On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex.

We study the asymptotic behavior of the competitive Leslie/Gower model (map) [Formula: see text]It is shown that T unconditionally admits a globally attracting 1-codimensional invariant hypersurface [Formula: see text], called carrying simplex, such that every nontrivial orbit is asymptotic to one in [Formula: see text]. More general and easily checked conditions to guarantee the existence of carrying simplex for competitive maps are provided. An equivalence relation is defined relative to local stability of fixed points on [Formula: see text] (the boundary of [Formula: see text]) on the space of all three-dimensional Leslie/Gower models. Using a formula on the sum of the indices of all fixed points on the carrying simplex for three-dimensional maps, we list the 33 stable equivalence classes in terms of simple inequalities on the parameters [Formula: see text] and [Formula: see text] and draw their orbits on [Formula: see text]. In classes 1-18, every nontrivial orbit tends to a fixed point on [Formula: see text]. In classes 19-25, each map possesses a unique positive fixed point which is a saddle on [Formula: see text], and hence Neimark-Sacker bifurcations do not occur. Neimark-Sacker bifurcation does occur within each of classes 26-31, while it does not occur in class 32. Each map from class 27 admits a heteroclinic cycle, which forms the boundary of [Formula: see text]. The criteria on the stability of heteroclinic cycles are also given. This classification makes it possible to further investigate various dynamical properties in respective class.

On heteroclinic cycles of competitive maps via carrying simplices.

We concentrate on the effects of heteroclinic cycles and the interplay of heteroclinic attractors or repellers on the boundary of the carrying simplices for low-dimensional discrete-time competitive systems. Based on the existence of the carrying simplex for the competitive mapping, we provide the criteria on stability of the heteroclinic cycle. This result can be seen as a discrete counterpart of that for the continuous-time systems. Several concrete discrete-time competition models are further analyzed, which do admit heteroclinic cycles. The criteria on the stability of the heteroclinic cycle for each model are also given, which are comparable with the corresponding continuous-time models.

Chaotic attractors in Atkinson-Allen model of four competing species.

We study the occurrence of chaos in the Atkinson-Allen model of four competing species, which plays the role as a discrete-time Lotka-Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson-Allen dynamics can lead to chaos.

Threshold conditions for west nile virus outbreaks.

In this paper, we study the stability and saddle-node bifurcation of a model for the West Nile virus transmission dynamics. The existence and classification of the equilibria are presented. By the theory of K-competitive dynamical systems and index theory of dynamical systems on a surface, sufficient and necessary conditions for local stability of equilibria are obtained. We also study the saddle-node bifurcation of the system. Explicit subthreshold conditions in terms of parameters are obtained beyond the basic reproduction number which provides further guidelines for accessing control of the spread of the West Nile virus. Our results suggest that the basic reproductive number itself is not enough to describe whether West Nile virus will prevail or not and suggest that we should pay more attention to the initial state of West Nile virus. The results also partially explained the mechanism of the recurrence of the small scale endemic of the virus in North America.

Coexistence of breakpoint cluster region-Abelson1 rearrangement and Janus kinase 2 V617F mutation in chronic myeloid leukemia: A case report.

BACKGROUND: The Janus kinase 2 (JAK2) V617F mutation is common in patients with breakpoint cluster region-Abelson1 (BCR-ABL1)-negative myeloproliferative neoplasms, including polycythemia vera, essential thrombocythemia and primary myelofibrosis, but is rarely detected in BCR-ABL1-positive chronic myeloid leukemia (CML) patients. Here, we report a CML patient with both a BCR-ABL1 rearrangement and JAK2 V617F mutation. CASE SUMMARY: A 45-year-old Chinese woman was admitted to our department with a history of significant thrombocytosis for 20 d. Color Doppler ultrasound examination showed mild splenomegaly. Bone marrow aspiration revealed a karyotype of 46, XX, t(9;22)(q34;q11.2) in 20/20 metaphases by cytogenetic analysis, rearrangement of BCR-ABL1 (32.31%) by fluorescent polymerase chain reaction (PCR) and mutation of JAK2 V617F (10%) by PCR and Sanger DNA sequencing. The patient was diagnosed with CML and JAK2 V617F mutation. Following treatment with imatinib for 3 mo, the patient had an optimal response and BCR-ABL1 (IS) was 0.143%, while the mutation rate of JAK2 V617F rose to 15%. CONCLUSION: Emphasis should be placed on the detection of JAK2 mutation when CML is diagnosed to distinguish JAK2 mutation-positive CML and formulate treatment strategies.

