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Tumor heterogeneity is a large obstacle for cancer study and treatment. Different cancer patients may involve different combinations of gene mutations or the distinct regulatory pathways for inducing the progression of tumor. Investigating the pathways of gene mutations which can cause the formation of tumor can provide a basis for the personalized treatment of cancer. Studies suggested that KRAS, APC and TP53 are the most significant driver genes for colorectal cancer. However, it is still an open issue regarding the detailed mutation order of these genes in the development of colorectal cancer. For this purpose, we analyze the mathematical model considering all orders of mutations in oncogene, KRAS and tumor suppressor genes, APC and TP53, and fit it on data describing the incidence rates of colorectal cancer at different age from the Surveillance Epidemiology and End Results registry in the United States for the year 1973–2013. The specific orders that can induce the development of colorectal cancer are identified by the model fitting. The fitting results indicate that the mutation order with KRAS → APC → TP53, APC → TP53 → KRAS and APC → KRAS → TP53 explain the age–specific risk of colorectal cancer with very well. Furthermore, eleven pathways of gene mutations can be accepted for the mutation order of genes with KRAS → APC → TP53, APC → TP53 → KRAS and APC → KRAS → TP53, and the alternation of APC acts as the initiating or promoting event in the colorectal cancer. The estimated mutation rates of cells in the different pathways demonstrate that genetic instability must exist in colorectal cancer with alterations of genes, KRAS, APC and TP53. Author summary: Cumulative mutations in driver genes are the essential cause of cancer disease. For the colorectal cancer, KRAS, APC and TP53 are the common driver genes, and approximately 15% patients with colorectal cancer carry all mutations of the three genes. Exploring the pathway of mutations in these gene is extremely useful for the diagnosis and treatment of cancer. However, the mutation orders of these genes may vary in different patients due to the heterogeneity of tumor. Hence, we discuss all possible mutation orders in the genes, KRAS, APC and TP53 by using the model with five hits and find out the mutation pathways of genes that can effectively fit the incidence rate of colorectal cancer at different age in this article. In addition, we give the estimated values of mutation rates in each pathway that can explain the procession of colorectal cancer. The results obtained can offer guidelines to the treatment strategy of colorectal cancer. [ABSTRACT FROM AUTHOR]

Roguing and replanting are the most common strategies to control plant diseases and pests. How to build the mathematical models of plant virus transmission and consider the impact of roguing and replanting strategies on plant virus eradication is of great practical significance. In the present paper, we propose the mathematical models for plant virus transmission with continuous and impulsive roguing control. For the model with continuous control strategies, the threshold values for the existences and stabilities of multiple equilibria have been given, and the effect of roguing strategies on the threshold values is also addressed. Furthermore, the model with impulsive roguing control tactics is proposed, and the existence and stability of the plant-only and disease-free periodic solutions of the model are investigated by calculating several threshold values. Moreover, when selecting the design control strategy to minimize the threshold, we systematically analyze the existence of the optimal times of roguing infected plants within a replanting cycle, which is of great significance to the design and optimization of the prevention and control strategy of plant virus transmission. Finally, numerical investigations are given to reveal the main conclusions, and the biological implications of the main results are briefly discussed in the last section. [ABSTRACT FROM AUTHOR]

We propose a Holling II predator–prey model with nonlinear pulse as state-dependent feedback control strategy and then provide a comprehensively qualitative analysis by using the theories of impulsive semi-dynamical systems. First, the Poincaré map is constructed based on the domains of impulsive and phase sets which are defined according to the phase portraits of the model. Second, the threshold conditions for the existence and stability of the semi-trivial periodic solution are given, and subsequently an order-1 periodic solution is generated through the transcritical bifurcation. Furthermore, the different parameter spaces for the existence and stability of an order-1 periodic solution are investigated. In addition, the existence and nonexistence of order- k ( k ≥ 2 ) periodic solutions have been studied theoretically. Moreover, the numerical investigations are presented in order to substantiate our theoretical results and show the complex dynamics of proposed model. Finally, some biological implications of the mathematical results are discussed in the conclusion section. [ABSTRACT FROM AUTHOR]

