Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell density.

We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection R 0 and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state E 0 is globally asymptotically stable if R 0 < 1 . When R 0 > 1 , then E 0 becomes unstable, and another basic reproduction number of CTL response R 1 becomes the dynamic threshold in the sense that if R 1 < 1 , then the CTL-inactivated steady state E 1 is globally asymptotically stable; and if R 1 > 1 , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state E 2 is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.

Dynamic optimal allocation of medical resources: a case study of face masks during the first COVID-19 epidemic wave in the United States.

In this paper, we propose a two-group SIR epidemic model to simulate the outcome of the stay-at-home policy and the imposed face mask policy during the first COVID-19 epidemic wave in the United States. Then, we use a dynamic optimal control approach (with the objective of minimizing total deaths) to find the optimal dynamical distribution of face masks between healthcare workers and the general public. It is not surprising that all face masks should be solely reserved for healthcare workers if the supply is short. However, when the supply is indeed sufficient, our numerical study indicates that the general public should share a large portion of face masks at the beginning of the epidemic wave to dramatically reduce the death toll. This interesting result partially contradicts the guideline advised by the US Surgeon General and the Centers for Disease Control and Prevention (CDC) in March 2020. The optimality of this sounding CDC guideline highly depends on the supply level of face masks, which changes frequently; hence, it should be adjusted according to the supply of face masks.

The importance of quarantine: modelling the COVID-19 testing process.

We incorporate the disease state and testing state into the formulation of a COVID-19 epidemic model. For this model, the basic reproduction number is identified and its dependence on model parameters related to the testing process and isolation efficacy is discussed. The relations between the basic reproduction number, the final epidemic and peak sizes, and the model parameters are further explored numerically. We find that fast test reporting does not always benefit the control of the COVID-19 epidemic if good quarantine while awaiting test results is implemented. Moreover, the final epidemic and peak sizes do not always increase along with the basic reproduction number. Under some circumstances, lowering the basic reproduction number increases the final epidemic and peak sizes. Our findings suggest that properly implementing isolation for individuals who are waiting for their testing results would lower the basic reproduction number as well as the final epidemic and peak sizes.

Viral dynamics with immune responses: effects of distributed delays and Filippov antiretroviral therapy.

In this paper, we propose a general viral infection model to incorporate two infection modes (virus-to-cell mode and cell-to-cell mode), the CTL immune response, and the distributed intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We investigate the existence, the uniqueness, and the global stability of three equilibria: infection-free equilibrium [Formula: see text], immune-inactivated equilibrium [Formula: see text] and immune-activated equilibrium [Formula: see text], respectively. We prove that the viral dynamics are determined by two threshold parameters: the basic reproduction number for infection [Formula: see text] and the basic reproduction number for immune response [Formula: see text]. We also numerically explore the viral dynamics beyond stability. We use bifurcation diagrams to show that increasing the delay in CTL immune cell recruitment can induce a switch in viral load from a stable constant level to sustained oscillations, and then back to a stable equilibrium. We also compare the contributions of the two infection modes to the total infection level and identify the key parameters that would affect the percentages of virus-to-cell infection and cell-to-cell infection. Finally, we explore how Filippov control can be applied in antiretroviral therapy to reduce the viral loads.

A New Class of Weighted CUSUM Statistics.

A change point is a location or time at which observations or data obey two different models: before and after. In real problems, we may know some prior information about the location of the change point, say at the right or left tail of the sequence. How does one incorporate the prior information into the current cumulative sum (CUSUM) statistics? We propose a new class of weighted CUSUM statistics with three different types of quadratic weights accounting for different prior positions of the change points. One interpretation of the weights is the mean duration in a random walk. Under the normal model with known variance, the exact distributions of these statistics are explicitly expressed in terms of eigenvalues. Theoretical results about the explicit difference of the distributions are valuable. The expansions of asymptotic distributions are compared with the expansion of the limit distributions of the Cramér-von Mises statistic and the Anderson and Darling statistic. We provide some extensions from independent normal responses to more interesting models, such as graphical models, the mixture of normals, Poisson, and weakly dependent models. Simulations suggest that the proposed test statistics have better power than the graph-based statistics. We illustrate their application to a detection problem with video data.

Threshold dynamics of a nonlocal and delayed cholera model in a spatially heterogeneous environment.

