Simulation and sequential dynamical systems

Computer simulations have a generic structure. Motivated by this the authors present a new class of discrete dynamical systems that captures this structure in a mathematically precise way. This class of systems consists of (1) a loopfree graph {Upsilon} with vertex set {l_brace}1,2,{hor_ellipsis},n{r_brace} where each vertex has a binary state, (2) a vertex labeled set of functions (F{sub i,{Upsilon}}:F{sub 2}{sup n} {yields} F{sub 2}{sup n}){sub i} and (3) a permutation {pi} {element_of} S{sub n}. The function F{sub i,{Upsilon}} updates the state of vertex i as a function of the states of vertex i and its {Upsilon}-neighbors and leaves the states of all other vertices fixed. The permutation {pi} represents the update ordering, i.e., the order in which the functions F{sub i,{Upsilon}} are applied. By composing the functions F{sub i,{Upsilon}} in the order given by {pi} one obtains the dynamical system (equation given in paper), which the authors refer to as a sequential dynamical system, or SDS for short. The authors will present bounds for the number of functionally different systems and for the number of nonisomorphic digraphs {Gamma}[F{sub {Upsilon}},{pi}] that can be obtained by varying the update order and applications of these to specific graphs and graph classes

On theoretical issues of computer simulations sequential dynamical systems

The authors study a class of discrete dynamical systems that is motivated by the generic structure of simulations. The systems consist of the following data: (a) a finite graph Y with vertex set {l_brace}1,...,n{r_brace} where each vertex has a binary state, (b) functions F{sub i}:F{sub 2}{sup n} {r_arrow} F{sub 2}{sup n} and (c) an update ordering {pi}. The functions F{sub i} update the binary state of vertex i as a function of the state of vertex i and its Y-neighbors and leave the states of all other vertices fixed. The update ordering is a permutation of the Y-vertices. They derive a decomposition result, characterize invertible SDS and study fixed points. In particular they analyze how many different SDS that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number. Finally, some specific SDS are investigated

Potential impact of annual vaccination with reformulated COVID-19 vaccines: Lessons from the US COVID-19 scenario modeling hub

Background AU Coronavirus Disease 2019 (COVID-19) continues to cause :significant hospitalizations and deaths in the United States. Its continued burden and the impact of annually reformulated vaccines remain unclear. Here, we present projections of COVID-19 hospitalizations and deaths in the United States for the next 2 years under 2 plausible assumptions about immune escape (20% per year and 50% per year) and 3 possible CDC recommendations for the use of annually reformulated vaccines (no recommendation, vaccination for those aged 65 years and over, vaccination for all eligible age groups based on FDA approval). Methods and findings The COVID-19 Scenario Modeling Hub solicited projections of COVID-19 hospitalization and deaths between April 15, 2023 and April 15, 2025 under 6 scenarios representing the intersection of considered levels of immune escape and vaccination. Annually reformulated vaccines are assumed to be 65% effective against symptomatic infection with strains circulating on June 15 of each year and to become available on September 1. Age- and state-specific coverage in recommended groups was assumed to match that seen for the first (fall 2021) COVID-19 booster. State and national projections from 8 modeling teams were ensembled to produce projections for each scenario and expected reductions in disease outcomes due to vaccination over the projection period. From April 15, 2023 to April 15, 2025, COVID-19 is projected to cause annual epidemics peaking November to January. In the most pessimistic scenario (high immune escape, no vaccination recommendation), we project 2.1 million (90% projection interval (PI) [1,438,000, 4,270,000]) hospitalizations and 209,000 (90% PI [139,000, 461,000]) deaths, exceeding pre-pandemic mortality of influenza and pneumonia. In high immune escape scenarios, vaccination of those aged 65+ results in 230,000 (95% confidence interval (CI) [104,000, 355,000]) fewer hospitalizations and 33,000 (95% CI [12,000, 54,000]) fewer deaths, while vaccination of all eligible individuals results in 431,000 (95% CI: 264,000–598,000) fewer hospitalizations and 49,000 (95% CI [29,000, 69,000]) fewer deaths. Conclusions COVID-19 is projected to be a significant public health threat over the coming 2 years. Broad vaccination has the potential to substantially reduce the burden of this disease, saving tens of thousands of lives each year

Impact of SARS-CoV-2 vaccination of children ages 5–11 years on COVID-19 disease burden and resilience to new variants in the United States, November 2021–March 2022: A multi-model study

