The role of natural recovery category in malaria dynamics under saturated treatment.

In the process of malaria transmission, natural recovery individuals are slightly infectious compared with infected individuals. Our concern is whether the infectivity of natural recovery category can be ignored in areas with limited medical resources, so as to reveal the epidemic pattern of malaria with simpler analysis. To achieve this, we incorporate saturated treatment into two-compartment and three-compartment models, and the infectivity of natural recovery category is only reflected in the latter. The non-spatial two-compartment model can admit backward bifurcation. Its spatial version does not possess rich dynamics. Besides, the non-spatial three-compartment model can undergo backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. For spatial three-compartment model, due to the complexity of characteristic equation, we apply Shengjin's Distinguishing Means to realize stability analysis. Further, the model exhibits Turing instability, Hopf bifurcation and Turing-Hopf bifurcation. This makes the model may admit bistability or even tristability when its basic reproduction number less than one. Biologically, malaria may present a variety of epidemic trends, such as elimination or inhomogeneous distribution in space and periodic fluctuation in time of infectious populations. Notably, parameter regions are given to illustrate substitution effect of two-compartment model for three-compartment model in both scenarios without or with spatial movement. Finally, spatial three-compartment model is used to present malaria transmission in Burundi. The application of efficiency index enables us to determine the most effective method to control the number of cases in different scenarios.

Complicated Bosworth fracture-dislocation: A case report and review of the literature.

Bosworth fracture and dislocation is relatively rare, accounting for about 1% of ankle fractures. It is characterized by the proximal fibula fracture embedded in the posterolateral distal tibia. Due to an insufficient understanding of this fracture, it is easy to cause missed diagnosis and misdiagnosis in clinical practice. Due to the insertion of the fracture, it is challenging to perform closed reduction, and improper treatment is easy to cause complications. Surgical treatment is recommended for this type of fracture. In order to improve the understanding of orthopedic surgeons about Bosworth fracture and dislocation, this paper reports the diagnosis and treatment of 2 cases of Bosworth fracture and dislocation, and reviews the literature on Bosworth fracture's mechanism, diagnosis, classification, complications, and treatment options in recent years.

Dynamics and optimal control of a stochastic Zika virus model with spatial diffusion.

Zika is an infectious disease with multiple transmission routes, which is related to severe congenital disabilities, especially microcephaly, and has attracted worldwide concern. This paper aims to study the dynamic behavior and optimal control of the disease. First, we establish a stochastic reaction-diffusion model (SRDM) for Zika virus, including human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, and prove the existence, uniqueness, and boundedness of the global positive solution of the model. Then, we discuss the sufficient conditions for disease extinction and the existence of a stationary distribution of positive solutions. After that, three controls, i.e. personal protection, treatment of infected persons, and insecticides for spraying mosquitoes, are incorporated into the model and an optimal control problem of Zika is formulated to minimize the number of infected people, mosquitoes, and control cost. Finally, some numerical simulations are provided to explain and supplement the theoretical results obtained.

Stability and bifurcation analysis of Alzheimer's disease model with diffusion and three delays.

A reaction-diffusion Alzheimer's disease model with three delays, which describes the interaction of ß-amyloid deposition, pathologic tau, and neurodegeneration biomarkers, is investigated. The existence of delays promotes the model to display rich dynamics. Specifically, the conditions for stability of equilibrium and periodic oscillation behaviors generated by Hopf bifurcations can be deduced when delay σ (σ=σ1+σ2) or σ3 is selected as a bifurcation parameter. In addition, when delay σ and σ3 are selected as bifurcation parameters, the stability switching curves and the stable region are obtained by using an algebraic method, and the conditions for the existence of Hopf bifurcations can also be derived. The effects of time delays, diffusion, and treatment on biomarkers are discussed via numerical simulations. Furthermore, sensitivity analysis at multiple time points is drawn, indicating that different targeted therapies should be taken at different stages of development, which has certain guiding significance for the treatment of Alzheimer's disease.

Dynamics of an impulsive reaction-diffusion mosquitoes model with multiple control measures.

