A network SIRX model for the spreading of COVID-19.

Infectious diseases, such as the current COVID-19, have a huge economic and societal impact. The ability to model its transmission characteristics is critical to minimize its impact. In fact, predicting how fast an infection is spreading could be a major factor in deciding on the severity, extent and strictness of the applied mitigation measures, such as the recent lockdowns. Even though modelling epidemics is a well studied subject, usually models do not include quarantine or other social measures, such as those imposed in the recent pandemic. The current work builds upon a recent paper by Maier and Brockmann (2020), where a compartmental SIRX model was implemented. That model included social or individual behavioural changes during quarantine, by introducing state X , in which symptomatic quarantined individuals are not transmitting the infection anymore, and described well the transmission in the initial stages of the infection. The results of the model were applied to real data from several provinces in China, quite successfully. In our approach we use a Monte-Carlo simulation model on networks. Individuals are network nodes and the links are their contacts. We use a spreading mechanism from the initially infected nodes to their nearest neighbours, as has been done previously. Initially, we find the values of the rate constants (parameters) the same way as in Maier and Brockmann (2020) for the confirmed cases of a country, on a daily basis, as given by the Johns Hopkins University. We then use different types of networks (random Erdos-Rényi, Small World, and Barabási-Albert Scale-Free) with various characteristics in an effort to find the best fit with the real data for the same geographical regions as reported in Maier and Brockmann (2020). Our simulations show that the best fit comes with the Erdos-Rényi random networks. We then apply this method to several other countries, both for large-size countries, and small size ones. In all cases investigated we find the same result, i.e. best agreement for the evolution of the pandemic with time is for the Erdos-Rényi networks. Furthermore, our results indicate that the best fit occurs for a random network with an average degree of the order of 〈 k 〉 ≈ 10-25, for all countries tested. Scale Free and Small World networks fail to fit the real data convincingly.

Tumor Hypoxia Heterogeneity Affects Radiotherapy: Inverse-Percolation Shell-Model Monte Carlo Simulations.

Tumor hypoxia was discovered a century ago, and the interference of hypoxia with all radiotherapies is well known. Here, we demonstrate the potentially extreme effects of hypoxia heterogeneity on radiotherapy and combination radiochemotherapy. We observe that there is a decrease in hypoxia from tumor periphery to tumor center, due to oxygen diffusion, resulting in a gradient of radiative cell-kill probability, mathematically expressed as a probability gradient of occupied space removal. The radiotherapy-induced break-up of the tumor/TME network is modeled by the physics model of inverse percolation in a shell-like medium, using Monte Carlo simulations. The different shells now have different probabilities of space removal, spanning from higher probability in the periphery to lower probability in the center of the tumor. Mathematical results regarding the variability of the critical percolation concentration show an increase in the critical threshold with the applied increase in the probability of space removal. Such an observation will have an important medical implication: a much larger than expected radiation dose is needed for a tumor breakup enabling successful follow-up chemotherapy. Information on the TME's hypoxia heterogeneity, as shown here with the numerical percolation model, may enable personalized precision radiation oncology therapy.

Tumor Biochemical Heterogeneity and Cancer Radiochemotherapy: Network Breakdown Zone-Model.

Breakdowns of two-zone random networks of the Erdos-Rényi type are investigated. They are used as mathematical models for understanding the incompleteness of the tumor network breakdown under radiochemotherapy, an incompleteness that may result from a tumor's physical and/or chemical heterogeneity. Mathematically, having a reduced node removal probability in the network's inner zone hampers the network's breakdown. The latter is described quantitatively as a function of reduction in the inner zone's removal probability, where the network breakdown is described in terms of the largest remaining clusters and their size distributions. The effects on the efficacy of radiochemotherapy due to the tumor micro-environment (TME)'s chemical make-up, and its heterogeneity, are discussed, with the goal of using such TME chemical heterogeneity imaging to inform precision oncology.

Fractal dimension of microbead assemblies used for protein detection.

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 µm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles.

Reaction efficiency effects on binary chemical reactions.

