🚀 Introduction

Mathematical models have long been used to simulate the spread of infectious diseases by dividing populations into compartments (groups) representing different stages of an epidemic. The classic SIR model, for instance, has three compartments – Susceptible (S) individuals who can catch the disease, Infectious (I) individuals who have the disease and can spread it, and Removed (R) individuals who have either recovered with immunity or died. In such models, people flow from one compartment to another (e.g. S → I → R) over time, governed by a set of ordinary differential equations (ODEs). While simple models like SIR (or its extension SEIR, which adds an Exposed (E) incubation stage can capture many epidemics, they assume that only symptomatic infected people drive the spread.

However, real diseases often defy this assumption. COVID-19 is a prime example: many people spread the virus before showing symptoms or without ever developing symptoms. Early in the pandemic, studies estimated that only 13–18% of infections resulted in noticeable symptoms – meaning the majority of infections were initially “silent” carriers. Moreover, nearly half or more of the virus transmission was found to occur before symptom onset (the presymptomatic phase). These “invisible” transmission routes led epidemiologists to expand their models. Ignoring asymptomatic or presymptomatic spread can make us dramatically underestimate how contagious a disease is. For example, when accounting for unseen spreaders, the basic reproduction number R₀ for SARS-CoV-2 was found to be significantly higher – in one analysis, potentially in the range of 3.2–4.4 rather than ~2–3. Even in pessimistic scenarios where asymptomatic individuals transmit at lower rates, at least 50% of new infections were still driven by presymptomatic and asymptomatic carriers.

To address these realities, epidemiologists introduced more complex compartmental frameworks. One such model is the SEPIAR model, which stands for Susceptible–Exposed–Presymptomatic–Infectious–Asymptomatic–Removed. This model extends the standard SEIR approach by explicitly including compartments for presymptomatic and asymptomatic infection. In essence, SEPIAR “opens up” the single infectious category of SEIR into three sub-compartments: individuals who are infectious before symptoms (P), those who are infectious with symptoms (I), and those who are infectious but will never show symptoms (A). By doing so, the model captures the full trajectory of an infection from exposure to outcome, including the hard-to-detect stages. This article will introduce the SEPIAR model in detail, explain its parameters and assumptions, and explore its variants. We will see how SEPIAR provides a more realistic picture for diseases with hidden spreaders – not only COVID-19, but any illness where people without symptoms can propagate an outbreak.

🧮 Model Description

In the SEPIAR framework, the population is divided into six compartments :

• S (Susceptible): Individuals who are healthy and vulnerable to infection.
• E (Exposed): Individuals who have been infected (exposed to the pathogen) but are not yet infectious. This is an incubation or latent stage during which the person harbors the pathogen but cannot transmit it.
• P (Presymptomatic Infectious): Individuals who have left the latent stage and become infectious to others before showing symptoms. These people will eventually develop symptoms, but during this presymptomatic phase they often feel well, making this group hard to detect.
• I (Infectious Symptomatic): Individuals who have become infectious and symptomatic – in other words, actively sick and capable of transmitting the disease. This compartment typically corresponds to confirmed cases that show symptoms.
• A (Asymptomatic Infectious): Individuals who are infectious but never develop symptoms. They carry and spread the pathogen without ever knowing (no symptoms). In some contexts, this group may include very mild, unreported infections. Asymptomatic carriers often have lower individual infectivity (for example, due to lower viral shedding), but they are dangerous because they are difficult to identify.
• R (Removed/Recovered): Individuals who have left the infectious pool, either by recovering (and gaining immunity, at least temporarily) or by being removed via isolation or death. They no longer participate in disease transmission. (In many models, “Removed” encompasses both recoveries and deaths, since neither can infect others.)

Each compartment’s size is a function of time (e.g. S(t), E(t), etc.). The SEPIAR model is governed by a system of ODEs that describe how individuals flow between these compartments. A standard form of the equations is given below:

dS/dt = – β * S * (I + κ_P * P + κ_A * A) / N                                 (1)
dE/dt = β * S * (I + κ_P * P + κ_A * A) / N – σ * E                           (2)
dP/dt = (1 – θ) * σ * E – κ * P                                              (3)
dA/dt = θ * σ * E – γ_A * A                                                  (4)
dI/dt = κ * P – γ_I * I                                                     (5)
dR/dt = γ_A * A + γ_I * I                                                   (6)

Let’s walk through these equations step by step (the notation here is one common choice – different sources may use slightly different symbols, but the concepts are equivalent):