It is well-known that mosquito control is one of the effective methods to reduce and prevent the transmission of mosquito-borne diseases. In this paper, we formulate a reaction-diffusion impulsive hybrid model incorporating Wolbachia, impulsively spraying of insecticides, spatial heterogeneity, and seasonality to investigate the control of mosquito population. The sufficient conditions for mosquito extinction or successful Wolbachia persistence in a population of natural mosquitoes are derived. More importantly, we give the estimations of the spraying times of insecticides during a period for achieving the mosquito extinction and population replacement in a special case. A global attractivity of the positive periodic solution is analyzed under appropriate conditions. Numerical simulations disclose that spatial heterogeneity and seasonality have significant impacts on the design of mosquitoes control strategies. It is suggested to combine biological control and chemical pulse control under certain situations to reduce the natural mosquitoes. Further, our results reveal that the establishment of a higher level of population replacement depends on the strain type of the Wolbachia and the high initial occupancy of the Wolbachia-infected mosquitoes.

Basic reproduction ratio of a mosquito-borne disease in heterogeneous environment.

To explore the influence of spatial heterogeneity on mosquito-borne diseases, we formulate a reaction-diffusion model with general incidence rates. The basic reproduction ratio [Formula: see text] for this model is introduced and the threshold dynamics in terms of [Formula: see text] are obtained. In the case where the model is spatially homogeneous, the global asymptotic stability of the endemic equilibrium is proved when [Formula: see text]. Under appropriate conditions, we establish the asymptotic profiles of [Formula: see text] in the case of small or large diffusion rates, and investigate the monotonicity of [Formula: see text] with respect to the heterogeneous diffusion coefficients. Numerically, the proposed model is applied to study the dengue fever transmission. Via performing simulations on the impacts of certain factors on [Formula: see text] and disease dynamics, we find some novel and interesting phenomena which can provide valuable information for the targeted implementation of disease control measures.

Dynamics of a multi-strain malaria model with diffusion in a periodic environment.

This paper mainly explores the complex impacts of spatial heterogeneity, vector-bias effect, multiple strains, temperature-dependent extrinsic incubation period (EIP) and seasonality on malaria transmission. We propose a multi-strain malaria transmission model with diffusion and periodic delays and define the reproduction numbers Ri and R^i (i = 1, 2). Quantitative analysis indicates that the disease-free ω-periodic solution is globally attractive when Ri<1, while if Ri>1>Rj (i≠j,i,j=1,2), then strain i persists and strain j dies out. More interestingly, when R1 and R2 are greater than 1, the competitive exclusion of the two strains also occurs. Additionally, in a heterogeneous environment, the coexistence conditions of the two strains are R^1>1 and R^2>1. Numerical simulations verify the analytical results and reveal that ignoring vector-bias effect or seasonality when studying malaria transmission will underestimate the risk of disease transmission.

Dynamic analysis of a malaria reaction-diffusion model with periodic delays and vector bias.

One of the most important vector-borne disease in humans is malaria, caused by Plasmodium parasite. Seasonal temperature elements have a major effect on the life development of mosquitoes and the development of parasites. In this paper, we establish and analyze a reaction-diffusion model, which includes seasonality, vector-bias, temperature-dependent extrinsic incubation period (EIP) and maturation delay in mosquitoes. In order to get the model threshold dynamics, a threshold parameter, the basic reproduction number $ R_{0} $ is introduced, which is the spectral radius of the next generation operator. Quantitative analysis indicates that when $ R_{0} < 1 $, there is a globally attractive disease-free $ \omega $-periodic solution; disease is uniformly persistent in humans and mosquitoes if $ R_{0} > 1 $. Numerical simulations verify the results of the theoretical analysis and discuss the effects of diffusion and seasonality. We study the relationship between the parameters in the model and $ R_{0} $. More importantly, how to allocate medical resources to reduce the spread of disease is explored through numerical simulations. Last but not least, we discover that when studying malaria transmission, ignoring vector-bias or assuming that the maturity period is not affected by temperature, the risk of disease transmission will be underestimate.

Analysis of a two-strain malaria transmission model with spatial heterogeneity and vector-bias.