We study the effect of the variation of reaction efficiency in binary reactions. We use the well-known A + B → 0 model, which has been extensively studied in the past. We perform simulations on this model where we vary the efficiency of reaction, i.e., when two particles meet they do not instantly react, as has been assumed in previous studies, but they react with a probability γ, where γ is in the range 0 < γ < 1. Our results show that at small γ values the system is reaction limited, but as γ increases it crosses over to a diffusion limited behavior. At early times, for small γ values, the particle density falls slower than for larger γ values. This fall-off goes over a crossover point, around the value of γ = 0.50 for high initial densities. Under a variety of conditions simulated, we find that the crossover point was dependent on the initial concentration but not on the lattice size. For intermediate and long times simulations, all γ values (in the depleted reciprocal density versus time plot) converge to the same behavior. These theoretical results are useful in models of epidemic reactions and epidemic spreading, where a contagion from one neighbor to the next is not always successful but proceeds with a certain probability, an analogous effect with the reaction probability examined in the current work.

An electrostatics method for converting a time-series into a weighted complex network.

This paper proposes a new method for converting a time-series into a weighted graph (complex network), which builds on electrostatics in physics. The proposed method conceptualizes a time-series as a series of stationary, electrically charged particles, on which Coulomb-like forces can be computed. This allows generating electrostatic-like graphs associated with time-series that, additionally to the existing transformations, can be also weighted and sometimes disconnected. Within this context, this paper examines the structural similarity between five different types of time-series and their associated graphs that are generated by the proposed algorithm and the visibility graph, which is currently the most popular algorithm in the literature. The analysis compares the source (original) time-series with the node-series generated by network measures (that are arranged into the node-ordering of the source time-series), in terms of a linear trend, chaotic behaviour, stationarity, periodicity, and cyclical structure. It is shown that the proposed electrostatic graph algorithm generates graphs with node-measures that are more representative of the structure of the source time-series than the visibility graph. This makes the proposed algorithm more natural rather than algebraic, in comparison with existing physics-defined methods. The overall approach also suggests a methodological framework for evaluating the structural relevance between the source time-series and their associated graphs produced by any possible transformation.

Priority diffusion model in lattices and complex networks.

We introduce a model for diffusion of two classes of particles (A and B ) with priority: where both species are present in the same site the motion of A's takes precedence over that of B's. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal, but the B particles are significantly slower due to the presence of the A particles. From the fraction of sites where the B particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where B's are free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes, leads to trapping of the low priority particles in scale-free networks.

Spontaneous repulsion in the A+B→0 reaction on coupled networks.

We study the transient dynamics of an A+B→0 process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions q of cross couplings, the concentration of A (or B) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time t_{x}. By numerical and analytical arguments, we show that for symmetric and homogeneous structures t_{x}∝(〈k〉/q)log(〈k〉/q) where 〈k〉 is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like 〈k〉/q, we identify the logarithmic slowing down in t_{x} to be the result of a spontaneous mechanism of repulsion between the reactants A and B due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

Improving immunization strategies.

We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need for knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model and find it to be significantly more effective.

Matrix Density Engineering of Hydrogel Nanoparticles with Simulation-Guided Synthesis for Tuning Drug Release and Cellular Uptake.

The use of a nanoparticle (NP)-based antitumor drug carrier has been an emerging strategy for selectively delivering the drugs to the tumor area and, thus, reducing the side effects that are associated with a high systemic dose of antitumor drugs. Precise control of drug loading and release is critical so as to maximize the therapeutic index of the NPs. Here, we propose a simple method of synthesizing NPs with tunable drug release while maintaining their loading ability, by varying the polymer matrix density of amine- or carboxyl-functionalized hydrogel NPs. We find that the NPs with a loose matrix released more cisplatin, with up to a 33 times faster rate. Also, carboxyl-functionalized NPs loaded more cisplatin and released it at a faster rate than amine-functionalized NPs. We performed detailed Monte Carlo computer simulations that elucidate the relation between the matrix density and drug release kinetics. We found good agreement between the simulation model and the experimental results for drug release as a function of time. Also, we compared the cellular uptake between amine-functionalized NPs and carboxyl-functionalized NPs, as a higher cellular uptake of NPs leads to improved cisplatin delivery. The amine-functionalized NPs can deliver 3.5 times more cisplatin into cells than the carboxyl-functionalized NPs. The cytotoxic efficacy of both the amine-functionalized NPs and the carboxyl-functionalized NPs showed a strong correlation with the cisplatin release profile, and the latter showed a strong correlation with the NP matrix density.