• Infection rate (Eq. 1): Susceptible individuals (S) become infected through contact with those who are infectious. The term β · S · (…)/N represents the new infections per unit time. Here β is the transmission rate – a parameter that encapsulates how often a susceptible-infectious contact leads to a new infection, combined with contact frequency. The infectious people that contribute to new infections are in P, I, and A. We scale each by a factor (κ_P or κ_A) reflecting relative infectiousness if needed. For example, presymptomatic individuals might be nearly as infectious as symptomatic (so we could set κ_P ≈ 1), whereas asymptomatic individuals might transmit less readily (e.g. κ_A could be 0.5 to indicate 50% relative infectiousness). The sum (I + κ_P·P + κ_A·A) is divided by N (the total population) to model a well-mixed random contact process. Thus, dS/dt is negative since S decreases as people leave susceptibility upon infection.
• Latent period (Eq. 2): The Exposed (E) compartment grows by new infections and shrinks as exposed individuals incubate the pathogen and become infectious. The parameter σ (sigma) is the rate at which exposed individuals progress to the next stage. Its inverse, 1/σ, is the average latent period (incubation time before infectiousness). For a disease like COVID-19, the incubation period was around 5–6 days on average, so σ might be ~0.17 per day. All exposed individuals eventually leave E; we assume they all become infectious in some form (there is no “subclinical cure” at this stage). Thus, dE/dt = new exposures – progression out of E.
• Presymptomatic vs. Asymptomatic split (Eqs. 3 & 4): A crucial aspect of SEPIAR is that when latent individuals become infectious, they split into two groups: those destined to show symptoms (enter P) and those who will remain asymptomatic (enter A). We denote by θ (theta) the fraction of infections that stay asymptomatic. Conversely, (1–θ) is the fraction that will eventually develop symptoms. For COVID-19, estimates of θ varied widely (e.g. one review found ~16% on average, with ranges 6–41%, though some studies suggested it could be higher). If we take θ = 0.3 (30% asymptomatic) as an example, then 70% go into P and 30% into A upon leaving E.
  • Equation (3) describes Presymptomatic (P) individuals. They are “born” from E at rate (1–θ)·σ·E (the fraction that is presymptomatic), and they leave P as they develop symptoms. We use κ (kappa) for the rate at which presymptomatic individuals become symptomatic. The average presymptomatic period is 1/κ. Studies suggest that people tend to develop symptoms a few days after becoming infectious. For instance, with COVID-19, individuals might spend ~2 days in the presymptomatic infectious phase. That would correspond to κ ≈ 0.5 per day.
  • Equation (4) governs Asymptomatic infectious (A) individuals. They arise from E at rate θ·σ·E (the fraction that is asymptomatic) and then are removed as they recover. Asymptomatic carriers typically recover without ever showing signs, often a bit faster than symptomatic cases. We denote γ_A as the recovery rate for asymptomatic infections (so 1/γ_A is the average duration an asymptomatic person remains infectious). If asymptomatic infections last, say, ~10 days on average, then γ_A ≈ 0.1 per day.
• Symptomatic infection (Eq. 5): Once a presymptomatic person develops illness, they move into the Infectious Symptomatic (I) compartment. Equation (5) shows dI/dt = κ·P – γ_I·I, meaning people enter I from P at rate κ and leave I upon recovery at rate γ_I. The recovery rate γ_I corresponds to the infectious period of a symptomatic case. Symptomatic COVID-19 cases, for example, were often considered infectious for about 7–10 days after symptom onset, so γ_I might be around 0.14–0.1 per day. In some models, a small fraction of I may also die rather than recover; if so, γ_I would be split into a recovery rate and a mortality rate (we’ll discuss this extension later).
• Removal (Eq. 6): Finally, dR/dt collects all individuals leaving the infectious compartments. Both asymptomatic and symptomatic individuals eventually transition to Removed (R), either by recovering with immunity or by other removal (like isolation or death). In this basic model, we assume removed individuals are effectively out of circulation and no longer infectious. Thus, R increases at rate γ_A·A + γ_I·I, and S + E + P + I + A + R remains constant (total population N, minus any deaths if those aren’t counted in R).

The SEPIAR model’s equations encapsulate a detailed infection lifecycle. An individual starts susceptible, becomes exposed (infected but not yet infectious), then enters an infectious stage which is either presymptomatic (followed by symptomatic illness) or asymptomatic. Finally, they end up removed. Figure 1 illustrates these flows. Importantly, both P and A contribute to new infections, alongside I, which is what distinguishes SEPIAR from simpler models. By tracking P and A, we can quantify their impact on transmission. For instance, if κ_P and κ_A in Eq. 1 are set based on empirical data, one can estimate what portion of R₀ is due to presymptomatic or asymptomatic spread. This level of granularity proved invaluable during COVID-19, as health policies depended on understanding hidden transmission. The SEPIAR model was used to demonstrate, for example, that even if asymptomatic individuals transmitted at only half the rate of symptomatic individuals, they and presymptomatics could still drive at least 50% of infections at the peak of an outbreak. In short, the model captures how “silent spreaders” fuel an epidemic.

🔑 Parameter Definitions and Typical Values

The SEPIAR model introduces several parameters that govern the transition rates between compartments. Understanding these parameters is key to interpreting the model and to using it for real diseases. Below we define each major parameter, along with typical values or ranges (especially in the context of COVID-19, which motivated the SEPIAR structure):

