Compartmental models like SIR (Susceptible-Infected-Recovered) and SEIR (Susceptible-Exposed-Infected-Recovered) are fundamental tools in epidemiology for understanding and predicting the spread of infectious diseases [1] . When represented using matrices, the core difference between a 3 × 3 matrix-based SIR model and a 4 × 4 matrix-based SEIR model lies directly in the number of compartments they include and, consequently, the transitions between these compartments that the matrices describe ( Figure 1 , Table 1 ).

At its heart, a matrix representation in this context, particularly in discrete-time models, provides a structured way to describe how individuals move between different disease states over a specific time interval. The dimensions of the matrix are determined by the number of compartments in the model.

The standard SEIR model is described by the following system of ODEs:

$\frac{\text{d}S}{\text{d}t}=-\beta \frac{SI}{N}$

$\frac{\text{d}E}{\text{d}t}=\beta \frac{SI}{N}-\sigma E$

$\frac{\text{d}I}{\text{d}t}=\sigma E-\gamma I$

$\frac{\text{d}R}{\text{d}t}=\gamma I$

Let the state vector be $X_{SEIR}\left(t\right)=\left(\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\text{}\end{array}\right)$. Then, $\frac{\text{d}{X}_{SEIR}}{\text{d}t}=M_{SEIR}\left(t\right)X_{SEIR}\left(t\right)$.

Based on the ODEs, the coefficients for $S,E,I,R$ in each rate equation are:

For $\frac{\text{d}S}{\text{d}t}$: The term is $-\beta \frac{I}{N}S$. Coefficient for S is $-\beta \frac{I}{N}$, others are 0.

For $\frac{\text{d}E}{\text{d}t}$: The terms are $+\beta \frac{I}{N}S$ and $-\sigma E$. Coefficient for S is $+\beta \frac{I}{N}$, for E is −σ, others are 0.

For $\frac{\text{d}I}{\text{d}t}$: The terms are $+\sigma E$ and $-\gamma I$. Coefficient for E is +σ, for I is −γ, others are 0.

For $\frac{\text{d}R}{\text{d}t}$: The term is $+\gamma I$. Coefficient for I is +γ, others are 0.

So, the matrix $M_{SEIR}\left(t\right)$ is:

$M_{SEIR}\left(t\right)=\left(\begin{array}{cccc}-\beta \frac{I\left(t\right)}{N}& 0& 0& 0\\ \beta \frac{I\left(t\right)}{N}& -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right)$,

Then, the 4 × 4 matrix-based SEIR model is:

$\left(\begin{array}{c}{S}_{t+1}\\ E_{t+1}\\ I_{t+1}\\ R_{t+1}\end{array}\right)=\left(\begin{array}{cccc}-\beta \frac{I\left(t\right)}{N}& 0& 0& 0\\ \beta \frac{I\left(t\right)}{N}& -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right)\left(\begin{array}{c}{S}_t\\ E_t\\ I_t\\ R_t\end{array}\right)$

where our provided parameters β = 0.03, σ = 0.2, γ = 0.1, and N = 1000.

We want to solve the matrix equation:

$\left(\begin{array}{c}{S}_{t+1}\\ E_{t+1}\\ I_{t+1}\\ R_{t+1}\end{array}\right)=\left(\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right)\left(\begin{array}{c}{S}_t\\ E_t\\ I_t\\ R_t\end{array}\right)$

These values are approximate and can vary based on specific strains and individual cases (15).

To compute the SEIR model values from time $t=0$ to $t=5$ using the Euler method with a variable step size $h=t+1$, we’ll proceed step by step. The matrix-based SEIR model is:

$\left(\begin{array}{c}\frac{\text{d}{S}_t}{\text{d}t}\\ \frac{\text{d}{E}_t}{\text{d}t}\\ \frac{\text{d}{I}_t}{\text{d}t}\\ \frac{\text{d}{R}_t}{\text{d}t}\end{array}\right)=\left(\begin{array}{cccc}-\beta \frac{I\left(t\right)}{N}& 0& 0& 0\\ \beta \frac{I\left(t\right)}{N}& -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right)\left(\begin{array}{c}{S}_t\\ E_t\\ I_t\\ R_t\end{array}\right)$

We’ll compute the values at each time step using the Euler method:

