed to suppress the spread of diseases.•Some numerical simulations are presented to illustrate the efficiency of the theoretical results.
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