• β (beta) – Transmission rate: This parameter determines how quickly the infection spreads in a fully susceptible population. It combines contact frequency and probability of transmission per contact. A higher β means an infectious person infects more people per day. β is usually calibrated to match a known basic reproduction number R₀ (approximately R₀ = β/γ in simple SIR, though for SEPIAR the formula is more involved). In early COVID-19, an R₀ around 3 meant that β was on the order of 0.3–0.5 per day (assuming an infectious period ~10 days). Interventions like social distancing or masks effectively reduce β by cutting down contacts or transmission probability. (In model extensions, β can be time-dependent to simulate interventions.)
• σ (sigma) – Progression rate from E to infectious: This is the inverse of the latent period. It dictates how fast exposed individuals become infectious. A larger σ means a shorter incubation. For COVID-19, σ was about 0.16 per day initially (incubation ~6 days). For other diseases with shorter incubation (e.g. influenza ~2 days), σ would be higher (~0.5). This parameter is often derived from clinical data on incubation periods.
• θ (theta) – Asymptomatic infection fraction: This fraction (0 ≤ θ ≤ 1) is the probability that an infected person never develops symptoms. It effectively splits the infectious population into asymptomatic vs. (eventually) symptomatic. θ varies by disease and context. COVID-19 estimates ranged widely; early on, θ was thought to be 0.4 or higher in some outbreaks, but later analyses (accounting for testing and serology) indicated maybe θ ~0.15–0.30 in many populations. For example, one study in NYC found only ~13–18% of infections were symptomatic (so θ as high as ~0.8), though that was an extreme scenario with many unreported cases. In modeling, θ might be adjusted if new data (like seroprevalence surveys) reveal more asymptomatic cases than initially detected. Age and other factors can influence θ – e.g., children had higher asymptomatic rates for COVID.
• κ (kappa) – Presymptomatic progression rate: This parameter controls how fast presymptomatic individuals become symptomatic. 1/κ is the presymptomatic infectious period length. If, say, people are presymptomatic and infectious for ~2 days on average, κ ≈ 0.5 per day. A higher κ means people show symptoms quickly after becoming infectious (short presymptomatic phase), whereas a lower κ would mean a prolonged contagious period before symptoms appear. Epidemiological studies of COVID-19 suggested that viral shedding (and peak infectiousness) often started ~2-3 days before symptom onset, implying a significant presymptomatic phase which SEPIAR models with this parameter.
• γ_I (gamma_I) – Symptomatic recovery rate: This is the rate at which symptomatic infectious individuals recover (or are removed). Its inverse is the average duration a symptomatic person remains infectious. For a moderate illness, one might be infectious for about a week or two; for COVID-19, health guidelines often used ~10 days of contagiousness for mild cases, so γ_I could be ~0.1 per day. If the disease causes long infectious periods, γ_I would be smaller. In practice, γ_I can be estimated from how long viral shedding lasts or how long patients isolate until they test negative.
• γ_A (gamma_A) – Asymptomatic recovery rate: Similarly, this is the removal rate for asymptomatic carriers. Asymptomatic individuals might clear the infection faster (some studies assumed ~10 days infectious for asymptomatic vs. ~14 days for symptomatic). For instance, if asymptomatics shed virus for ~8–10 days, γ_A might be around 0.125–0.1 per day. In some models, one might set γ_A > γ_I to reflect slightly faster recovery, although data on asymptomatic clearance can be limited.
• κ_P, κ_A – Relative infectiousness multipliers: These are scaling factors we included in Eq. (1) (often denoted differently in literature) to adjust the infectivity of P and A relative to I. Instead of having separate β’s for presymptomatic and asymptomatic, many formulations use a common β but multiply P and A by constants. For example, if asymptomatic carriers are only half as likely to infect someone (due to, say, lower viral load or less coughing), we set κ_A = 0.5. If presymptomatics are thought to be highly infectious (some studies suggested presymptomatic COVID cases could be as contagious as symptomatic ones), we might use κ_P = 1 (or even >1 if presymptomatic individuals have high contact rates before they realize they’re ill). These parameters are often tuned to fit contact tracing data or viral load measurements. For instance, one modeling study set a lower transmission coefficient for asymptomatics based on WHO’s early assertion of lower asymptomatic transmissibility.
• N – Total population size: While not a rate parameter, N is the population count used to normalize infection contacts. In a closed-population model (no births/deaths except disease), N = S+E+P+I+A+R remains constant over the epidemic (assuming “Removed” includes deaths). If modeling a specific region, N would be that region’s population. This parameter is important for scaling the model to real numbers of people.

Typical values: To give a concrete sense, consider a plausible parameter set for COVID-19 in 2020 (pre-vaccines) often used in simulations:

• β ≈ 0.4 per day (R₀ around 2.5–3.5 depending on other parameters).
• 1/σ ≈ 5–6 days (incubation period).
• θ ≈ 0.2–0.4 (20–40% infections asymptomatic, recognizing uncertainty).
• 1/κ ≈ 2 days (presymptomatic infectious phase length).
• 1/γ_I ≈ 7–10 days (infectious period for symptomatic cases).
• 1/γ_A ≈ 5–8 days (infectious period for asymptomatic cases, shorter due to quicker recovery or removal).
• κ_P = 1 (presymptomatic equally infectious as symptomatic), κ_A ≈ 0.5 (asymptomatic about half as infectious).

These values would be adjusted for different diseases. For example, a disease like measles has no asymptomatic infection (θ ≈ 0) and a shorter incubation, whereas a disease like Ebola has essentially no presymptomatic spread (people are contagious only after symptoms, so one could set κ_P very high to model an almost immediate symptom onset upon infectiousness).