A nonlocal and delayed cholera model with two transmission mechanisms in a spatially heterogeneous environment is derived. We introduce two basic reproduction numbers, one is for the bacterium in the environment and the other is for the cholera disease in the host population. If the basic reproduction number for the cholera bacterium in the environment is strictly less than one and the basic reproduction number of infection is no more than one, we prove globally asymptotically stability of the infection-free steady state. Otherwise, the infection will persist and there exists at least one endemic steady state. For the special homogeneous case, the endemic steady state is actually unique and globally asymptotically stable. Under some conditions, the basic reproduction number of infection is strictly decreasing with respect to the diffusion coefficients of cholera bacteria and infectious hosts. When these conditions are violated, numerical simulation suggests that spatial diffusion may not only spread the infection from high-risk region to low-risk region, but also increase the infection level in high-risk region.

Resonance of Periodic Combination Antiviral Therapy and Intracellular Delays in Virus Model.

There is a substantial interest in detailed models of viral infection and antiviral drug kinetics in order to optimize the treatment against viruses such as HIV. In this paper, we study within-viral dynamics under general intracellular distributed delays and periodic combination antiviral therapy. The basic reproduction number [Formula: see text] is established as a global threshold determining extinction versus persistence, and spectral methods are utilized for analytical and numerical computations of [Formula: see text]. We derive the critical maturation delay for virus and optimal phase difference between sinusoidally varying drug efficacies under various intracellular delays. Furthermore, numerical simulations are conducted utilizing realistic pharmacokinetics and gamma-distributed viral production delays for HIV. Our results demonstrate that the relative timing of the key viral replication cycle steps and periodic antiviral treatment schedule involving distinct drugs all can interact to critically affect the overall viral dynamics.

Numerical optimal control of a size-structured PDE model for metastatic cancer treatment.

In this paper, we propose a unified size-structured PDE model for the growth of metastatic tumors, which extends a well-known coupled ODE-PDE dynamical model developed and studied in the literature. A treatment model based on the proposed unified PDE model is investigated via optimal control theory, where its first-order necessary optimality system characterizing the optimal control is derived. We prove that the uniqueness of the optimal control depends on the chosen objective functional, and the optimal control is of bang-bang type when it is unique. For obtaining its efficient numerical solutions, a projection gradient descent algorithm based on the characteristic scheme is developed for solving the established optimal treatment model. Several numerical examples are provided to validate our mathematical analysis and numerical algorithm, and also illustrate the biologically interesting treatment outcomes of different models and control strategies. Our simple model reveals that: (i) only the total drug dosage matters if one just cares about the final treatment output; (ii) given the same total drug dosage, the optimal bang-bang treatment plan outperforms the others in the sense that it maximally reduces the total tumor sizes during the whole period of treatment, although their final tumor sizes are the same.

Traveling waves of di usive disease models with time delay and degeneracy.

In this paper, we propose a diffusive epidemic model with a standard incidence rate and distributed delays in disease transmission. We also consider the degenerate case when one of the diffusion coe cients vanishes. By establishing existence theory of traveling wave solutions and providing sharp lower bound for the wave speeds, we prove linear determinacy of the proposed model system. Sensitivity analysis suggests that disease propagation is slowed down by transmission delay but fastened by spatial diffusion. The existence proof is based on the construction of a suitable convex set which is invariant under the integral map of traveling wave equations. An innovative argument is formulated to study the boundary value problems of nonlinear elliptic equations satisfied by the traveling wave solutions, which enables us to prove that there does not exist a positive traveling wave connecting two nontrivial equilibria.

Dual-Gated Transistor Platform for On-Site Detection of Lead Ions at Trace Levels.

On-site monitoring of heavy metals in drinking water has become crucial because of several high profile instances of contamination. Presently, reliable techniques for trace level heavy metal detection are mostly laboratory based, while the detection limits of contemporary field-based methods are barely meeting the exposure limits set by regulatory bodies such as the World Health Organization (WHO). Here, we show an on-site deployable, Pb2+ sensor on a dual-gated transistor platform whose lower detection limit is 2 orders of magnitude better than the traditional sensor and 1 order of magnitude lower than the exposure limit set by WHO. The enhanced sensitivity of our design is verified by numerically solving PNP (Planck-Nernst-Poisson) model. We demonstrate that the enhanced sensitivity is due to the suppression of ionic flux. The simplicity and the robustness of the design make it applicable for on-site screening, thereby facilitating rapid response to contamination events.

Ebola outbreak in West Africa: real-time estimation and multiple-wave prediction.

