Embedded systems in IoT are expected to be run by reliable, resource-efficient software. Devices on the edge are typically required to communicate with central nodes, and in some setups with each other, constituting a distributed system. The Erlang language, favored for its constructs that support building fault-tolerant, distributed systems, offers solutions to these challenges. Its dynamic type system and higher-level abstractions enable fast development, while also featuring tools for building highly available and fault-tolerant applications. To study the viability of using Erlang in embedded systems, we analyze the solutions the language offers, contrasting them with the challenges of developing embedded systems, with a particular focus on resource use. We measure the footprint of the language's constructs in executing tasks characteristic of end devices, such as gathering, processing and transmitting sensor data. We conduct our experiments with constructs and data of varying sizes to account for the diversity in software complexity of real-world applications. Our measured data can serve as a basis for future research, supporting the design of the software stack for embedded systems. Our results demonstrate that Erlang is an ideal technology for implementing software on embedded systems and a suitable candidate for developing a prototype for a real-world use case.
This study aimed to estimate the variance, heritability, and genetic correlation of growth traits in Inner Mongolia white cashmere goats (Erlangshan type) (IMWCG-ER). Data collected from the Erlang Mountain Ranch in 2022-2023 were analyzed. The traits studied included birth weight (BW), weaning weight (WW), pre-weaning daily gain (ADG), and the 12-month weight (12W). In animal models, single and multi-trait analyses were performed using the restricted maximum likelihood (REML) method to estimate genetic parameters, and six different models were fitted for each trait by ignoring or including maternal permanent environmental effects, maternal additive genetic effects, and the interaction with individual additive genetic effects. The accuracy and suitability of each model were tested using the likelihood ratio and AIC and BIC tests. The heritability estimates of birth weight, weaning weight, daily gain before weaning, and the 12-month weight were 0.3884, 0.2951, 0.2749, and 0.2192, respectively. The absolute value of the genetic correlation coefficient between traits ranged from -0.8700 to 0.5529. The highest genetic association was between birth weight and pre-weaning daily gain (-0.8700), followed by birth weight and the 12-month weight (-0.6256). The absolute value of phenotypic correlation coefficients ranged from -0.7906 to 0.3562. The highest phenotypic correlation was between birth weight and daily gain before weaning, and the correlation coefficient was -0.7906.
This paper discusses the estimation of the discrete mixed Poisson-Erlang distribution (DMPED). Compared to many traditional discrete distributions, DMPED offers several surprising benefits, especially when examining count data with high variation and that are positively skewed. We have explored several statistical characteristics of the assumed distribution, such as moments, the moment-generating function, the failure rate function, the monotonicity of the probability mass function, and a couple of descriptive measures (central tendency and dispersion). We have used the maximum likelihood estimation technique to estimate the parameters of the DMPED. We conducted a simulation study to validate the proposed estimators. Finally, four applications related to cancer diseases have been discussed, where DMPED (especially DMPEIID) fits the number of doses required for treatment, remission times, and therapy type comparisons.
The purpose of this study was to investigate the effects of non-genetic factors on the growth and development performance of Inner Mongolia white cashmere goats (Erlanghan type), such as birth weight (BW), weaning weight (WW), 6-month weight (6 WT), 12-month weight (12 WT), body height (BH), and body length (BL), and wool production performance, such as cashmere fineness (CF), cashmere thickness (CT), and cashmere yield (CY). The research objects were 4654 kids produced by 45 buck goats and 2269 doe goats in the Erlang Mountain Ranch of Beiping Textile Co., Ltd., Inner Mongolia, from 2020 to 2023. Based on the generalized linear model, ANOVA was used to analyze the effects of non-genetic factors, such as birth year (Y), birth month (M), sex (S), birth type (T), birth herd (H), assay flock (F), age at measurement (MA), and the age of doe goats at lambing (DLA), on growth and development traits and cashmere traits. The results show that the birth weight (BW), weaning weight (WW), 6-month weight (6 WT), 12-month weight (12 WT), body length (BL), body height (BH), chest depth (CD), chest width (CW), chest circumference (CC), cannon circumference (CNC), wool length (WL), and cashmere yield (CY) of buck goats were significantly higher than those of doe goats (p < 0.01), and the fineness of the cashmere produced by doe goats was significantly finer than that produced by buck goats (p < 0.01). The birth weight, weaning weight, and 6-month weight of single kids were significantly higher than those of multiple kids (p < 0.01), but the effect on the 12-month weight was not significant (p > 0.05). The age of doe goats at lambing had significant effects on birth weight, weaning weight, and 6-month weight (p < 0.01). Assay flock and age at measurement had significant effects on cashmere fineness, cashmere thickness, and cashmere yield (p < 0.01). This study will provide a basis for the scientific breeding and management of cashmere goats and lay a foundation for the setting of fixed effects in the genetic evaluation model of Inner Mongolia white cashmere goats (Erlangshan type).