Background: The COVID-19 Scenario Modeling Hub convened nine modeling teams to project the impact of expanding SARS-CoV-2 vaccination to children aged 5–11 years on COVID-19 burden and resilience against variant strains. Methods: Teams contributed state- and national-level weekly projections of cases, hospitalizations, and deaths in the United States from September 12, 2021 to March 12, 2022. Four scenarios covered all combinations of 1) vaccination (or not) of children aged 5–11 years (starting November 1, 2021), and 2) emergence (or not) of a variant more transmissible than the Delta variant (emerging November 15, 2021). Individual team projections were linearly pooled. The effect of childhood vaccination on overall and age-specific outcomes was estimated using meta-analyses. Findings: Assuming that a new variant would not emerge, all-age COVID-19 outcomes were projected to decrease nationally through mid-March 2022. In this setting, vaccination of children 5–11 years old was associated with reductions in projections for all-age cumulative cases (7.2%, mean incidence ratio [IR] 0.928, 95% confidence interval [CI] 0.880–0.977), hospitalizations (8.7%, mean IR 0.913, 95% CI 0.834–0.992), and deaths (9.2%, mean IR 0.908, 95% CI 0.797–1.020) compared with scenarios without childhood vaccination. Vaccine benefits increased for scenarios including a hypothesized more transmissible variant, assuming similar vaccine effectiveness. Projected relative reductions in cumulative outcomes were larger for children than for the entire population. State-level variation was observed. Interpretation: Given the scenario assumptions (defined before the emergence of Omicron), expanding vaccination to children 5–11 years old would provide measurable direct benefits, as well as indirect benefits to the all-age U.S. population, including resilience to more transmissible variants. Funding: Various (see acknowledgments)

Evaluation of the US COVID-19 Scenario Modeling Hub for informing pandemic response under uncertainty

Our ability to forecast epidemics far into the future is constrained by the many complexities of disease systems. Realistic longer-term projections may, however, be possible under well-defined scenarios that specify the future state of critical epidemic drivers. Since December 2020, the U.S. COVID-19 Scenario Modeling Hub (SMH) has convened multiple modeling teams to make months ahead projections of SARS-CoV-2 burden, totaling nearly 1.8 million national and state-level projections. Here, we find SMH performance varied widely as a function of both scenario validity and model calibration. We show scenarios remained close to reality for 22 weeks on average before the arrival of unanticipated SARS-CoV-2 variants invalidated key assumptions. An ensemble of participating models that preserved variation between models (using the linear opinion pool method) was consistently more reliable than any single model in periods of valid scenario assumptions, while projection interval coverage was near target levels. SMH projections were used to guide pandemic response, illustrating the value of collaborative hubs for longer-term scenario projections

NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

In this paper we study the evolution of sequential dynamical systems $(\mathsf{SDS})$ as a result of the erroneous replication of the SDS words. An $\mathsf{SDS}$ consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The $\mathsf{SDS}$ over the word w and Y is the composed map: $[\mathfrak{F}_Y,w]=\prod_{i=1}^{k} F_{w_i}$. The word w represents the genotype of the $\mathsf{SDS}$ in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the $\mathsf{SDS}$. We introduce combinatorial properties of $\mathsf{SDS}$ which allow us to construct a new distance measure $\mathsf{D}$ for words. We show that $\mathsf{D}$ captures the similarity of corresponding $\mathsf{SDS}$. We will use the distance measure $\mathsf{D}$ to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.Sequential dynamical systems, neutral evolution, error thresholds, acyclic orientations

c ○ World Scientific Publishing Company NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

In this paper we study the evolution of sequential dynamical systems (SDS) asaresult of the erroneous replication of the SDS words. An SDS consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,...,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The SDS over the word w and Y is the composed map: [FY,w] = Qk i=1 Fwi.Thewordwrepresents the genotype of the SDS in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the SDS. We introduce combinatorial properties of SDS which allow us to construct a new distance measure D for words. We show that D captures the similarity of corresponding SDS. We will use the distance measure D to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring

REDUCTION OF DISCRETE DYNAMICAL SYSTEMS OVER GRAPHS

In this paper we study phase space relations in a certain class of discrete dynamical systems over graphs. The systems we investigate are called Sequential Dynamical Systems (SDSs), which are a class of dynamical systems that provide a framework for analyzing computer simulations. Specifically, an SDS consists of (i) a finite undirected graph Y with vertex set {1,2,…,n} where each vertex has associated a binary state, (ii) a collection of Y-local functions (Fi,Y)i∈v[Y] with $F_{i,Y}: \mathbb{F}_2^n\to \mathbb{F}_2^n$ and (iii) a permutation π of the vertices of Y. The SDS induced by (i)–(iii) is the map $$[F_Y,\pi] = F_{\pi(n),Y} \circ \cdots \circ F_{\pi(1),Y}\,.$$ The paper is motivated by a general reduction theorem for SDSs which guarantees the existence of a phase space embedding induced by a covering map between the base graphs of two SDSs. We use this theorem to obtain information about phase spaces of certain SDSs over binary hypercubes from the dynamics of SDSs over complete graphs. We also investigate covering maps over binary hypercubes, $Q_2^n$, and circular graphs, Circn. In particular we show that there exists a covering map $\phi: Q_2^n\to K_{n+1}$ if and only if 2n≡0 mod n+1. Furthermore, we provide an interpretation of a class of invertible SDSs over circle graphs as right-shifts of length n-2 over {0,1}2n-2. The paper concludes with a brief discussion of how we can extend a given covering map to a covering map over certain extended graphs.Sequential dynamical systems, graph morphisms, covering maps, phase space embeddings, reduction