It’s worth noting that parameter values can change with new variants or conditions. During the COVID-19 pandemic, for instance, more transmissible variants effectively had a higher β, and some evidence suggested different asymptomatic fractions or durations. The SEPIAR model (like any ODE model) assumes parameters are average constants, but one can piecewise-adjust them to simulate changes (e.g. increasing β for a more contagious strain, or changing θ if a variant causes more asymptomatic infections).

In summary, the parameters of SEPIAR have concrete interpretations: β controls how fast the fire spreads, σ and κ set the timing of when people become fire-starters, θ splits the hidden embers from the visible flames, and γ’s determine how quickly infections burn out. By plugging in estimates from epidemiological data, the model can mimic the trajectory of an outbreak under various scenarios.

🤔 Assumptions and Applicability

Every model rests on assumptions. For SEPIAR, the underlying assumptions are similar to those of other basic compartmental models, with additional considerations for the new compartments:

• Homogeneous Mixing: The SEPIAR model assumes that individuals mix uniformly in the population – any susceptible person is equally likely to contact any infectious person. This mass-action mixing means we do not explicitly account for social network structure or geographic clustering. In reality, people interact in clusters (families, workplaces, etc.), but homogeneous mixing is a useful first approximation. It implies that the infection terms (like S * I / N) are averaged over the whole population. If the population is very heterogeneous or segregated, a basic SEPIAR model may not capture local outbreak dynamics well (though it can be extended with spatial or network structure).
• Fixed Population (No Demographics): Typically, SEPIAR is applied to a single epidemic wave over a relatively short period. It often ignores births and natural deaths during the outbreak. The total population N is treated as constant (except for disease-related removals). This is reasonable for an epidemic unfolding over months, but for very long-term endemic modeling, one might add birth/death terms. The model also usually assumes no immigration or emigration – it’s a closed population scenario unless modified.
• No re-infection (short term): Once individuals move to R (Removed/Recovered), they are assumed immune (at least for the model’s timeframe). This means the model does not allow people in R to return to S (which would be an S–E–P–I–A–R–S cycle). For diseases that confer lasting immunity (e.g. measles, or COVID-19 in the short term), this is fine. If immunity wanes quickly or re-infections are common, the model can be extended to an SEPIARS structure, where recovered individuals gradually become susceptible again. In COVID-19, immunity was imperfect and temporary for some, but for modeling waves, SEPIAR often assumed immunity over the wave and considered re-infection outside that window.
• Disease Stages Are Sequential: The model assumes each individual goes through the compartments in a set order: S → E → (P or A) → possibly I → R. We assume no shortcuts and no backtracking. For example, a person cannot skip the exposed stage or go from asymptomatic to symptomatic mid-infection – in SEPIAR, asymptomatic means truly never symptomatic. This is a simplification; in reality a small fraction of “asymptomatic” might develop late symptoms, or mild cases could be misclassified. The model uses θ to essentially hard-code two distinct paths (asymptomatic vs. symptomatic). We also assume that presymptomatic individuals will definitely develop symptoms (100% of P goes to I). Again, reality is messier – symptoms can be a spectrum – but the compartments enforce these binary categories.
• Constant Transition Rates: SEPIAR uses constant rates (σ, κ, γ, etc.), which implicitly assume exponentially distributed stage durations. This means, for instance, that the chance of leaving E is the same at any moment and some people might leave E much faster or slower than the average. In reality, incubation periods have a more bell-shaped distribution, not a perfect exponential. To better fit real data, modelers sometimes use multiple E compartments (a chain of E’s) to approximate a more realistic distribution. The SEPIAR framework itself doesn’t dictate this, but it’s a common tweak – using, say, E₁, E₂,… Eₘ in series instead of one E gives a Gamma-distributed latent period. Similarly, one could use multiple stages in P or I if needed. The standard SEPIAR assumes one stage each, for simplicity.
• Instantaneous Symptom Onset: In the model, the moment a person leaves P, they are in I (symptomatic). There’s no intermediate gradation – symptoms are assumed to appear right when the compartment changes. Realistically, symptom onset is a point in time indeed, but severity can increase over time. The model doesn’t capture severity or progression of illness beyond contagiousness.
• Removed means Removed: Once in R, individuals are taken out of circulation. We assume they no longer transmit the disease at all. For COVID-19, recovered individuals were generally immune for some months and not infectious, so that fits. The model also lumps any fatalities into R (or one could split out a separate D compartment for deaths). If analyzing fatalities, one might add a separate death compartment D that branches out from I (and possibly A). In many SEPIAR implementations, however, “Removed” implicitly includes those who died (since they are removed from the population). This is acceptable if the fraction dying is small, but if mortality is substantial, tracking D explicitly is useful for public health planning.
• No Explicit Age or Risk Structure (unless extended): The basic SEPIAR model treats the population as homogeneous in terms of risk. Everyone shares the same parameters (β, θ, etc.). In truth, factors like age can hugely influence these values (older people might be less likely to be asymptomatic, for instance). The model can be stratified into subpopulations to handle this – e.g., one can create separate SEPIAR compartments for different age groups. In fact, researchers did this during COVID-19 to account for children vs. adults having different θ and β, etc.. But if not explicitly stated, a single SEPIAR model uses average parameters for the whole population. So, applying it assumes no group has a disproportionate role in spreading or different disease course. For many scenarios (like a broad community spread), this is a reasonable approximation. If one were modeling, say, a university campus vs. general public, two linked SEPIAR models might be better.