The complete classification for dynamics in a homosexually-transmitted disease model.

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population was proposed by Li et al. in (J Math Biol 10:1037-1052, 1986). The sufficient and necessary conditions for coexistence and the sufficient conditions for stability of the boundary equilibria were provided. This paper will present a thorough classification of dynamics for this model in terms of the first and second so called reproductive numbers of infection in strains I and J. This classification not only solves a conjecture proposed in (Li et al., J Math Biol 10:1037-1052, 1986) but also gives the sufficient and necessary conditions for the competitive exclusion.

Preface.

In the new century, with rapid population growth, large-scale urbanization, global warming and many other factors, we are facing unprecedented ecological, health, social, and other challenges and issues. These include biological invasion, environmental degradation, sharp increase in cancer morbidity, high frequency of emerging and re-emerging infectious diseases, which pose a grave threat to biological diversity, public health, economic development and so on. Based on the solid research in population dynamics and disease dynamics, mathematical modeling, analysis and simulation have been widely used over the past decades to study various problems in life sciences and medicine, from the expression of gene sequence to the pathogenesis of cancer, from the control of molecular organisms to the resistance of bacteria and viruses, from immune response to diseases to the design and evaluation of treatments, and so on. To provide a platform for researchers in mathematical biology and related fields to present latest findings and research trends, to exchange ideas and approaches, and to enhance communication and cooperation, we organized a workshop entitled "Current Topics in Mathematical Biology (CTMB)" at Shanghai Normal University, December 18-20, 2015. We acknowledge the support from the Mathematics and Science College at Shanghai Normal University and Shanghai Gaofeng Project for University Academic Development Program.

Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM.

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly 24 hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411--2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate.

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold R0 which completely governs the disease dynamics: when R0 < 1, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when R0 > 1, the disease is permanent. It is interesting that the threshold value R0 bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.

Prognostic prediction and diagnostic role of intercellular adhesion molecule-1 (ICAM1) expression in clear cell renal cell carcinoma.

The intercellular adhesion molecule-1 (ICAM1) has been reported to function in multiple malignancies, but its effect on clear cell renal cell carcinoma (ccRCC) hasn't been discussed yet. This study aimed to identify the potential role of ICAM1 in prognostic prediction and early diagnosis of ccRCC. ICAM1 expression was inspected by immunohistochemistry and correlated with clinicopathologic variables. Association between protein expression and cancer-specific survival (CSS) of ccRCC patients was evaluated and the value of area under the receiver operating characteristics (ROC) curve (AUC) was calculated to measure the protein's diagnostic accuracy. ICAM1 was positively immunostained in 83.2% of 173 ccRCC tissues, but negatively immunostained in all the para-cancerous normal epitheliums of renal tubules. High ICAM1 expression was significantly related to male sex (P = 0.00241), T3/T4 stage (P = 0.02249), non-N0M0 stage (P = 0.03797) and positive renal pelvis invasion (P = 0.04227). Kaplan-Meier survival analysis illustrated that high ICAM1 expression was significantly correlated to a decreased CSS (P = 0.00006). Multivariate Cox analysis indicated that ICAM1 was an independent predictor for CSS of patients (P = 0.00451). Furthermore, the AUC value of ICAM1 in diagnosing ccRCC was 0.916 (P < 0.00001). In conclusion, high ICAM1 expression on tumor cells indicates a poor outcome of patients and ICAM1 is likely to be an independent predictor for the prognosis of ccRCC. Moreover, ICAM1 has a high AUC value and may be a potential and useful diagnostic marker.

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model.

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.