Among diseases, cancer exhibits the fastest global spread, presenting a substantial challenge for patients, their families, and the communities they belong to. This paper is devoted to modeling such a disease as a special case. A newly proposed distribution called the binomial-discrete Erlang-truncated exponential (BDETE) is introduced. The BDETE is a mixture of binomial distribution with the number of trials (parameter [Formula: see text]) taken after a discrete Erlang-truncated exponential distribution. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, Quantile function, moment generating function, Shannon entropy, order statistics, and stress-strength reliability, are provided. The distribution's parameters are estimated using the maximum likelihood method. Two real-world lifetime count data sets from the cancer disease, both of which are right-skewed and over-dispersed, are fitted using the proposed BDETE distribution to evaluate its efficacy and viability. We expect the findings to become standard works in probability theory and its related fields.
The study has attempts to use ERLANG method to find out if a change is needed in the existing manpower with respect to physicians in various departments. The methodology involves calculating the time taken by the doctor of a selected department to carry out multiple activities like consultation of patients, performing minor or major surgeries, other procedures as per the requirement of the department by observation, interaction with staff and accessing required information from Hospital Information System for the departments of surgery, General Medicine, Orthopedics and Emergency Medicine.
It is widely believed that cancers develop upon acquiring a particular number of (epi) mutations in driver genes, but the law governing the kinetics of this process is not known. We have previously shown that the age distribution of incidence for the 20 most prevalent cancers of old age is best approximated by the Erlang probability distribution. The Erlang distribution describes the probability of several successive random events occurring by the given time according to the Poisson process, which allows an estimate for the number of critical driver events. Here we employ a computational grid search method to find global parameter optima for five probability distributions on the CDC WONDER dataset of the age distribution of childhood and young adulthood cancer incidence. We show that the Erlang distribution is the only classical probability distribution we found that can adequately model the age distribution of incidence for all studied childhood and young adulthood cancers, in addition to cancers of old age. This suggests that the Poisson process governs driver accumulation at any age and that the Erlang distribution can be used to determine the number of driver events for any cancer type. The Poisson process implies the fundamentally random timing of driver events and their constant average rate. As waiting times for the occurrence of the required number of driver events are counted in decades, and most cells do not live this long, it suggests that driver mutations accumulate silently in the longest-living dividing cells in the body-the stem cells.
Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, then each cell can be imagined as sampling from a probability density of times to division and death. The exponential density is the most mathematically and computationally convenient choice. It has the advantage of satisfying the memoryless property, consistent with a Markov process, but it overestimates the probability of short division times. With the aim of preserving the advantages of a Markovian framework while improving the representation of experimentally-observed division times, we consider a multi-stage model of cellular division and death. We use Erlang-distributed (or, more generally, phase-type distributed) times to division, and exponentially distributed times to death. We classify cells into generations, using the rule that the daughters of cells in generation n are in generation [Formula: see text]. In some circumstances, our representation is equivalent to established models of lymphocyte dynamics. We find the growth rate of the cell population by calculating the proportions of cells by stage and generation. The exponent describing the late-time cell population growth, and the criterion for extinction of the population, differs from what would be expected if N steps with rate [Formula: see text] were equivalent to a single step of rate [Formula: see text]. We link with a published experimental dataset, where cell counts were reported after T cells were transferred to lymphopenic mice, using Approximate Bayesian Computation. In the comparison, the death rate is assumed to be proportional to the generation and the Erlang time to division for generation 0 is allowed to differ from that of subsequent generations. The multi-stage representation is preferred to a simple exponential in posterior distributions, and the mean time to first division is estimated to be longer than the mean time to subsequent divisions.