When is SEPIAR appropriate? This model is most useful under conditions where asymptomatic or presymptomatic transmission is significant. COVID-19 is the archetypal example: people could spread the virus days before showing symptoms and many never showed symptoms at all. Other diseases where SEPIAR-like thinking could apply include influenza (which has a short presymptomatic infectious period of about 1 day), chickenpox (people are contagious a couple days before the rash), and diseases with asymptomatic carriers like polio (the majority of polio infections are asymptomatic, yet infectious). In such cases, a simple SEIR model (which lumps all infectious individuals together and assumes they’re symptomatic) would underestimate the stealth spread. SEPIAR, by contrast, can capture those stealth dynamics.

On the other hand, if a disease has negligible asymptomatic infection and little to no presymptomatic transmission, SEPIAR might be overkill. For example, Ebola virus disease is not significantly transmissible until symptoms (especially severe symptoms) occur, and essentially all infected persons become visibly ill. A regular SEIR or SIR model (with an incubation period but no asymptomatic compartment) might suffice for Ebola. SEPIAR would in that case just add complexity without much benefit (you might set θ≈0 and κ very large to mimic “no presymptomatic spread,” effectively collapsing P and A compartments to trivial roles).

Model validity and data: The SEPIAR model assumes that presymptomatic and asymptomatic individuals have the potential to spread disease and that we can estimate their relative infectiousness. These assumptions must be backed by data – e.g., contact tracing studies showing transmission from presymptomatic persons, or viral load measurements in asymptomatic carriers. If such data are unreliable, there is uncertainty in θ, κ_A, etc., which can make model predictions uncertain. Indeed, one challenge noted in COVID-19 modeling was identifying just how infectious asymptomatic cases truly are. But the strength of SEPIAR is that it allows analysts to test different assumptions – one can run the model assuming asymptomatics are, say, 50% as infectious, vs 0% or 100%, to see how outcomes change.

In summary, SEPIAR is applicable to epidemics where “invisible spreaders” matter. It assumes a structured but still simplified progression of disease states. As with all models, “all models are wrong, but some are useful.” SEPIAR sacrifices some simplicity to gain realism in scenarios like COVID-19. When its assumptions align with reality, it becomes a very useful tool in an epidemiologist’s toolbox – for forecasting outbreaks, evaluating control strategies (like isolation of presymptomatic cases via contact tracing), and estimating unobserved infections. If those assumptions don’t hold, a simpler model or a differently extended model may be more appropriate.

🧩 Model Extensions and Variants

The SEPIAR model did not emerge in isolation – it’s part of a family of compartmental models, each developed to address specific epidemiological features. We now discuss some direct variants and extensions related to SEPIAR, explaining how they modify the core model and why.

1. SIR and SEIR – The Baseline Models: Before asymptomatic transmission was a major concern, the go-to frameworks were SIR and SEIR. The SIR model (Susceptible–Infectious–Removed) dates back to 1927 and is the foundation for most compartmental modeling. It assumes an infectious person is symptomatic and infectious immediately upon infection (no latent period). The SEIR model adds an Exposed (E) compartment to incorporate an incubation period during which individuals are infected but not yet infectious. SEIR’s compartments are a subset of SEPIAR’s: essentially SEIR merges the P and A back into I or E. In SEIR, once an individual leaves E, they go straight to I (infectious and symptomatic). This is appropriate for diseases where either no asymptomatic spread occurs or it’s negligible, and where presymptomatic infectiousness is either absent or can be approximated by saying that people become infectious at symptom onset. SEIR is a special (simpler) case of SEPIAR where θ = 0 (no asymptomatics) and κ is effectively infinite (meaning the moment someone is infectious, they are symptomatic). Many early epidemic models for COVID-19 started as SEIR, but it became clear that additional compartments were needed to capture COVID’s dynamics.

2. SEIAR – Introducing Asymptomatic Infectious: One straightforward extension from SEIR is the SEIAR model (Susceptible–Exposed–Infectious–Asymptomatic–Removed). In this model, once the exposed incubation period ends, individuals split into two groups: I (symptomatic infectious) and A (asymptomatic infectious). There is no distinct presymptomatic compartment; presymptomatic time is usually considered part of E or simply not explicitly modeled. In SEIAR, asymptomatic individuals are infectious but never show symptoms, similar to SEPIAR’s A, and the I compartment includes only symptomatic cases. This model was used by many researchers in early COVID-19 to estimate the impact of asymptomatic carriers. Conceptually, SEIAR is like setting κ_P = 0 (no separate presymptomatic phase) in SEPIAR and absorbing that time either into E or A. Its equations are a bit simpler – essentially: S → E → I or A → R. What SEIAR captures: the existence of asymptomatic spreaders. What it misses: the nuance of presymptomatic transmission (it assumes either that infectiousness starts with symptoms for those who get them, or that any infectious period prior to symptoms is negligible or folded into E). This model is adequate if presymptomatic infectiousness is minor or if one primarily wants to distinguish reported (I) vs. unreported (A) cases. For instance, some COVID-19 modeling in China used SEIAR to simulate the early outbreak, considering asymptomatic (or unreported mild) cases as a separate class contributing to spread.