Characteristics of reaction-diffusion on scale-free networks.

We examine some characteristic properties of reaction-diffusion processes of the A+A-->0 type on scale-free networks. Due to the inhomogeneity of the structure of the substrate, as compared to usual lattices, we focus on the characteristics of the nodes where the annihilations occur. We show that at early times the majority of these events take place on low-connectivity nodes, while as time advances the process moves towards the high-connectivity nodes, the so-called hubs. This pattern remarkably accelerates the annihilation of the particles, and it is in agreement with earlier predictions that the rates of reaction-diffusion processes on scale-free networks are much faster than the equivalent ones on lattice systems.

Effect of a slit-shaped trap on depletion kinetics within a microchannel.

The diffusion-limited trapping reaction kinetics of the growth of the depletion zone within and around a "slit-shaped" trap in a flat microchannel was studied experimentally and numerically. In the experiment, an ellipse-shaped laser beam acted as a slit trap in a long, flat capillary, and the trapping reaction is photobleaching of fluorescein dye. The parameter studied was the theta distance, i.e., the distance from the trap to the point where the reactant concentration has been locally depleted to the specific survival fraction [theta] of its initial bulk value. When the trap is perfect, then, due to the geometry of the trap and the reactor, as many as three time regimes can be found, with up to two crossover transitions. The number of crossovers is determined by the relative sizes of the trap and the microreactor. In the case of two crossovers, we show that the first crossover relates to the length of the trap, while the second crossover relates to the width of the reactor. When the slit trap is imperfect and its width cannot be neglected, as is the case in the experiments, a nontrivial early behavior is observed, followed by two regions in time, separated by a single crossover only.

Reaction-diffusion processes on correlated and uncorrelated scale-free networks.

We compare reaction-diffusion processes of the A+A-->0 type on scale-free networks created with either the configuration model or the uncorrelated configuration model. We show via simulations that except for the difference in the behavior of the two models, different results are observed within the same model when the minimum number of connections for a node varies from k(min) =1 to k(min) =2 . This difference is attributed to the varying local properties of the two systems. In all cases, we are able to identify a power-law behavior of the density decay with time, with an exponent f>1 , considerably larger than its lattice counterpart.

Depletion kinetics in the photobleaching trapping reaction inside a flat microchannel.

The diffusion-limited kinetics of the growth of a depletion zone around a static point trap in a thin, long channel geometry was studied using a laser photobleaching experiment of fluorescein dye inside a flat rectangular capillary. The dynamics of the depletion zone was monitored by the theta distance, defined as the distance from the trap to the point where the reactant concentration has been locally depleted to the specified survival fraction (theta) of its initial bulk value. A dimensional crossover from two dimensions to one dimension, due to the finite width of the reaction zone, was observed. We define a "parallel" and a "perpendicular" theta distance, along the slab long and short dimensions, respectively, and study their time development as a means to study the asymmetrical nature of the slab geometry. For all theta values, the crossover occurs concurrently for both theta distances when the depletion zone touches the boundary for the first time. We derive theoretical expressions for this geometry and compare them with the experimental data. We also obtain important insight from the ratio of the reactant concentration profiles in the parallel and perpendicular directions. Exact enumeration and Monte Carlo simulations support the anomalous depletion scaling results. Nevertheless, the crossover time (tau(c)) is still found to scale with the width (W) of the rectangular reaction zone as tau(c) approximately W2 , as expected from the basic Einstein diffusion law.

Modeling and Monte Carlo simulations in oral drug absorption.

Drug dissolution, release and uptake are the principal components of oral drug absorption. All these processes take place in the complex milieu of the gastrointestinal tract and they are influenced by physiological (e.g. intestinal pH, transit time) and physicochemical factors (e.g. dose, particle size, solubility, permeability). Due to the enormous complexity issues involved, the models developed for drug dissolution and release attempt to capture their heterogeneous features. Hence, Monte Carlo simulations and population methods have been utilized since both dissolution and release processes are considered as time evolution of a population of drug molecules moving irreversibly from the solid state to the solution. Additionally, mathematical models have been proposed to determine the effect of the physicochemical properties, solubility/dose ratio and permeability on the extent of absorption for regulatory purposes, e.g. biopharmaceutics classification. The regulatory oriented approaches are based on the tube model of the intestinal lumen and apart from the drug's physicochemical properties, take into account the formulation parameters the dose and the particle size.