Human behavioral responses to changes in risks are often delayed. Methods for estimating these delayed responses either rely on rigid assumptions about the delay distribution (e.g., Erlang distribution), producing a poor fit, or yield period-specific estimates (e.g., estimates from the Autoregressive Distributed Lag (ARDL) model) that are difficult to integrate into simulation models. We propose a hybrid ARDL-Erlang approach that yields an interpretable summary of behavioral responses suitable for incorporation into simulation models. We apply the ARDL-Erlang approach to estimate the effect of COVID-19 deaths on mobility across US counties from October 2020 to July 2021. A standard panel autoregressive distributed lag (ARDL) model first estimates the effect of past deaths and past mobility on current mobility. The ARDL model is then transformed into an Infinite Distributed Lag (IDL) model consisting of only past deaths. The coefficients of the past deaths are aggregated into an overall effect and fit to an Erlang distribution, summarized by average delay length and shape parameter. Our results show that on the national level, a one-standard-deviation permanent increase in weekly deaths per 100,000 population (log-transformed) is associated with a 0.46-standard-deviation decrease in human mobility in the long run, where the delay distribution follows a first-order Erlang distribution, and the average delay length is about 3.2 weeks. However, there is much heterogeneity across states, with first- to third-order Erlang delays and 2 to 18 weeks of average delay providing a theoretically cogent summary of how mobility followed changes in deaths during the first year and a half of the pandemic. This study provides a novel approach to estimating delayed human responses to health risks using a hybrid ARDL-Erlang model. Our findings highlight significant variability in the impact and timing of responses across states, underscoring the need for tailored public health policies. This study can also serve as guidelines and an example for identifying delayed human behavior in other settings.
The stochastic theory of chromatography describes solute migration as the cumulative result of random retention events superimposed on convective transport and axial dispersion. Classical Poisson-based formulations offer analytical transparency but are limited in their ability to represent heterogeneous, multistep, or multipathway adsorption kinetics increasingly revealed by single-molecule measurements. Here, we reformulate stochastic chromatography within a Lévy-Khintchine characteristic function framework and extend the underlying event structure to phase-type Markov renewal processes, including Erlang and hyper-Erlang waiting-time distributions. This representation preserves analytical tractability through matrix-exponential evaluation while enabling flexible descriptions of heterogeneous adsorption-desorption pathways. We further develop an extended classical characteristic function Fourier inversion method that operates directly in the first-passage domain and incorporates multisite log-normal sojourn heterogeneity, γ distributed stationary residence times, and inverse-Gaussian treatment of mobile-phase dispersion. Application to experimental DNA chromatograms demonstrates accurate reconstruction of peak position, width, asymmetry, and tailing behavior. A hybrid Markov renewal Monte Carlo simulator was also introduced as a mechanistic first-passage benchmark. Identifiability analysis indicated that effective sojourn times and transport parameters are robustly constrained, whereas several microscopic kinetic constants remain structurally nonidentifiable from single chromatograms alone. Overall, the proposed Lévy and Fourier-inversion framework links ensemble chromatographic peak shapes with microscopic adsorption statistics and provides a practical analytical route for modeling heterogeneous stationary phases, biomolecular separations, and single-molecule-informed chromatographic method development.
Avoidance behaviors can have a substantial influence on both the spread of an infectious disease and on the long-term dynamics of substance use disorder. However, this behavior is typically reactive based on delayed information related to personal risk as well as both the length of exposure to and retention of that information. This suggests that in epidemic models, feedback delays should be strong - that is, not exponentially distributed (Erlang-1) but instead peaking at a point strictly in the past (Erlang-2 or greater). However, almost all studies of infectivity feedback delays are exponential with qualitatively different results than are seen with strong delays. To address this gap, we analyze two compartmental models for infectious disease epidemiology. Our results demonstrate that sustained oscillations in the total number of active cases may appear even as early as the first two years of an outbreak, suggesting that human behavior may be an explanatory factor for periodic fluctuations evident in recent pandemic time-series data (e.g. COVID-19). We then extend our analysis to a study of the role of information feedback on substance use disorder epidemiology, with a focus on both the transient and asymptotic dynamics of drug waves. We show that under certain conditions, oscillations in substance use disorder can become sustained and that models without strong feedback delays can fail to produce important qualitative transient behavior in substance use incidence rates. To our knowledge, this work represents the first mathematical model exhibiting oscillations with non-contact (linear) pathways to substance use disorder.
