CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy.First,a stochastic impulsive model describing the interaction and competition among normal cells,tumor cells and immune cells under periodically pulsed chemotherapy is established.Then,sufficient conditions for the extinction,non-persistence in the mean,weak and strong persistence in the mean of tumor cells are obtained.Finally,numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features.The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries.In clinical practice,doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.

The dynamics of delayed models for interactive wild and sterile mosquito populations

The sterile insect technique(SIT)has been applied as an alternative method to reduce or eradicate mosquito-borne diseases.To explore the impact of the sterile mosquitoes on controlling the wild mosquito populations,in this paper,we further extend the work in[J.Li,New revised simple models for interactive wild and sterile mosquito populations and their dynamics,J.Biol.Dyn.11(S2)(2017)316-333]and formulate delayed models for interactive wild and sterile mosquitoes,which can depict wild mosquito population undergoing distinct stages of development during a lifetime.By performing mathematical analysis,the threshold dynamics of the proposed models are explored,respectively.In particular,Hopf bifurcation phenomena are observed as the delay T is varying.Numerical examples illustrateourfindings.

Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated.

A class of metrics and foliations on tangent bundle of Finsler manifolds

Let （M, F） be a Finsler manifold, and let TMo be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of （TMo, G） and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.

Dynamical behavior of a mosquito population suppression model composed of two sub-models

In this paper,a new mosquito population suppression model with stage and sex structure is constructed,which is composed of two sub-models switching each other.Sterile mosquitoes are released with period T and remain sexually active for time T.For the case T<T,three thresholds T^(*),m^(*)and c^(*)are determined for the release period T and release amount c.According to the values of T and c in different ranges determined by these thresholds,we study the dynamical behavior of the system for different release strategies,mainly including the existence and stability of the mosquito-extinction equilibrium and positive periodic solutions.Finally,some numerical simulations are performed to illustrate our results.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

In this paper, a susceptible-vazcinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

Bifurcation analysis of a fractional-order SIQR model with double time delays

In this paper,a fractional-order delayed SIQR model with nonlinear incidence rate is investigated.Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle.By analyzing the associated characteristic equations,the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases.Besides,the critical values of time delays at which a Hopf bifurcation occurs are obtained,and the influence of the fractional order on the dynamics behavior of the system is also investigated.Numerically,it has been shown that when the endemic equilibrium is locally stable,the convergence rate of the system becomes slower with the increase of the fractional order.Besides,our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one.Furthermore,we find in all these three cases,the bifurcation values are very sensitive to the change of the fractional order,and they decrease with the increase of the order,which means the Hopf bifurcation gradually occurs in advance.

On U(n)-invariant strongly convex complex Finsler metrics

In this paper,we obtain a necessary and sufficient condition for a U(n)-invariant complex Finsler metric F on domains in C^(n) to be strongly convex,which also makes it possible to investigate the relationship between real and complex Finsler geometries via concrete and computable examples.We prove a rigid theorem which states that a U(n)-invariant strongly convex complex Finsler metric F is a real Berwald metric if and only if F comes from a U(n)-invariant Hermitian metric.We give a characterization of U(n)-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of the U(n)-invariant strongly pseudoconvex complex Finsler metric.Finally,we prove that the real geodesics of some U(n)-invariant complex Finsler metric restricted on the unit sphere S^(2n-1)■C^(n) share a specific property as that of the complex Wrona metric on C^(n).

Modeling impulsive resource inputs in host-parasitoid interactions with time delays

For the interaction of parasitoids and their insect hosts in the laboratory environment, a novel mathematical model with impulsive resource inputs, stage-structure, maturation delays and negative binomial distribution is proposed. Based on the adaptability of the insect host to the environment, we study the permanence of the system in two cases and gain conditions under which the host and parasitoid species can coexist with impulsive resource inputs. We also discuss the existence of the positive periodic solution when the system is permanent by applying a fixed point theory. Besides, we perform numerical simulations which not only confirm but also further enhance our theoretical results. The simulations show that when total input of resource is fixed, smaller input amounts with shorter periods of impulsive delivery produce smaller oscillation amplitudes for both the host and parasitoid populations at the juvenile stage. However, both the densities of adult host and adult parasitoid are not affected by the resource management strategy. Furthermore, we also reconfirm that larger maturation delays, either the host or the parasitoid＇s delay, lead to any more individuals staying at the inmature stage of the species, while the adult populations decline dramatically at the same time. On the other hand, larger host maturation delays promote the parasitoid＇s population growth at both stages, and the impact of parasitoid maturation delay on the host population is almost the same but not as dramatic. These findings give us a deeper understanding about the host parasitoid interaction in laboratory environment.