3. SEPIR – Introducing Presymptomatic Infectiousness: Another variant is the SEPIR model (Susceptible–Exposed–Presymptomatic–Infectious–Recovered). This model, as the name suggests, focuses on presymptomatic transmission. It splits the infectious class into presymptomatic (P) and symptomatic (I), but does not include an asymptomatic compartment. In other words, SEPIR assumes everyone will eventually show symptoms; it’s essentially SEPIAR with θ = 0 (no A’s). The benefit of SEPIR is isolating the time window when individuals are infectious but have no symptoms. This was critical for COVID-19, since control measures like contact tracing needed to catch people in P before they progressed to I. The SEPIR model was studied to quantify how presymptomatic spread alone could drive an epidemic. In SEPIR’s dynamics, after incubation (E), an individual goes into P (now infectious) and later to I (when symptoms appear). All infectious individuals do become symptomatic eventually. The equations for SEPIR are essentially the SEPIAR equations minus anything involving A (and setting θ=0). One study phrased it as: “The compartment P denotes those who are infectious but pre-symptomatic, whilst I denotes those who are infectious and symptomatic. The original I of SEIR is thus split into P and I.”. SEPIR can capture scenarios like pandemic influenza where virtually everyone gets symptoms, but significant transmission happens in the day or two before coughing and fever start. Use case: If asymptomatic truly means non-infectious (or there are few asymptomatic infections), but presymptomatic spread is observed, SEPIR is an appropriate model. It’s simpler than SEPIAR but richer than SEIR.

To illustrate the difference between SEIAR and SEPIR: The SEIAR model treats asymptomatics as a separate branch, but it doesn’t explicitly carve out presymptomatic time for those who will be symptomatic – it effectively assumes once you’re infectious, you either have symptoms or you’re an asymptomatic case. The SEPIR model treats everyone the same until they become infectious, then splits their infectious period into a symptom-free part (P) and a symptomatic part (I), but assumes no one skips the symptomatic part. The SEPIAR model combines both ideas – it allows some people to skip symptoms entirely (A) and for those who don’t, it separates their presymptomatic infectious phase (P). Thus, SEPIAR contains SEIAR and SEPIR as special cases. In fact, one publication described SEPIAR as “a combination of the first two” models (referring to SEPIR and SEIAR), including them as nested particular cases.

4. Adding a Death or Hospitalization compartment: The basic SEPIAR model doesn’t explicitly track disease outcomes beyond removal. But during severe outbreaks, it’s often useful to model how many people are in hospital or have died. One extension, applied in multiple COVID-19 studies, is the SEPIARD** model (or SEPIARD), where D is a compartment for deceased individuals. In such a model, a fraction of the I (symptomatic infected) flow would go to D instead of R. For example, one could have an additional equation for dD/dt = μ * I, and correspondingly dR/dt = γ_I * I – μ * I (if μ is the mortality rate out of I). The compartment D has no outflow (it’s absorbing). This allows the model to predict fatalities over time, which was important for healthcare planning. In the scientific reports model we discussed earlier, the authors explicitly included a fatality compartment D with age-specific mortality rates. Because fatality doesn’t feed back into transmission (the deceased don’t interact), adding D doesn’t change the infection dynamics, but it adds insight for public health (how deadly the epidemic is). Similarly, one might add a H (Hospitalized) compartment for those infectious individuals who are removed from circulation via hospitalization. This was done in some models; for instance, the SAPHIRE model for COVID-19 included H (isolated in hospital) in addition to asymptomatic and presymptomatic compartments. In SAPHIRE: Susceptible, Exposed, Pre-symptomatic (P), Asymptomatic (A), Symptomatic Infected (I), Hospitalized (H), and Removed (R) were all tracked. Hospitalization (H) can be considered a removed subclass that still might contribute a tiny bit to spread (e.g., infections in hospital), or we treat H as removed from community spread. Either way, adding H is useful to model healthcare load.

These extensions (D, H, etc.) are directly built on SEPIAR – they don’t change the infection pathways; they just add more granularity to outcomes. They are particularly relevant when modeling an epidemic’s impact on health systems (H) or in assessing interventions like isolation (which effectively moves people to a quarantined compartment that doesn’t infect others).

5. Age-structured or multi-group SEPIAR: Another improvement, briefly mentioned, is replicating the SEPIAR structure for different demographics and allowing interactions between them. Researchers often split the population into age groups because susceptibility, contact patterns, and symptomatic rates differ by age. For example, one might have separate S_E, E_E, P_E, I_E, A_E, R_E for the elderly, and S_Y, E_Y, P_Y, I_Y, A_Y, R_Y for youth, etc., with different θ or β values. The equations then include terms for infections across groups (like a contact matrix). This goes beyond a single SEPIAR model but is conceptually a parallel set of SEPIAR equations for each group with cross-coupling. During COVID-19, such age-structured SEPIAR-like models helped estimate, for instance, how closing schools (reducing contacts for children) might affect the overall spread, given children had different asymptomatic proportions and lower susceptibility. The trade-off is complexity and the need for more data (e.g., contact rates between each age group). Still, age-structured models remain “SEPIAR at heart” – they just handle multiple interacting subpopulations. A study by D. Neofotistos et al. (2020) in Scientific Reports used an age-stratified SEPIAR (with additional compartments) to simulate community spread and the effects of targeted lockdowns.