The Erlang B equation is directly applicable to smaller hospital departments such as maternity and paediatrics departments. The bed occupancy margin is directly linked to size and not 'efficiency'. A figure of 0.1% turn-away has been recommended as a planning target, i.e., only 1 in a thousand admissions suffer a delay before a bed can be found. Two bed calculators are provided which can be used for paediatric, obstetric, maternity, midwife-led, birthing wards and neonatal/paediatric critical care capacity. The negative effects of turn-away are likely to be context specific, hence, critical care > theatres > birthing unit > maternity unit. The uncertainty regarding future births is discussed along with the variable nature of seasonality in births. For paediatrics, much of bed demand is also influenced by the trend in births. Weighted population density (WPD) is associated with the size distribution of hospitals/units within countries and regions. This influences the average cost per birth/admission. The USA has a low WPD and a significant problem with small hospitals/departments. Only 10% of countries have WPD higher than England. Some countries choose to operate with even more hospitals than needed and this acts to elevate costs. Suggestions are made for a pragmatic approach to bed planning, especially where a dispersed population dictates a need for small hospitals, and hence, issues regarding size and costs. For maternity/paediatrics admissions (and other relatively short-stay admissions) the majority of overhead/indirect costs and most staffing costs should be apportioned based on admissions, and not LOS. Apportionment based on LOS creates the spurious illusion that LOS is the major cost driver and that reducing LOS will immediately save costs. Below 20 beds, Poisson statistical variation plus environment-induced randomness in daily arrivals imply that staff costs may become increasingly fixed irrespective of LOS. Around >30 beds, it looks possible to save costs by reducing LOS. Allocating total organizational costs to individual units and then to patients is less precise than realized and can be done in different ways, which all heavily rely on the steady-state assumption. When bed availability is the bottleneck, then reducing LOS may increase throughput per bed and increase income; however, is this for the benefit of the patient or for the benefit of the organization, and does it lead to higher unanticipated total costs including patient harm? The older economy-of-scale literature has been demonstrated to be flawed, with a recent focus on economy of scale at the department level being entirely consistent with the application of the Erlang B equation. A list of nine catastrophic pitfalls is given for doctors to identify dubious capacity advice from managers and external experts.
We study an infection-age structured epidemic model in which both the infectivity and the rate of loss of immunity depend on the time-since-infection. The model can be equivalently viewed as a nonlinear renewal equation for the incidence of infection or as a partial differential equation for the density of infected individuals. We explicitly consider gamma, rather than Erlang, distributed durations of infection using a combination of ODE approximations and numerical bifurcation methods. We show that the shape of this distribution strongly influences stability of the endemic equilibrium, even when the basic reproduction number R0 and the mean duration of infectiousness are fixed. Moreover, we establish the existence of regions of bistability, where a stable endemic equilibrium coexists with a stable periodic orbit. To our knowledge, this provides the first example of bistability in infection-age structured models with waning immunity alone. Finally, we show how common compartmental modelling approaches, which impose implicit assumptions on the distribution of the duration of infection, can lead to spurious dynamical outcomes. Taken together, our analysis underscores the crucial role of distributional structure in epidemic modelling and provides new insights into the rich dynamics of infection-age structured SIS/SIRS models.
Aiming at the problem of matching scarce resources among donors, recipients, and medical institutions in organ transplantation, a stable three-sided matching method is proposed. Firstly, in view of the preference structure characteristics of the problems in the context of organ transplantation, a mixed preference structure from the three-sided matching problem is introduced for description, and the stability conditions under this structure are also provided, intuitionistic fuzzy information was utilized to quantify uncertain indicators during organ transplantation. The Erlang distribution was introduced to describe the randomness of the occurrence of organ donors. Coping strategies for possible false reporting behaviors of organ recipients were designed, and a three-sided matching model of donors, recipients, and medical institutions was constructed. And proved the stability of this matching; Secondly, based on the NSGA-III algorithm, the initial search strategy of the Bird Swarm algorithm was introduced, a new mutation method was designed, and its feasibility was theoretically analyzed. Finally, the feasibility of the model and algorithm was verified through simulation examples in the context of organ transplantation, and the performance of the algorithm was analyzed and verified in combination with algorithm complexity, effectiveness, and ablation experiments. The experimental results show that the model and algorithm can effectively improve the stability of the matching results and increase the efficiency of resource allocation.