Based on the reported data until 18 March 2015 and numerical fitting via a simple formula of cumulative case number, we provide real-time estimation on basic reproduction number, inflection point, peak time and final outbreak size of ongoing Ebola outbreak in West Africa. From our simulation, we conclude that the first wave has passed its inflection point and predict that a second epidemic wave may appear in the near future.

Serum endocan levels are correlated with the presence and severity of coronary artery disease in patients with hypertension.

BACKGROUND: Endothelial dysfunction is one of the most important early indicators of atherosclerosis in hypertension (HT) patients. Endocan has been reported to play a role in the pathophysiology of endothelial dysfunction. OBJECTIVE: We sought to assess whether serum endocan levels are correlated with the presence and severity of coronary artery disease (CAD) in patients with HT. METHODS: We measured endocan levels in 164 patients with HT and in 55 controls. The severity of CAD was assessed by the coronary atherosclerosis index scores. RESULTS: Serum endocan levels were independently correlated with the presence and severity of CAD in HT patients. CONCLUSION: Endocan might function as a useful biomarker for monitoring the development and progression of CAD in HT patients.

Global stability analysis of a delayed susceptible-infected-susceptible epidemic model.

We study a susceptible-infected-susceptible model with distributed delays. By constructing suitable Lyapunov functionals, we demonstrate that the global dynamics of this model is fully determined by the basic reproductive ratio R0. To be specific, we prove that if R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable. On the other hand, if R0>1, then the endemic equilibrium is globally asymptotically stable. It is remarkable that the model dynamics is independent of the probability of immunity lost.

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems.

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Our methodology thus provides a standard procedure for solving the PNP system and we illustrate this by solving some practical examples. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.

In vitro study of the cytotoxicities of two mixed-ligand oxovanadium complexes on human hepatoma cells.

The cytotoxicities of two oxovanadium complexes, VOI [VO(satsc)(phen)] (satsc = salicylaldehyde thiosemicarbazone, phen = 1,10-phenanthroline) and VOII [VO(3,5-dibrsatsc)(phen)](3,5-dibrsatsc = 3,5-dibromosalicylaldehyde thiosemicarbazone), were studied by performing MTT assays on human hepatoma cell lines BEL-7402, HUH-7 and HepG2. The results showed that both the VOI and VOII complexes possess significant anti-proliferative effects. In addition, the anti-proliferative mechanism of the complexes was analyzed by cell cycle analysis and an apoptosis assay and by detecting the mitochondrial membrane potential (delta psi m). The experimental results showed that the complexes can cause a G0/G1 phase cell cycle arrest and can significantly decrease delta psi m, causing depolarization of the mitochondrial membrane. Notably, the two complexes induced apoptosis in BEL-7402 cells and displayed typical morphological apoptotic characteristics. The cytotoxicities of the VOII complex are significantly stronger than that of the VOI complex, suggesting that the cytotoxic effects of oxovanadium complexes may be associated with the electronic effects of the complexes.

Richards model revisited: validation by and application to infection dynamics.

Ever since Richards proposed his flexible growth function more than half a century ago, it has been a mystery that this empirical function has made many incredible coincidences with real ecological or epidemic data even though one of its parameters (i.e., the exponential term) does not seem to have clear biological meaning. It is therefore a natural challenge to mathematical biologists to provide an explanation of the interesting coincidences and a biological interpretation of the parameter. Here we start from a simple epidemic SIR model to revisit Richards model via an intrinsic relation between both models. Especially, we prove that the exponential term in the Richards model has a one-to-one nonlinear correspondence to the basic reproduction number of the SIR model. This one-to-one relation provides us an explicit formula in calculating the basic reproduction number. Another biological significance of our study is the observation that the peak time is approximately just a serial interval after the turning point. Moreover, we provide an explicit relation between final outbreak size, basic reproduction number and the peak epidemic size which means that we can predict the final outbreak size shortly after the peak time. Finally, we introduce a constraint in Richards model to address over fitting problem observed in the existing studies and then apply our method with constraint to conduct some validation analysis using the data of recent outbreaks of prototype infectious diseases such as Canada 2009 H1N1 outbreak, GTA 2003 SARS outbreak, Singapore 2005 dengue outbreak, and Taiwan 2003 SARS outbreak. Our new formula gives much more stable and precise estimate of model parameters and key epidemic characteristics such as the final outbreak size, the basic reproduction number, and the turning point, compared with earlier simulations without constraints.