6. Network or spatial SEPIAR: In some cases, rather than splitting by demography, modelers incorporate spatial spread. One approach was a metapopulation SEPIAR where each city or region had its own SEPIAR compartments and movement between communities was modeled. Another approach is network-based, where individuals are nodes; some recent studies used agent-based simulations akin to SEPIAR but on contact networks. These aren’t new compartments per se, but they modify how the infection term is calculated (not purely S*I/N, but via network connections). These models relax the homogeneous mixing assumption for greater realism. They are advanced extensions requiring more computation.

Summary of variants: To keep it organized, here’s a quick list of variants and their defining features:

• SIR: No E, no P, no A. Simple but ignores incubation and asymptomatic carriers.
• SEIR: Adds incubation (E) but assumes once infectious, individuals are symptomatic. No P or A.
• SEIAR: Adds an asymptomatic compartment (A) alongside I. Captures asymptomatic spread but not explicitly presymptomatic spread (infectiousness assumed to coincide with symptoms for those who get them).
• SEPIR: Adds a presymptomatic compartment (P) before I. Captures presymptomatic spread but assumes everyone eventually symptomatic (no A compartment).
• SEPIAR: Full model with both P and A. Individuals can either be presymptomatic→symptomatic or asymptomatic. This encompasses both added complexities.
• SEPIARD / SEPIAH(R): Further splits removed into outcomes like D (deaths) or H (hospitalized/isolated). Useful for modeling fatalities or healthcare interventions.
• Stratified SEPIAR (Age, etc.): Multiple SEPIAR models running in parallel for different groups, exchanging infections between them. Improves realism at the cost of more parameters.
• Stochastic SEPIAR: Not a structural variant, but worth noting – all the above can be implemented stochastically (e.g. as jump processes or agent-based models) instead of deterministic ODEs. Stochastic versions account for chance and randomness (important in small populations or early outbreak stages), whereas the ODE version gives average behavior. Many COVID-19 analyses used particle filtering or stochastic simulations on SEPIAR to fit uncertain data.

Each variant is chosen based on the epidemiological question at hand. For instance, if one is studying optimal vaccination strategies and essential workers’ role, one might use a stratified SEPIAR with compartments for workers vs. others. In fact, Buckner et al. (2021) utilized an ODE-based SEPIAR model with an added dimension for “essential workers” to evaluate vaccine prioritization strategies. By doing so, they could account for the higher contact rates and exposure risk of essential workers within the SEPIAR framework, demonstrating who should get vaccinated first to best reduce spread. This exemplifies how flexible the model is: you can tailor compartments or split groups as needed, while preserving the fundamental P and A structure that accounts for hidden transmission.

Another extension worth mentioning is incorporating control measures directly into the model. While not new compartments, parameters like β can be made time-dependent or linked to policy variables (mask wearing, mobility reduction, etc.). Some SEPIAR models introduced factors for quarantine or isolation that move individuals out of circulation faster (effectively increasing removal rates or shifting people to “isolated” compartments). For example, one could add a compartment Q for quarantined infected individuals who are no longer spreading disease. During COVID-19, models that included testing and isolation could shunt a portion of P, I, or A into a quarantined state upon detection. This again shows how SEPIAR can be extended: the compartments we discussed can be entry or exit points for interventions (like detection moving someone from P/I/A to an isolated category).

The bottom line is that SEPIAR serves as a kind of modular template for many infectious disease models in the modern era. If a new pathogen emerges and we find that a significant number of cases are asymptomatic or that people transmit before symptoms, epidemiologists will likely deploy a SEPIAR or SEIAR/SEPIR type model to understand it. If the pathogen has other complexities (multiple strains, long-term carriers, etc.), additional compartments will be bolted on accordingly.

🎯 Conclusion

The SEPIAR model represents a significant evolution of classical epidemic modeling, reflecting the hard lessons learned from diseases like COVID-19. By explicitly accounting for presymptomatic and asymptomatic infectious individuals, SEPIAR provides a more comprehensive and realistic picture of outbreaks where a substantial amount of transmission occurs under the radar. This popular science overview has walked through the model’s structure – from Susceptibles to Removed, with all the subtle stages in between – and introduced the meaning of each equation and parameter in plain language.

In a nutshell, SEPIAR recognizes that an infected person’s journey is not always straightforward. There may be a stealth phase (E) when they don’t even know they’re infected, a ninja phase (P) when they are spreading disease invisibly before symptoms, or they might remain a silent carrier throughout (A). Only after these phases might the alarm bells of illness ring (I), eventually leading to outcome (R). By capturing these phases, the model allows scientists and public health officials to estimate how many “hidden” infections are likely occurring for each detected case, how quickly an outbreak might grow even if many cases aren’t caught, and what fraction of transmission control efforts (like contact tracing or testing) need to focus on people without symptoms.

