Global stability of a multiple infected compartments model for waterborne diseases

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number $R_0$, we prove that, if $R_0⩽1$, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if $R_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.     

On the compositum of integral closures of valuation rings

It is well known that if $K_1,K_2$ are algebraic number fields with coprime discriminants, then the composite ring $A_{K_1}{A}_{K_2}$ is integrally closed and $K_1,K_2$ are linearly disjoint over the field of rationals, $A_{K_i}$ being the ring of algebraic integers of $K_i$. In an attempt to prove the converse of the above result, in this paper we prove that if $K_1,K_2$ are finite separable extensions of a valued field $(K,v)$ of arbitrary rank which are linearly disjoint over $K=K_1\cap K_2$ and if the integral closure $S_i$ of the valuation ring $R_v$ of v in $K_i$ is a free $R_v$-module for $i=1,2$ with $S_1{S}_2$ integrally closed, then the discriminant of either $S_1/R_v$ or of $S_2/R_v$ is the unit ideal. We quickly deduce from this result that for algebraic number fields $K_1,K_2$ linearly disjoint over $K=K_1\cap K_2$ for which $A_{K_1}{A}_{K_2}$ is integrally closed, the relative discriminants of $K_1/K$ and $K_2/K$ must be coprime.     

Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay

In this paper, we study the dynamics of an SIR epidemic model with a logistic process and a distributed time delay. We first show that the attractivity of the disease-free equilibrium is completely determined by a threshold $R_0$. If $R_0?1$, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if $R_0>1$, then the disease-free equilibrium is unstable, and meanwhile there exists uniquely an endemic equilibrium. We then prove that for any time delay $h>0$, the delayed SIR epidemic model is permanent if and only if there exists an endemic equilibrium. In other words, $R_0>1$ is a necessary and sufficient condition for the permanence of the epidemic model. Numerical examples are given to illustrate the theoretical results. We also make a distinction between the dynamics of the distributed time delay system and the discrete time delay system.     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.