Pre-exposure prophylaxis (PrEP) is an effective strategy for HIV prevention, offering individual protection and broader public health benefits. Enrollment in PrEP programs not only provides access to HIV prevention but also serves as a strategic entry point for the diagnosis of other sexually transmitted infections (STIs). Because participation requires regular HIV testing and routine STI screening (e.g., for syphilis), PrEP implementation facilitates early detection and treatment of coexisting infections, strengthening integrated sexual health surveillance and control efforts. We developed a mathematical model capturing syphilis dynamics, incorporating PrEP as a mechanism for diagnosis and treatment engagement. The model considers coinfection via high-risk sexual contact, partial protection of PrEP (HIV but not syphilis), and diagnostic pathways linked to PrEP program entry. Independent analysis of syphilis (without PrEP) established population persistence, basic reproduction numbers, and stability of disease-free and endemic equilibria. Integrating PrEP, we derived conditions under which PrEP-related parameters—particularly diagnostic access—positively influence syphilis transmission dynamics. Sensitivity analysis showed that higher PrEP adherence reduces reproduction numbers for syphilis and coinfection. Computational simulations using literature-based parameters confirmed these findings: increased PrEP use and lower discontinuation rates decreased new infections and improved treatment outcomes. These results highlight the role of PrEP in improving the detection and treatment of syphilis and HIV–syphilis coinfection.

The Boolean logic of subsets, usually presented as 'propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical."But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense.Our purpose here is to explore the notions of implication and negation in that other mathematical logic of partitions.

<p>The Euler-Sombor index <span class="math inline">\(EU\)</span> is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices <span class="math inline">\(u,v\)</span> of the underlying graph, of the terms <span class="math inline">\(\sqrt{d_u^2+d_v^2+d_u\,d_v}\)</span>, where <span class="math inline">\(d_u\)</span> and <span class="math inline">\(d_v\)</span> are the degrees of the vertices <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span>, respectively. For a real number <span class="math inline">\(\lambda\)</span>, a variable version of <span class="math inline">\(EU\)</span> is constructed, denoted by <span class="math inline">\(EU(\lambda)\)</span>, defined via <span class="math inline">\(\sqrt{d_u^2+d_v^2+\lambda\,d_u\,d_v}\)</span>. Its special cases for <span class="math inline">\(\lambda=2,\,-2,\,0\)</span>, and 1 are, respectively, the first Zagreb, Albertson, Sombor, and the ordinary Euler-Sombor indices. The basic properties of <span class="math inline">\(EU(\lambda)\)</span> are determined, including a method for its approximate calculation and bounds in terms of minimum degree, maximum degree, order and size for several graph products. It is shown how to find values of <span class="math inline">\(\lambda\)</span> for which <span class="math inline">\(EU(\lambda)\)</span> is optimal with regard to predicting molecular properties.</p>

In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which can be inferred straightforwardly from our results.

In this article, we establish fixed point outcomes for mappings that are asymptotically regular within the context of b -metric spaces. These findings broaden and enhance the familiar outcomes found in existing literature. Additionally, we present corollaries to demonstrate that our results are more encompassing compared to the established findings in the literature.

In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for \(0 < x < \frac{\pi}{2}\) and any \(n \geq 5\) $$0 < P_n(x)\ <\ (\sin x)^2- x^3\cot x < P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} - \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} $$ where \(P_n(x) = \sum_{3=k}^n a_k x^{2k+1}\) is a \(n\)-polynomial, with positive coefficients (\(k \geq 5\)), \(a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),\) \( B_{2k} \) are Bernoulli numbers. This improves a lot of lower bounds of \( \frac{\sin(x)}{x}\) and generalizes inequalities chains. Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function $$g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 - \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 1 $$

Motivated by Euler-Goldbach and Shallit-Zikan theorems, we introduce zeta-one functions with infinite sums of \(n^{s}\pm1\) as an analogy of the Riemann zeta function. Then we compute values of these functions at positive even integers by the residue theorem.

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one extra free parameter to them. In particular, we generalize generating functions for the continuous \(q\)-ultraspherical/Rogers, little \(q\)-Laguerre/Wall, and \(q\)-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and \(q\)-integrals.

Let \(S\) be a dominating set of a graph \(G\). The set \(S\) is called a pendant dominating set of \(G\) if the induced subgraph of \(S\) contains a minimum of one node of degree one. The minimum cardinality of the pendant dominating set in \(G\) is referred to as the pendant domination number of \(G\), indicated by \(\gamma_{pe}(G)\). This article analyzes the effect of \(\gamma_{pe}(G)\) when an arbitrary node or edge is removed.