Triapine is a potent small-molecule ribonucleotide reductase inhibitor investigated in combination with radiation and/or chemotherapy for the treatment of advanced stage solid cancers. The aim of this study is to develop a population pharmacokinetic-pharmacodynamic (PK/PD) model for triapine to describe the PK parameters, the effect of smoking on exposure, and the relationship between methemoglobin concentrations and exposure. A total of 36 patients with advanced stage cervical or neuroendocrine cancers from two phase I studies were included in the population PK/PD model building. Triapine and methemoblogin plasma concentrations were sampled over an 8-hr or 24-hour period. Data were analyzed by a nonlinear mixed-effects modelling approach. Simulations were performed to optimize the dosing strategy for oral triapine. A two-compartment model with two-transit compartment Erlang absorption and first-order elimination best described the PK of triapine, and an effect compartment model best described the PD effect of triapine on methemoglobin concentrations. The final model described triapine PK/PD well, and a 38% increase in triapine clearance was estimated due to smoking. Simulations suggest that dose adjustments may be necessary, as increasing the oral dose for smokers from 100 to 125 mg resulted in exposures matching those observed in nonsmokers. This study provides a quantitative model characterizing the relationship between triapine exposure and methemoglobin concentrations. The developed PK/PD model can be used to optimize the dosing regimen for oral triapine, illustrating how population PK/PD modeling can inform decision-making throughout the triapine drug development lifecycle.
We study the global dynamics of a two-patch epidemic model that integrates spatial migration, Erlang-distributed delays, and environmental stochasticity. The two patches are coupled by migration whose intensities are modulated by an Ornstein-Uhlenbeck (OU) process, capturing stochasticity but mean-reverting fluctuations in population movement, while the infection progression is described by a distributed delay structure represented through a linear-chain formulation. For the deterministic system, we establish threshold-type results that characterize disease extinction and persistence: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R 0 < 1 , whereas infection persists when R 0 > 1 . For the corresponding stochastic model, we derive explicit sufficient conditions for almost sure exponential extinction by constructing Lyapunov functions and exploiting the Metzler structure of the infected subsystem. Moreover, we prove that the stochastic system admits a stationary distribution when stochastic threshold R 0 s > 1 , which serves as a stochastic analogue of the endemic equilibrium and describes persistent random fluctuations of infection levels. Numerical simulations are provided to validate the theoretical findings and to quantify how distributed delays and OU-driven migration randomness shape long-term infection burden and stationary prevalence across patches. Notably, adjusting the noise intensity and the mean-reversion rate can redistribute the long-term infection burden between patches, which highlights how migration-driven randomness fundamentally reshapes spatial epidemic patterns.
Chikungunya fever (CF), an arthropod-borne disease caused by Chikungunya virus (CHIKV), is a serious public health threat globally. In July 2025, an outbreak occurred in Foshan City, Guangdong Province, China. A large number of these cases involved cross-city movements, which complicated containment efforts and provided a unique opportunity to study transmission patterns and key epidemiological parameters. We obtained information on 400 confirmed cases of CF with cross-city exposure histories reported across Guangdong Province in southern China by 21 August 2025. Demographic, clinical, and mobility data were mainly obtained from the National Notifiable Infectious Disease Reporting System and supplemented by epidemiological investigations. The incubation period was estimated using a parametric accelerated failure time model, with log-normal, gamma, Weibull, and Erlang distributions used to fit the model. Subgroup analysis was performed based on mobility patterns and exposure windows. Significant demographic differences were observed compared with local Foshan cases: cross-city cases had higher proportions of males (58.3%), individuals within the age groups of 15-24, 25-34, and 35-44,and individuals with occupations of general staff/workers, business/service workers, and students. Fever (86.4%), arthralgia (79.3%), and rash (61.2%) were the most common symptoms. Mobility analysis revealed that Foshan and Guangzhou were the major sources of infection, with cases spreading mainly to cities within the Pearl River Delta and provinces such as Guangxi (43.3%) and Hunan (15.4%). The median incubation period was estimated to be 5.4 days (95% CI 5.0-5.7), with 2.5th and 97.5th percentiles of 2.5 days and 11.4 days, respectively. This study underscores the central role of population mobility in the spread of CHIKV and highlights distinct epidemiological characteristics of cross-city cases. The estimated median incubation period of 5.4 days provides important evidence for surveillance and response strategies during chikungunya outbreaks. Notably, students and migrant workers accounted for a higher proportion of cross-city cases, suggesting that highly mobile populations may contribute to inter-regional transmission. These findings highlight the importance of strengthened surveillance and coordination across regions for the prevention and control of future outbreaks in Guangdong and other high-risk areas in China.
