Our study investigates the dynamics of disease interaction and persistence within populations, exploring various epidemic scenarios, including backward bifurcation and cross-immunity effects. We establish conditions under which the disease-free equilibrium of the model demonstrates local or global asymptotic stability, contingent on the efficacy of quarantine measures. Notably, we find that a strain with a quarantine reproduction number greater than 1 will out-compete a strain with a quarantine reproduction number less than 1, leading to its extinction under complete immunity conditions. Additionally, we identify scenarios where diseases persist in a sub-critical coexistence endemic equilibrium, despite one control reproduction number being below one. Our exploration of backward bifurcation reveals the model's capacity to accommodate the coexistence of the disease-free equilibrium with up to four endemic equilibria. Moreover, we demonstrate that the existence of cross-immunity enhances the coexistence of two strains. However, co-infections and imperfect quarantine measures pose significant challenges in containing outbreaks, sustaining the outbreak potential even with successful control of individual virus strains. Conversely, controlling outbreaks becomes more manageable in the absence of co-infections, especially with perfect quarantine measures. We conclude by advocating for public health strategies that address the complexities posed by co-infections, emphasizing the importance of simultaneously tackling multiple pathogens.

In a previous paper, the authors established a link between generalized grey Brownian motions (ggBm) and generalized grey Ornstein-Uhlenbeck processes as an extension of the results in by using a representation of ggBm as a product of a positive and time-independent random variable and an fBm. The results in this publication are wrong, since they just considered the moving average representation of fractional Brownian motion, omitting the residual part. In this short note, we fix this gap and give corrected formulas for the representation of ggBm using an infinite dimensional superposition of generalized grey Ornstein-Uhlenbeck processes.

The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras L_k(E) and L_k(F) are graded Morita equivalent, i.e., Gr-L_k(E) approximate to _gr Gr-L_k(F), if and only if, their graded Grothendieck groups are isomorphic K₀^gr(L_k(E)) congruent to K₀^gr(L_k(F)) as order-preserving Z[x,x^-1]-modules. Furthermore, if under this isomorphism, the class [L_k(E)] is sent to [L_k(F)] then the algebras are graded isomorphic, i.e., L_k(E) congruent to _gr L_k(F). In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving Z[x,x^-1]-module isomorphism K₀^gr(L_k(E)) congruent to K₀^gr(L_k(F)) gives that the categories of locally finite dimensional graded modules of L_k(E) and L_k(F) are equivalent, i.e., gr^Z-L_k(E) approximate to _grgr^Z-L_k(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod- L_k(E) approximate to mod L_k(F) and gr-L_k(E) approximate to _gr gr-L_k(F).

In this paper we show variant of the spectral theorem using an algebraic Jordan-Schwinger map. The advantage of this approach is that we don't have restriction of normality on the class of operators we consider.

The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand-Kirillov dimension and the entropy. We show that path algebras over finite graphs can be classified to be of finite dimension, finite Gelfand-Kirillov dimension or finite algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence but perhaps for a different filtration. In addition we give several examples of the entropy in path algebras and Leavitt path algebras.

The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein-Uhlenbeck process.

In this note we prove that the algebras and L_K(E) have the same entropy. Entropy is always referred to the standard filtrations in the corresponding kind of algebra. The main argument leans on (1) the holomorphic functional calculus; (2) the relation of entropy with suitable norm of the adjacency matrix; and (3) the Cohn path algebras which yield suitable bounds for the algebraic entropies.

In this paper, we study properties such as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand-Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.

Current estimates of pandemic SARS-CoV-2 spread in Germany using infectious disease models often do not use age-specific infection parameters and are not always based on age-specific contact matrices of the population. They also do usually not include setting- or pandemic phase-based information from epidemiological studies of reported cases and do not account for age-specific underdetection of reported cases. Here, we report likely pandemic spread using an age-structured model to understand the age- and setting-specific contribution of contacts to transmission during different phases of the COVID-19 pandemic in Germany. We developed a deterministic SEIRS model using a pre-pandemic contact matrix. The model was optimized to fit age-specific SARS-CoV-2 incidences reported by the German National Public Health Institute (Robert Koch Institute), includes information on setting-specific reported cases in schools and integrates age- and pandemic period-specific parameters for underdetection of reported cases deduced from a large population-based seroprevalence studies. Taking age-specific underreporting into account, younger adults and teenagers were identified in the modeling study as relevant contributors to infections during the first three pandemic waves in Germany. For the fifth wave, the Delta to Omicron transition, only age-specific parametrization reproduces the observed relative and absolute increase in pediatric hospitalizations in Germany. Taking into account age-specific underdetection did not change considerably how much contacts in schools contributed to the total burden of infection in the population (up to 12% with open schools under hygiene measures in the third wave). Accounting for the pandemic phase and age-specific underreporting is important to correctly identify those groups of the population in which quarantine, testing, vaccination, and contact-reduction measures are likely to be most effective and efficient. Age-specific parametrization is also highly relevant to generate informative age-specific output for decision makers and resource planers.

In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the generators of the talented monoid. With the help of this, we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. This then translates to a correspondence between the composition series of the talented monoid and the so-called matrix composition series of the adjacency matrix. In addition, we discuss the number of cycles in a graph. In particular, we give an equivalent characterization of acyclic graphs via the adjacency matrix, the talented monoid and the Leavitt path algebra. Finally, we compute the number of linearly independent paths of certain length in the Leavitt path algebra via adjacency matrices.