The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and obtain sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order- k ( $k\geq3$) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control. [ABSTRACT FROM AUTHOR]

Spiking neuron models which exhibit rich dynamics are usually defined by hybrid dynamical systems. It is revealed that mathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitative analysis for a quadratic integrate-and-fire model by using the theories of hybrid dynamical system. Firstly, the exact impulsive and phase sets are defined according to the phase portraits of the proposed model, and then the Poincaré map is constructed. Furthermore, the conditions for the existence and stability of an order 1 periodic solution are provided. Moreover, the existence and nonexistence of an order k  k≥2 periodic solution have been studied theoretically and numerically, and the results show that the system has periodic solutions with any period. Finally, some biological implications of the mathematical results are discussed. [ABSTRACT FROM AUTHOR]

A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the interesting global sliding bifurcations sequentially: touching → buckling → crossing → sliding homoclinic orbit to a pseudo-saddle → crossing → touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold. [ABSTRACT FROM AUTHOR]

Objective: To project the HIV/AIDS epidemics among men who have sex with men (MSM) under different combinations of HIV testing and linkage to care (TLC) interventions including antiretroviral therapy (ART) in Beijing, China. Design: Mathematical modeling. Methods: Using a mathematical model to fit prevalence estimates from 2000–2010, we projected trends in HIV prevalence and incidence during 2011–2020 under five scenarios: (S1) current intervention levels by averaging 2000–2010 coverage; (S2) increased ART coverage with current TLC; (S3) increased TLC/ART coverage; (S4) increased condom use; and (S5) increased TLC/ART plus increased condom use. Results: The basic reproduction number based upon the current level of interventions is significantly higher than 1 ( confidence interval (CI), 1.83–2.35), suggesting that the HIV epidemic will continue to increase to 2020. Compared to the 2010 prevalence of 7.8%, the projected HIV prevalence in 2020 for the five prevention scenarios will be: (S1) Current coverage: 21.4% (95% CI, 9.9–31.7%); (S2) Increased ART: 19.9% (95% CI, 9.9–28.4%); (S3) Increased TLC/ART: 14.5% (95% CI, 7.0–23.8%); (S4) Increased condom use: 13.0% (95% CI, 9.8–28.4%); and (S5) Increased TLC/ART and condom use: 8.7% (95% CI, 5.4–11.5%). HIV epidemic will continue to rise () for S1–S4 even with hyperbolic coverage in the sensitivity analysis, and is expected to decline () for S5. Conclusion: Our transmission model suggests that Beijing MSM will have a rapidly rising HIV epidemic. Even enhanced levels of TLC/ART will not interrupt epidemic expansion, despite optimistic assumptions for coverage. Promoting condom use is a crucial component of combination interventions. [ABSTRACT FROM AUTHOR]

Abstract: A non-smooth Gause predator–prey model with a constant refuge is proposed and analyzed. Firstly, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between the existence of a regular equilibrium and a pseudo-equilibrium are studied, and the results indicate that the two types of equilibria cannot coexist. The sufficient and necessary conditions for the global stability of limit cycle, sliding touching cycle, canard cycle, focus point and pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node (focus) and touching bifurcations were investigated by employing theoretical and numerical techniques. Finally, we compare our results with previous studies on a non-smooth Gause predator–prey model without involving a carrying capacity for the prey population, and some biological implications are discussed. [Copyright &y& Elsevier]

Abstract: The present paper aims to provide a detailed qualitative analysis of a non-smooth Gause predator–prey model. In this model, the saturating functional response function with a discontinuity at a critical prey density was employed to show the effects of a prey refuge on the population dynamic behavior. Analysis of this model revealed rich dynamics including locally (or globally) stable canard cycles, a locally (globally) stable pseudo-equilibrium, unbounded trajectories in which both populations go to infinity or the prey goes to infinity and the predator dies out eventually. The main purpose of the present work is to carry out a completely qualitative analysis for this model. In particular, two sets of sufficient conditions drive both populations to approach infinity and the sufficient and necessary conditions for all of the other main results are presented. [Copyright &y& Elsevier]