The upper Yellow River is a crucial ecological barrier and water conservation zone in the Yellow River Basin. Since 2000, extensive afforestation in this region has markedly improved the carbon sequestration capacity. Nevertheless, the responses of soil C:N:P stoichiometry and microbial diversity to different vegetation restoration patterns remain insufficiently understood. This study was conducted at a representative restoration site (Erlang Mountain) in the upper Yellow River. Abandoned farmland (Af, abandoned for more than three years) was used as the background, and six types of plantations with a uniform recovery period of 20 years were selected: mixed forest (Mf), Hippophae rhamnoides (Hr), Picea crassifolia (Pc), Prunus sibirica (Ps), Larix gmelinii (Lg), and Pinus tabuliformis (Pt). These tree species are widely used for afforestation in the upper Yellow River region and represent the dominant vegetation restoration strategies. Using 16S rRNA and ITS amplicon sequencing, this study clarified the mechanisms underlying soil microbial diversity and C:N:P stoichiometry across these restoration types. The results revealed the following. (1) Soil C (SOC) and N (TN) contents and stocks were the highest in Mf but the lowest in Hr (P < 0.05). Except for Mf, the soil C and N contents and stocks in the Lg forests exceeded those in the other stands (P < 0.05), whereasthe soil P (TP) in Pt was the lowest (P < 0.05). (2) The soil bacterial α-diversity in Mf and Lg was greater than that in other afforestation lands (P < 0.05), with Mf exhibiting the highest α-diversity, dominated by Acidobacteriota and Pyrinomonadaceae. The soil fungal α-diversity in Cl exceeded that of the other land use types (P < 0.05). The abundance of Acidobacteriota in Hr soils was lower than that in other forests, whereas the abundances of Ascomycota and Mortierellaceae were higher. (3) Precipitation exerted a negative effect on soil C:N:P stoichiometry but a positive effect on bacterial α-diversity (P < 0.05). pRDA analysis indicated that the vegetation restoration type significantly influenced soil C:N:P stoichiometry and microbial diversity (P < 0.05). This study demonstrated that vegetation restoration reshaping microbial community composition and improving soil C and N contents, with Mf and Lg exerting a stronger effect than other vegetation restoration types, thereby providing insights into rational afforestation in these regions.
The Erlang loss formula, also known as the Erlang B formula, has been known for over a century and has been used in a wide range of applications, from telephony to hospital intensive care unit management. It provides the blocking probability of arriving customers to a loss system involving a finite number of servers without a waiting room. Because of the need to introduce priorities in many services, an extension of the Erlang B formula to the case of a loss system with preemptive priority is valuable and essential. This paper analytically establishes the consistency between the global balance (steady state) equations for a loss system with preemptive priorities and a known result obtained using traffic loss arguments for the same problem. This paper, for the first time, derives this known result directly from the global balance equations based on the relevant multidimensional Markov chain. The paper also addresses the question of whether or not the well-known insensitivity property of the Erlang loss system is also applicable to the case of a loss system with preemptive priorities, provides explanations, and demonstrates through simulations that, except for the blocking probability of the highest priority customers, the blocking probabilities of the other customers are sensitive to the service time distributions and that a larger service time variance leads to a lower blocking probability of the lower priority traffic.
This manuscript introduces a new Erlang-distributed SEIR model. The model incorporates asymptomatic spread through a subdivided exposed class, distinguishing between asymptomatic ( E a ) and symptomatic ( E s ) cases. The model identifies two key parameters: relative infectiousness, β SA , and the percentage of people who become asymptomatic after being infected by a symptomatic individual, κ . Lower values of these parameters reduce the peak magnitude and duration of the infectious period, highlighting the importance of isolation measures. Additionally, the model underscores the need for strategies addressing both symptomatic and asymptomatic transmissions.