In this paper, we introduce a reaction-diffusion malaria model which incorporates vector-bias, spatial heterogeneity, sensitive and resistant strains. The main question that we study is the threshold dynamics of the model, in particular, whether the existence of spatial structure would allow two strains to coexist. In order to achieve this goal, we define the basic reproduction number [Formula: see text] and introduce the invasion reproduction number [Formula: see text] for strain [Formula: see text]. A quantitative analysis shows that if [Formula: see text], then disease-free steady state is globally asymptotically stable, while competitive exclusion, where strain i persists and strain j dies out, is a possible outcome when [Formula: see text] [Formula: see text], and a unique solution with two strains coexist to the model is globally asymptotically stable if [Formula: see text], [Formula: see text]. Numerical simulations reinforce these analytical results and demonstrate epidemiological interaction between two strains, discuss the influence of resistant strains and study the effects of vector-bias on the transmission of malaria.

CT Analysis of a Potential Safe Zone for Placing External Fixator Pins in the Humerus.

BACKGROUND: Iatrogenic radial nerve injures are a common complication during the placement of external fixator pins at the lateral aspect of the humeral shaft. This study uses a three-dimensional measurement technique to locate a safe entry point for humeral pins when externally fixating the elbow. Methods: We fixed a guide wire to the radial nerve by a suture string, and used computed tomography (CT) to scan the upper limbs of cadaver specimens. Then, we measured the deviation angles of the radial nerve on the CT scans, and the distance from the radial nerve to the "elbow rotation center" (ERC). Result: The average distance from the radial nerve to the ERC was 87.3 ± 8.5 mm (range: 68-100 mm), 58.3 ± 11.3 mm (range: 32.12-82.84 mm), 106.3 ± 5.8 mm (range: 86.93-115.08 mm), and 113.9 ± 4.8 mm (range: 97.93-120.22 mm) at radial nerve deviation angles of 0°, -30°, 30°, and 45°, respectively. The average radial nerve deviation angle was -37.7° ± 7.7° and 123.9° ± 19.9° at 50 and 150 mm, respectively. Relative to 0°, the distance between the radial nerve and the ERC at radial nerve deviation angles of -30°, 30°, and 45° showed a significant difference (t = 18.20, p < 0.05; Z = 6.07, p < 0.001; Z = 6.40, p < 0.001, respectively). Conclusions: Pins inserted into the proximal humerus should be about 150 mm from the ERC with a radial nerve deviation angle of 30° anteriorly, and 50 mm from the ERC with a deviation angle of 30°-45° posteriorly.

Modeling and Dynamics Analysis of Zika Transmission with Limited Medical Resources.

Zika virus, a reemerging mosquito-borne flavivirus, posed a global public health emergency in 2016. Brazil is the most seriously affected country. Some measures have been implemented to control the Zika transmission, such as spraying mosquitoes, developing vaccines and drugs. However, because of the limited medical resources (LMRs) in the country, not every infected patient can be treated in time when infected with Zika virus. We aim to build a deterministic Zika model by introducing a piecewise smooth treatment recovery rate to research the effect of LMRs on the transmission and control of Zika. For the model without treatment, we analyze the global stability of equilibria. For the model with treatment, the model exhibits complex dynamics. We prove that the model with treatment undergoes backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2. It means that the model with LMRs is sensitive to parameters and initial conditions, which has important significance for control of Zika. We also apply the model to estimate the basic and control reproduction numbers for the Zika transmission by using the data on weekly reported accumulated Zika cases from March 25, 2016, to April 14, 2018, in Brazil.

Partial differential equation modeling of rumor propagation in complex networks with higher order of organization.

Mathematical modeling is an important approach to research rumor propagation in online social networks. Most of prior work about rumor propagation either carried out empirical studies or focus on ordinary differential equation models with only consideration of temporal dimension; little attempt has been given on understanding rumor propagation over both temporal and spatial dimensions. This paper primarily addresses an issue related to how to define a spatial distance in online social networks by clustering and then proposes a partial differential equation model with a time delay to describing rumor propagation over both temporal and spatial dimensions. Theoretical analysis reveals the existence of equilibrium points, a priori bound of the solution, the local stability and the global stability of equilibrium points of our rumor propagation model. Finally, numerical simulations have analyzed the possible influence factors on rumor propagation and proved the validity of the theoretical analysis.

Modelling and analysis of HFMD with the effects of vaccination, contaminated environments and quarantine in mainland China.