Efficiency of message transmission using biased random walks in complex networks in the presence of traps.

We study the problem of a particle or message that travels as a biased random walk towards a target node in a network in the presence of traps. The bias is represented as the probability p of the particle to travel along the shortest path to the target node. The efficiency of the transmission process is expressed through the fraction f(g) of particles that succeed to reach the target without being trapped. By relating f(g) with the number S of nodes visited before reaching the target, we first show that, for the unbiased random walk, f(g) is inversely proportional to both the concentration c of traps and the size N of the network. For the case of biased walks, a simple approximation of S provides an analytical solution that describes well the behavior of f(g), especially for p>0.5. Also, it is shown that for a given value of the bias p, when the concentration of traps is less than a threshold value equal to the inverse of the mean first passage time (MFPT) between two randomly chosen nodes of the network, the efficiency of transmission is unaffected by the presence of traps and almost all the particles arrive at the target. As a consequence, for a given concentration of traps, we can estimate the minimum bias that is needed to have unaffected transmission, especially in the case of random regular (RR), Erdos-Rényi (ER) and scale-free (SF) networks, where an exact expression (RR and ER) or an upper bound (SF) of the MFPT is known analytically. We also study analytically and numerically, the fraction f(g) of particles that reach the target on SF networks, where a single trap is placed on the highest degree node. For the unbiased random walk, we find that f(g)∼N(-1/(γ-1)), where γ is the power law exponent of the SF network.

Analysis of Case II drug transport with radial and axial release from cylinders.

Analysis is presented for Case II drug transport with axial and radial release from cylinders. The previously reported [J. Control Release 5 (1987) 37] relationships for radial release from films and slabs are special cases of the general solution derived in this study. The widely used exponential relation M(t)/M(infinity) = kt(n) describes nicely the first 60% of the fractional release curve when Case II drug transport with axial and radial release from cylinders is operating.

Diffusion-driven spreading phenomena: the structure of the hull of the visited territory.

We study the hull of the territory visited by N random walkers after t time steps. The walkers move on two-dimensional substrates, starting all from the same position. For the substrate, we consider (a). a square lattice and (b). a percolation cluster at criticality. On the square lattice, we (c). also allow for birth and death processes, where at every time step, alphaN walkers die and are removed from the substrate, and simultaneously the same number of walkers is added randomly at the positions of the remaining walkers, such that the total numbers of walkers is constant in time. We perform numerical simulations for the three processes and find that for all of them, the structure of the hull is self-similar and described by a fractal dimension d(H) that slowly approaches, with an increasing number of time steps, the value d(H)=4/3. For process (c), however, the time to approach the asymptotic value increases drastically with increasing fraction of N/alpha, and can be observed numerically only for sufficiently small values of N/alpha.

Anomalous growth of the depletion zone in the photobleaching trapping reaction.

We study the anomalous growth of the depletion zone at a single trap, as observed in a photobleaching trapping reaction in confined geometry. We provide experimental evidence for a nonuniversal growth of this depletion. We also find an early-time behavior of the depletion zone, owing to the finite size of the trap. Various laser powers are used in order to study the effects of trapping strength, interpreted theoretically in terms of an imperfect trap. The results are supported by numerical calculations. Comparison with other trapping reactions provides insight into finite-size traps.

Dynamics of the depletion zone at a finite-sized imperfect trap in two dimensions: photobleaching experiments and simulations.

The kinetics of the growth of depletion zones around a static trap in an effective two-dimensional geometry were studied experimentally with photobleaching of fluorescein dye by a focused laser beam. The phototrap served as an imperfect trap with a finite size. The growth of the depletion zone was monitored by the theta distance, defined as the distance from the trap to the point where the concentration of the reactants reaches a given arbitrary fraction theta (0