Thresholds for extinction and proliferation in a stochastic tumour-immune model with pulsed comprehensive therapy

Periodical applications of immunotherapy and chemotherapy play significant roles in cancer treatment and studies have shown that the evolution of tumour cells is subject to random events. In order to capture the effects of such noise we developed a stochastic tumour-immune dynamical model with pulsed treatment to describe combinations of immunotherapy with chemotherapy. By using theorems of the impulsive stochastic dynamical equation, the tumour free solution and the global positive solution of the proposed system were investigated. We then show that the expectations of the solutions are bounded. Furthermore, threshold conditions for extinction, non-persistence in the mean, weak persistence and stochastic persistence of tumour cells are provided. The results reveal that comprehensive therapy or noise can dominate the evolution of tumours. Finally, biological implications are addressed and a conclusion is presented

Complex dynamics of an impulsive chemostat model

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincar´e map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-k (k ≥ 2) periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed

Modelling effects of a chemotherapeutic dose response on a stochastic tumour-immune model

A stochastic tumour-immune dynamical system with pulsed chemotherapeutic dose response is proposed to study how environmental noise affects the evolution of tumours. Firstly, the explicit expression of a tumour-free solution is obtained and then we show that the proposed system exists with a globally asymptotically stable positive solution under certain conditions. Secondly, threshold criteria ensuring the eradication and persistence of tumours are provided. Numerical investigations were carried out to address the effects of key factors on the tumours. The results reveal that environmental noise can dominate all of the tumour dynamics, but comprehensive therapy can not only accelerate the eradication of tumours, but also avoid the disadvantages of a single therapy

Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and \widetilde{\mathcal{R}}_0 < 1 , smokers will eventually become extinct. If \widetilde{\mathcal{R}}_0 > 1 , smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as R_0^s > 1 . Finally, conclusions are explained by numerical simulations

Bifurcation analysis of a tumour-immune model with nonlinear killing rate as state-dependent feedback control

Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincar´e map of the proposed model and then address the existence and orbital asymptotically stability of the model’s tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincar´e mapping, we investigate the model’s transcritical and subcritical pitchfork bifurcations with respect to the key parameter

Threshold dynamics of a stochastic model of intermittent androgen deprivation therapy for prostate cancer

Intermittent androgen deprivation therapy is often used to treat prostate cancer, but there are few mathematical modelling studies of it. To explore the mechanisms of such therapy, we describe intermittent therapy with impulsive differential equations, then we propose a novel mathematical model of intermittent androgen deprivation therapy with white noise. We first studied the model’s basic properties including the existence and uniqueness of the solution. By using the theory of stochastic differential equations, we investigated the thresholds for the extinction and persistence of prostate cancer cells, which are markedly affected by antigenicity of tumours and noise parameters. Moreover, sufficient conditions for the stationary distribution and ergodicity of the system are provided. The results show that reducing the period of pulsed interventions or increasing the dosages (or frequencies) of the therapy will be helpful for curing prostate cancer

Dynamic Behavior of an Interactive Mosquito Model under Stochastic Interference

For decades, mosquito-borne diseases such as dengue fever and Zika have posed serious threats to human health. Diverse mosquito vector control strategies with different advantages have been proposed by the researchers to solve the problem. However, due to the extremely complex living environment of mosquitoes, environmental changes bring significant differences to the mortality of mosquitoes. This dynamic behavior requires stochastic differential equations to characterize the fate of mosquitoes, which has rarely been considered before. Therefore, in this article, we establish a stochastic interactive wild and sterile mosquito model by introducing the white noise to represent the interference of the environment on the survival of mosquitoes. After obtaining the existence and uniqueness of the global positive solution and the stochastically ultimate boundedness of the stochastic system, we study the dynamic behavior of the stochastic model by constructing a series of suitable Lyapunov functions. Our results show that appropriate stochastic environmental fluctuations can effectively inhibit the reproduction of wild mosquitoes. Numerical simulations are provided to numerically verify our conclusions: the intensity of the white noise has an effect on the extinction and persistence of both wild and sterile mosquitoes

Robust stability analysis of impulsive complex-valued neural networks with mixed time delays and parameter uncertainties

Abstract The robust stability for the impulsive complex-valued neural networks with mixed time delays is considered in this paper. Based on the homeomorphic mapping theorem, some sufficient conditions are proposed for the existence and uniqueness of the equilibrium point. By constructing appropriate Lyapunov–Krasovskii functions and employing modulus inequality techniques, the global robust stability theorem is obtained for the neural networks in complex domain. Finally, numerical simulations confirm the stability of the system and manifest that the complex-valued neural networks work efficiently on storing and retrieving the image patterns

Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties

Abstract The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results

Modeling and analysis of a multilayer solid tumour with cell physiological age and resource limitations

We study an avascular spherical solid tumour model with cell physiological age and resource constraints in vivo. We divide the tumour cells into three components: proliferating cells, quiescent cells and dead cells in necrotic core. We assume that the division rate of proliferating cells is nonlinear due to the nutritional and spatial constraints. The proportion of newborn tumour cells entering directly into quiescent state is considered, since this proportion can respond to the therapeutic effect of drug. We establish a nonlinear age-structured tumour cell population model. We investigate the existence and uniqueness of the model solution and explore the local and global stabilities of the tumour-free steady state. The existence and local stability of the tumour steady state are studied. Finally, some numerical simulations are performed to verify the theoretical results and to investigate the effects of different parameters on the model