Currently, hand, foot, and mouth disease (HFMD) is widespread in mainland China and seriously endangers the health of infants and young children. Recently in mainland China, preventing the spread of the disease has entailed vaccination, isolation measures, and virus clean-up in the contaminated environment. However, quantifying and evaluating the efficacy of these strategies on HFMD remains challenging, especially because relatively little research analyses the impact of EV71 vaccination for this disease. To assess the effectiveness of these strategies, we propose a new mathematical model that considers vaccination, contaminated environment, and quarantine simultaneously. Unlike the previous studies for HFMD, in which the basic reproduction number R0 is the only threshold to decide whether the disease is extinct or not, our results show that another threshold value is needed:R̂0 < 1 (R0 <= R̂0 < 1) such that disease is extinct; i.e., the disease-free equilibrium is globally asymptotically stable. Moreover, numerical experiments show that our model may have positive equilibriums even if the basic reproduction number R0 is less than 1. In designing a new algorithm based on a BP network to estimate the unknown parameters, this proposed model is put forward to individually fit the HFMD reported cases annually in mainland China from 2015 to 2017. At last, the sensitivity analyses and numerical experiments show that increasing the rate of virus clearance, the vaccinated rate of infants and young children, and the quarantined rate of infectious individuals can effectively control the spread of HFMD in mainland China. Nevertheless, it remains difficult to eliminate the disease. Specifically, our results show that the current vaccine measures starting in 2016 have reduced the total number of patients in 2016 and 2017 by 17% and 22%, respectively.

Modeling the transmission and control of Zika in Brazil.

Zika virus, a reemerging mosquito-borne flavivirus, started spread across Central and Southern America and more recently to North America. The most serious impacted country is Brazil. Based on the transmission mechanism of the virus and assessment of the limited data on the reported suspected cases, we establish a dynamical model which allows us to estimate the basic reproduction number R 0 = 2.5020. The wild spreading of the virus make it a great challenge to public health to control and prevention of the virus. We formulate two control models to study the impact of releasing transgenosis mosquitoes (introducing bacterium Wolbachia into Aedes aegypti) on the transmission of Zika virus in Brazil. Our models and analysis suggest that simultaneously releasing Wolbachia-harboring female and male mosquitoes will achieve the target of population replacement, while releasing only Wolbachia-harboring male mosquitoes will suppress or even eradicate wild mosquitoes eventually. We conclude that only releasing male Wolbachia mosquitoes is a better strategy for control the spreading of Zika virus in Brazil.

The spatial dynamics of a zebrafish model with cross-diffusions.

This paper investigates the spatial dynamics of a zebrafish model with cross-diffusions. Sufficient conditions for Hopf bifurcation and Turing bifurcation are obtained by analyzing the associated characteristic equation. In addition, we deduce amplitude equations based on multiple-scale analysis, and further by analyzing amplitude equations five categories of Turing patterns are gained. Finally, numerical simulation results are presented to validate the theoretical analysis. Furthermore, some examples demonstrate that cross-diffusions have an effect on the selection of patterns, which explains the diversity of zebrafish pattern very well.

Dynamics analysis of a delayed reaction-diffusion predator-prey system with non-continuous threshold harvesting.

This paper deals with a delayed reaction-diffusion predator-prey model with non-continuous threshold harvesting. Sufficient conditions for the local stability of the regular equilibrium, the existence of Hopf bifurcation and Turing bifurcation are obtained by analyzing the associated characteristic equation. By utilizing upper-lower solution method and Lyapunov functions the globally asymptotically stability of a unique regular equilibrium and asymptotically stability of a unique pseudoequilibrium are studied respectively. Further, the boundary node bifurcations are studied. Finally, numerical simulation results are presented to validate the theoretical analysis.

Dynamic analysis of a fractional order delayed predator-prey system with harvesting.

In the study, we consider a fractional order delayed predator-prey system with harvesting terms. Our discussion is divided into two cases. Without harvesting, we investigate the stability of the model, as well as deriving some criteria by analyzing the associated characteristic equation. With harvesting, we investigate the dynamics of the system from the aspect of local stability and analyze the influence of harvesting to prey and predator. Finally, numerical simulations are presented to verify our theoretical results. In addition, using numerical simulations, we investigate the effects of fractional order and harvesting terms on dynamic behavior. Our numerical results show that fractional order can affect not only the stability of the system without harvesting terms, but also the switching times from stability to instability and to stability. The harvesting can convert the equilibrium point, the stability and the stability switching times.