Why is this important for the general public and policymakers? Models like SEPIAR have shown that failing to account for asymptomatic and presymptomatic spread can lead to a false sense of security. Early in the COVID-19 pandemic, some regions underestimated the virus because only symptomatic cases were considered; SEPIAR-type analyses revealed that the virus was spreading faster and wider than symptomatic case counts suggested. This understanding justified more aggressive and broad interventions (like masking everybody, not just the sick). It also underscored the importance of widespread testing – including of people without symptoms – since otherwise the A compartment would continue to fuel transmission unnoticed.

From a scientific perspective, SEPIAR and its variants have improved our ability to forecast and manage epidemics. They enable calculation of refined metrics like an effective reproduction number that changes over time, taking into account unobserved infections. They help evaluate strategies: for example, What if we isolate presymptomatic cases faster? (This reduces P → I transitions and P’s contribution to new infections.) Or What if a vaccine prevents not just symptoms but infection altogether? (This effectively reduces the S → E transition probability for those vaccinated, or changes θ because fewer infections occur asymptomatically among the vaccinated). By incorporating more biology and behavior into the model, we get closer to reality, while still keeping the model mathematically and computationally tractable.

In an Epidemiological ODE toolbox, the SEPIAR model would likely be one of the more complex preset models available – but also one of the most insightful for diseases like COVID-19, SARS, influenza, etc. It strikes a balance between simplicity and realism. On the one hand, it’s still a relatively low-dimensional ODE system that can be analyzed and simulated quickly. On the other, it captures the key epidemiological features that simpler models miss: the timing of infectiousness vs. symptoms and the existence of hard-to-detect carriers. As we saw, the model can be extended further (to SEPIAR plus other compartments or stratifications) if needed, illustrating its flexibility.

Looking forward, the experience with SEPIAR during COVID-19 has equipped public health experts with a better appreciation of “silent” epidemics. Should a new pathogen emerge, one of the first questions will be: “How much asymptomatic or presymptomatic transmission is happening?” If the answer is “a lot,” the SEPIAR framework will be dusted off and put to work to guide the response. Its equations might appear intimidating at first, but behind each term is a logical story of infection progression. And as we hope this article has conveyed, you don’t need to be a mathematician to grasp that story – it’s the story of how a virus or bacterium can spread in the shadows, and why shining light on those shadows with good data and models is crucial.

In conclusion, the SEPIAR model is a powerful tool in modern epidemiology, born out of necessity in understanding complex diseases like COVID-19. It reminds us that what we don’t see (in terms of symptomless spread) can deeply influence an outbreak’s course. By formalizing that insight, SEPIAR enables better predictions and more effective interventions, ultimately helping save lives. And that is the true measure of any epidemiological model’s success – not just fitting curves on a graph, but informing actions in the real world.

References:

en.wikipedia.orgen.wikipedia.org Compartmental models overview – basic idea of S, I, R compartments and their meaning.

nature.com Definition of SEIR model compartments (S, E, I, R) and constant population assumption.

nature.com Evidence of significant presymptomatic transmission of COVID-19 (up to ~48–62% of transmissions in studies from Singapore and China).

researchgate.net Study showing low proportion of symptomatic cases (only ~13–18%) and the need for larger R₀ to account for asymptomatic/presymptomatic spread.

researchgate.net Even if asymptomatics transmit poorly, presymptomatic + asymptomatic cases together caused ≥50% of infections at the outbreak peak (COVID-19, NYC).

ecp.ep.liu.se Definition of SEPIAR compartments by Norway’s public health institute: P = presymptomatic infectious, I = symptomatic infectious, A = asymptomatic infectious – essentially an extended SEIR model.

nature.com Description of SEPIAR structure in a Nature Communications study: Susceptible, Exposed, pre-symptomatic (post-latent), symptomatic, asymptomatic, recovered compartments.

nature.com Typical incubation period example: SARS-CoV-2 average ~6 days (range ~2–27 days).

nature.com Estimated fraction of asymptomatic cases (e.g. ~16% on average, with 6–41% range) in COVID-19 from reviews; highlights variability and uncertainty.

jgfleischer.com Assumption from prior modeling (Hsu & Hsieh 2008) that asymptomatic carriers may be ~50% as infectious as symptomatic individuals.

nature.com Example of assumed infectious period for asymptomatic COVID-19 cases (~10 days, yielding recovery rate γ_A = 0.1 per day).

covid-19.conacyt.mx Use of an ODE-based SEPIAR model with added structure (essential workers) in Buckner et al. 2021 to study optimal vaccine allocation – demonstrating a practical extension of SEPIAR.

nature.com Establishment of SEIAR models for COVID-19 due to many asymptomatic infections, noting asymptomatic transmission reports in various countries.

journals.plos.org Description of the SEPIR model’s infection stages: susceptible → exposed (latent) → pre-symptomatic infectious → symptomatic infectious → recovered, highlighting presymptomatic and symptomatic phases.

nature.com Inclusion of a mortality compartment (D) in an extended model – acknowledging COVID-19’s high mortality warranted a dedicated fatality compartment with no further interactions.

nature.com Introduction of age-structured compartments (Children, Young Adults, Adults, Elderly) in a COVID-19 model – an example of stratifying SEPIAR by age with different parameters and interactions.