Optimal control theory of an age-structured coronavirus disease model and the dynamical analysis of the underlying ordinary differential equation model having constant parameters

This paper addresses the dynamics of COVID-19 using the approach of age-structured modeling. A particular case of the model is presented by taking into account age-free parameters. The sub-model consisting of ordinary differential equations (ODEs) is investigated for possible equilibria, and qualitative aspects of the model are rigorously presented. In order to control the spread of the disease, we considered two age- and time-dependent non-pharmaceutical control measures in the age-structured model, and an optimal control problem using a general maximum principle of Pontryagin type is achieved. Finally, sample simulations are plotted which support our theoretical work.     

Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms

We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has, in every column, blocks of consecutive 1s, each corresponding to a shift. We focus on three types of shift scheduling problems classified according to the column structure in the constraint matrix: columns of consecutive 1s, columns of cyclical 1s, and columns of k consecutive blocks. In particular, the complexity of the cyclical scheduling problem, where the matrix satisfies the property of cyclical 1s in each column, was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2-approximation algorithm for the non-uniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem—a problem the complexity status of which is known to be randomized polynomial (RP). We then investigate the three types of shift scheduling problems and show that, while the consecutive ones version is polynomial and the k-block columns version is NP-hard (for k≥2), for the k-blocks problem we give a simple k-approximation algorithm, which is the first approximation algorithm determined for the problem.     

Testing satisfiability

Let Φ be a set of general boolean functions on n variables, such that each function depends on exactly k variables, and each variable can take a value from [1,d]. We say that Φ is ε-far from satisfiable, if one must remove at least εn^k functions in order to make the set of remaining functions satisfiable. Our main result is that if Φ is ε-far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k,d)/ε², are not satisfiable, where c(k,d) depends only on k and d. Using the above claim, we obtain similar results for k-SAT and k-NAEQ-SAT.Assume we relax the decision problem of whether an instance of one of the above mentioned problems is satisfiable or not, to the problem of deciding whether an instance is satisfiable or ε-far from satisfiable. While the above decision problems are NP-hard, our result implies that we can solve their relaxed versions, that is, distinguishing between satisfiable and ε-far from satisfiable instances, in randomized constant time.From the above result we obtain as a special case, previous results of Alon and Krivelevich, and of Czumaj and Sohler, concerning testing of graphs and hypergraphs colorability. We also discuss the difference between testing with one-sided and two-sided error.     

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 ? o(1))np Hamilton cycles C₁,…,C_t, each of which is oriented arbitrarily, a digraph ～(n, p) w.h.p. contains edge disjoint copies of C₁,…,C_t, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ～(n, p) w.h.p. contains (1 ± o(1))n!pⁿ copies of C, provided .     

On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Let f_r(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemerédi and of Erdős, Frankl and R?dl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤k<r, we have

Critical percolation on random regular graphs

The behavior of the random graph G(n,p) around the critical probability p_c = is well understood. When p = (1 + O(n^1/3))p_c the components are roughly of size n^2/3 and converge, when scaled by n^?2/3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 ‐ ?(n))p_c with ?(n)n^1/3 →∞ (the subcritical regime) the largest component is concentrated around 2?^?2 log(?³n). When p = (1 + ?(n))p_c with ?(n)n^1/3 →∞ (the supercritical regime), the largest component is concentrated around 2?n and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d‐regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non‐critical regimes) extends to random d‐regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Weighted matching with pair restrictions

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.     

Self‐Assembled Cyclic d,l‐α‐Peptides as Generic Conformational Inhibitors of the α‐Synuclein Aggregation and Toxicity: In Vitro and Mechanistic Studies

Many peptides and proteins with large sequences and structural differences self‐assemble into disease‐causing amyloids that share very similar biochemical and biophysical characteristics, which may contribute to their cross‐interaction. Here, we demonstrate how the self‐assembled, cyclic d,l ‐α‐peptide CP‐2 , which has similar structural and functional properties to those of amyloids, acts as a generic inhibitor of the Parkinson′s disease associated α‐synuclein (α‐syn) aggregation to toxic oligomers by an ?off‐pathway“ mechanism. We show that CP‐2 interacts with the N‐terminal and the non‐amyloid‐β component region of α‐syn, which are responsible for α‐syn′s membrane intercalation and self‐assembly, thus changing the overall conformation of α‐syn. CP‐2 also remodels α‐syn fibrils to nontoxic amorphous species and permeates cells through endosomes/lysosomes to reduce the accumulation and toxicity of intracellular α‐syn in neuronal cells overexpressing α‐syn. Our studies suggest that targeting the common structural conformation of amyloids may be a promising approach for developing new therapeutics for amyloidogenic diseases.