Abstract: Pulse-like pest management actions such as spraying pesticides and killing a pest instantly and the release of natural enemies at critical times can be modelled with impulsive differential equations. In practice, many pesticides have long-term residual effects and, also, both pest and natural enemy populations may have delayed responses to pesticide applications. In order to evaluate the effects of the duration of the residual effectiveness of pesticides and of delayed responses to pesticides on a pest management strategy, we developed novel mathematical models. These combine piecewise-continuous periodic functions for chemical control with pulse actions for releasing natural enemies in terms of fixed pulse-type actions and unfixed pulse-type actions. For the fixed pulse-type model, the stability threshold conditions for the pest eradication periodic solution and permanence of the model are derived, and the effects of key parameters including killing efficiency rate, decay rate, delayed response rate, number of pesticide applications and number of natural enemy releases on the threshold values are discussed in detail. The results indicate that there exists an optimal releasing period or an optimal number of pesticide applications which maximizes the threshold value. For unfixed pulse-type models, the effects of the killing efficiency rate, decay rate and delayed response rate on the pest outbreak period, and the frequency of control actions are also investigated numerically. [Copyright &y& Elsevier]

This paper proposes and analyzes a mathematical model of an infectious disease system with a piecewise control function concerning threshold policy for disease management strategy. The proposed models extend the classic models by including a piecewise incidence rate to represent control or precautionary measures being triggered once the number of infected individuals exceeds a threshold level. The long-term behaviour of the proposed non-smooth system under this strategy consists of the so-called sliding motion-a very rapid switching between application and interruption of the control action. Model solutions ultimately approach either one of two endemic states for two structures or the sliding equilibrium on the switching surface, depending on the threshold level. Our findings suggest that proper combinations of threshold densities and control intensities based on threshold policy can either preclude outbreaks or lead the number of infecteds to a previously chosen level. [ABSTRACT FROM AUTHOR]

Abstract: Assuming that a fish population follows the continuous logistic growth or the discrete Beverton–Holt model, several optimal impulsive harvesting policies for the maximum stock level of the fish at the end of a fishing season are investigated under the condition of fixed intensity and frequency of impulsive harvesting. The optimal impulsive harvesting moments for all cases considered are given analytically and the related numerical simulations are also provided. Furthermore, the methods employed can also be used to investigate the optimal timing of chemical control in pest management. Our results confirm that the optimal timing of pesticide applications such that the density of the pest population is minimal at any time during a planting season or the average of density of the pest population over the planting season is minimal is the beginning of the planting season. In practice, the results can be used to guide the fisherman to manage fisheries and guide farmers to control pests. [Copyright &y& Elsevier]

Abstract: Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest–natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided. Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications. Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics. [Copyright &y& Elsevier]

A striking and defining feature of circadian clocks is the small variation in period over a physiological range of temperatures. This is referred to as temperature compensation, although recent work has suggested that the variation observed is a specific, adaptive control of period. Moreover, given that many biological rate constants have a Q10 of around 2, it is remarkable that such clocks remain rhythmic under significant temperature changes. We introduce a new mathematical model for the Neurospora crassa circadian network incorporating experimental work showing that temperature alters the balance of translation between a short and long form of the FREQUENCY (FRQ) protein. This is used to discuss period control and functionality for the Neurospora system. The model reproduces a broad range of key experimental data on temperature dependence and rhythmicity, both in wild-type and mutant strains.We present a simple mechanism utilising the presence of the FRQ isoforms (isoform switching) by which period control could have evolved, and argue that this regulatory structuremay also increase the temperature range where the clock is robustly rhythmic. [ABSTRACT FROM AUTHOR]

Sufficient conditions are obtained for the global attractivity of the positive equilibrium of the delay-logistic equation with impulsive effect N˙(t)=r(t)N(t)K−Np(t−τ)/K+c(t)Np(t−τ),N(τk+)=N(τk)expIklnN(τk)/Kp,N(τk+)=N(τk)expIklnN(τk)/Kp, where r(t), c(t) are continuous functions, p, K, τ are positive constants, and {τk}k=1∞ denotes the sequence of impulsive time which assumed to be strictly increasing. Some special cases are also considered. [Copyright &y& Elsevier]

