Nilpotent Singularities and Periodic Perturbation of a $$GI\beta $$ Model: A Pathway to Glucose Disorder

Tao, Yiwen; Sun, Yutong; Zhu, Huaiping; Lyu, Jiangnan; Ren, Jingli

doi:10.1007/s00332-023-09907-z

Cited by 10 publications

(2 citation statements)

References 40 publications

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“…However, tumor-immune system interactions are very complicated, being highly nonlinear, and evolve with time to a certain extent [15]. So it is necessary to model tumor-immune system interactions using nonlinear response functions, which are widely used in the modeling of tumor-immune interactions [16], the ecological system [17], virus dynamics [18], the metabolic system [19] and so on.…”

Section: Introductionmentioning

confidence: 99%

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Liu,

Ma,

Yang

et al. 2024

Symmetry

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Tumor–immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor–immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated.

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Section: Introductionmentioning

confidence: 99%

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Liu,

Ma,

Yang

et al. 2024

Symmetry

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“…Since the application of Routh-Hurwitz criterion in the high dimensional system is very difficult, we have used the method of polar form [22] and graphs to obtain the precise conditions of Hopf bifurcation for any delay order [21]. High codimensional bifurcation and nonlineality are important issues in dynamical system theory [23,24] In this paper, we aim to study the stability of the coexistence equilibrium of the two-patch logistic model, considering that one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. The two patches are connected by diffusion, which is proportional to the difference of population densities between the two patches.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a two-patch logistic model with diffusion and time delay

Sawada,

Takeuchi,

Dong

2024

Nonlinearity

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In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.

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Mathematical Analysis on a General Delayed HBV Model with Capsids and Two Infection Routes

Liu,

Wang,

2024

Acta Math. Appl. Sin. Engl. Ser.

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Nilpotent Singularities and Periodic Perturbation of a $$GI\beta $$ Model: A Pathway to Glucose Disorder

Cited by 10 publications

References 40 publications

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Dynamics of a two-patch logistic model with diffusion and time delay

Mathematical Analysis on a General Delayed HBV Model with Capsids and Two Infection Routes

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