Synchronized bifurcation and stability in a ring of diffusively coupled neurons with time delay.

In this study, we consider a ring of diffusively coupled neurons with distributed and discrete delays. We investigate the synchronized stability and synchronized Hopf bifurcation of this system, as well as deriving some criteria by analyzing the associated characteristic transcendental equation and by taking τ and ß as the bifurcation parameters, which are parameters that measure the discrete delay and the strength of nearest-neighbor connection, respectively. Our simulations demonstrated that the numerically observed behaviors were in excellent agreement with the theoretically predicted results. In addition, using numerically simulations, we investigated the effects of τ and ß, as well as the diffusion on dynamic behavior. Our numerical results showed that the addition diffusion to a stable delay-differential equation (DDE) system may make it unstable and that the diffusion may make the system synchronous, whereas it is asynchronous without the diffusion term.

Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters.

This paper deals with a delayed reaction-diffusion three-species Lotka-Volterra model with interval biological parameters and harvesting. Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Then an optimal control problem has been considered. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality.

Function-preserving reduction and fixation of unstable Jefferson fractures using a C1 posterior limited construct.

STUDY DESIGN: This is a retrospective, clinical, and radiologic study of posterior reduction and fusion of the C1 arch in the treatment of unstable Jefferson fractures. OBJECTIVE: The aim of the study was to describe a new motion-preserving surgical technique in the treatment of unstable Jefferson fracture. SUMMARY OF BACKGROUND DATA: The management of unstable Jefferson fractures remains controversial. The majority of C1 fractures can be effectively treated nonoperatively with external immobilization unless there is an injury to the transverse atlantal ligament (TAL). Conservative treatment usually involves immobilization for a long time in Halo vest, whereas surgical intervention generally involves C1-C2 fusion, eliminating the range of motion of the upper cervical spine. We propose a novel method for the treatment of unstable Jefferson fractures without restricting the range of motion. METHODS: A retrospective review of 12 patients with unstable C1 fractures between April 2008 and October 2011 was performed. They were treated by inserting bilateral posterior C1 pedicle screws or lateral mass screws interconnected by a transversal rod to achieve internal fixation. There were 8 men and 4 women, with an average age of 35.6 years (range, 20-60 y). Presenting symptoms included neck pain, stiffness, and decreased range of motion but none had neurological injury. Seven patients had bilateral posterior arch fractures associated with unilateral anterior arch fractures (posterior 3/4 Jefferson fracture, Landells type II), and 5 had unilateral anterior and posterior arch fractures (half-ring Jefferson fracture, Landells type II). Seven patients had intact TAL, and 5 patients had fractures and avulsion of the attachment of TAL (Dickman type II). RESULTS: A total of 24 screws were inserted. Five cases had screws placed in the lateral mass: 3 because of posterior arch breakage, and 2 because the height of the posterior arch at the entry point was <4 mm. The remaining 7 cases had pedicle screw fixation. One patient had venous plexus injury during exposure of lower margin of the posterior arch; however, successful hemostasis was achieved with Gelfoam. Postoperative x-ray and computed tomography scan showed partial breach of the transverse foramen caused by a screw in 1 case, and breach of the inner cortex of the pedicle caused by screw displacement in 1 case; however, no spinal cord injury or vertebral artery injury was found. The remaining screws were in good position. Patients were followed up for 6-40 months (average, 22 mo). All cases had recovery of range of motion of the cervical spine to the preinjury level by 3-6 months after surgery, with resolution of pain. At 6 months follow-up, plain radiographs and computed tomography scans revealed satisfactory cervical alignment, no implant failure, and satisfactory bony fusion of the fractures; no C1-C2 instability was observed on the flexion-extension radiographs. CONCLUSIONS: C1 posterior limited construct is a valid technique and a feasible method for treating unstable Jefferson fractures, which allows preservation of the function of the craniocervical junction, without significant morbidity.