We develop a mathematical model to closely mimic the integrated program of impulsive vector control (every Friday afternoon since the initiation of the program) and continuous patient treatment and isolation implemented in the Guangdong Province of China during its 2014 dengue outbreak. We fitted the data of accumulated infections and used the parameterized model to carry out a retrospective analysis to estimate the basic reproduction number 1.7425 (95% CI 1.4443–2.0408), the control reproduction number 0.1709, and the mosquito-killing ratios 0.1978, 0.2987, 0.6158 and 0.5571 on October 3, 10, 17 and 24, respectively. This suggests that integrated intervention is highly effective in controlling the dengue outbreak. We also simulated outbreak outcomes under different variations of the implemented interventions. We showed that skipping one Friday for vector control would not result in raising the control reproduction number to the threshold value 1 but would lead to significant increase in the accumulated infections at the end of the outbreak. The findings indicate that quick and persistent impulsive implementation of vector control result in an effective reduction in the control reproduction number and hence lead to significant decline of new infections. [ABSTRACT FROM AUTHOR]

• A novel mathematical model is proposed to describe the heterogeneity of control interventions. • Xi'an presents a lower transmission probability but a higher reproduction number. • Enhancing closed-off management, tracing and testing intensities, are effective. • Citywide lockdown is not essential when the closed-off management is strictly implemented. • Enhancing tracing strategy is more effective than enhancing testing strategy in the early stage. Several local outbreaks have occurred and been suppressed with the dynamic zero-COVID policy and widely promoted vaccination program implemented in China. The epidemic duration and final size vary significantly in different cities, which may be attributed to different implementation patterns and intensities of non-pharmaceutical interventions (NPIs). It's important to capture the underlying mechanism to explore more efficient implementation patterns of NPIs in order to prevent the future resurgence. In this study, outbreaks caused by Delta variant in Xi'an, Yangzhou and Guangzhou in 2021 are chosen as the examples. A novel model dividing the population into three groups is proposed to describe the heterogeneity of control interventions. The model is calibrated and key parameters related to NPIs are estimated by using multi-source epidemic data. The estimation results show a lower transmission probability but a higher initial reproduction number in Xi'an. Sensitivity analysis are conducted to investigate the impact of various control measures in different epidemic phases. The results identify the vital role of enhancing closed-off management, strengthening tracing and testing intensities, on shortening the epidemic durations and reducing the final size. Further, we find that sufficiently implemented closed-off management would prevent the city from lockdown. Strengthening the tracing other than the testing strategy in the initial stage is more effective on containing the epidemic in a shorter duration with less infections. [ABSTRACT FROM AUTHOR]

Abstract: We propose and analyze a mathematical model to mimic its transmission dynamics to assess the transmission potential of the novel avian influenza A(H7N9) virus. By fitting the model to data of the confirmed human cases we estimate the reproduction number for human-to-human transmission as 0.467 (95% CI 0.387–0.651). Simulation results indicate that approximate twofold of the current human-to-human transmission rate or periodic outbreaks of avian influenza in poultry may induce an outbreak in human. Through the recent limited transmission potential of the novel avian influenza A(H7N9) virus, a new outbreak may be possible due to virus mutation and adaption or periodic outbreaks in poultry, and hence careful surveillance and persistent intervention strategies in poultry have to be required. [Copyright &y& Elsevier]

Abstract: We review mathematical modeling and related statistical issues of HIV dynamics primarily in response to antiretroviral drug therapy in this article. We start from a basic model of virus infection and then review a number of more advanced models with consideration of pharmacokinetic factors, adherence and drug resistance. Specifically, we illustrate how mathematical models can be developed and parameterized to understand the effects of long-term treatment and different treatment strategies on disease progression. In addition, we discuss a variety of parameter estimation methods for differential equation models that are applicable to either within- or between-host viral dynamics. [Copyright &y& Elsevier]