$S\left(t+1\right)=S\left(t\right)+h\cdot \frac{\text{d}S}{\text{d}t}$

$E\left(t+1\right)=E\left(t\right)+h\cdot \frac{\text{d}E}{\text{d}t}$

$I\left(t+1\right)=I\left(t\right)+h\cdot \frac{\text{d}I}{\text{d}t}$

$R\left(t+1\right)=R\left(t\right)+h\cdot \frac{\text{d}R}{\text{d}t}$

At t = 0, h = 1:

$\frac{\text{d}S}{\text{d}t}=-0.3\cdot \frac{998\cdot 1}{1000}=-0.2994$

$\frac{\text{d}E}{\text{d}t}=0.3\cdot \frac{998\cdot 1}{1000}-0.2\cdot 1=0.0994$

$\frac{\text{d}I}{\text{d}t}=0.2\cdot 1-0.3\cdot 1=-0.1$

$\frac{\text{d}R}{\text{d}t}=0.3\cdot 1=0.3$

Updating the compartments:

$S\left(1\right)=998+1\cdot \left(-0.2994\right)=997.7006$

$E\left(1\right)=1+1\cdot 0.0994=1.0994$

$I\left(1\right)=1+1\cdot \left(-0.1\right)=0.9$

$R\left(1\right)=0+1\cdot 0.3=0.3$

At t = 1, h = 2:

$\frac{\text{d}S}{\text{d}t}=-0.3\cdot 997.7006\cdot 0.91000\approx -0.2694$

$\frac{\text{d}E}{\text{d}t}=0.3\cdot 997.7006\cdot 0.91000-0.2\cdot 1.0994\approx 0.0495$

$\frac{\text{d}I}{\text{d}t}=0.2\cdot 1.0994-0.3\cdot 0.9\approx 0.0399$

$\frac{\text{d}R}{\text{d}t}=0.3\cdot 0.9=0.27$

Updating the compartments:

$S\left(2\right)=997.7006+2\cdot \left(-0.2694\right)=997.1618$

$E\left(2\right)=1.0994+2\cdot 0.0495=1.1984$

$I\left(2\right)=0.9+2\cdot 0.0399=0.9798$

$R\left(2\right)=0.3+2\cdot 0.27=0.84$

At t = 2, h = 3:

$\text{d}S/\text{d}t=-0.3\cdot 997.1618\cdot 0.97981000\approx -0.2930$

$\text{d}E/\text{d}t=0.3\cdot 997.1618\cdot 0.97981000-0.2\cdot 1.1984\approx 0.0533$

$\text{d}I/\text{d}t=0.2\cdot 1.1984-0.3\cdot 0.9798\approx 0.0659$

$\text{d}R/\text{d}t=0.3\cdot 0.9798=0.2939$

Updating the compartments:

$S\left(3\right)=997.1618+3\cdot \left(-0.2930\right)=996.2828$

$E\left(3\right)=1.1984+3\cdot 0.0533=1.3583$

$I\left(3\right)=0.9798+3\cdot 0.0659=1.1775$

$R\left(3\right)=0.84+3\cdot 0.2939=1.7217$

At t = 3, h = 4:

$\text{d}S/\text{d}t=-0.3\cdot 996.2828\cdot 1.17751000\approx -0.3519$

$\text{d}E/\text{d}t=0.3\cdot 996.2828\cdot 1.17751000-0.2\cdot 1.3583\approx 0.0802$

$\text{d}I/\text{d}t=0.2\cdot 1.3583-0.3\cdot 1.1775\approx 0.0056$

$\text{d}R/\text{d}t=0.3\cdot 1.1775=0.3532$

Updating the compartments:

$S\left(4\right)=996.2828+4\cdot \left(-0.3519\right)=994.8752$

$E\left(4\right)=1.3583+4\cdot 0.0802=1.6791$

$I\left(4\right)=1.1775+4\cdot 0.0056=1.1999$

$R\left(4\right)=1.7217+4\cdot 0.3532=3.1345$

At t = 4, h = 5:

$\text{d}S/\text{d}t=-0.3\cdot 994.8752\cdot 1.19991000\approx -0.3581$

$\text{d}E/\text{d}t=0.3\cdot 994.8752\cdot 1.19991000-0.2\cdot 1.6791\approx 0.0223$

$\text{d}I/\text{d}t=0.2\cdot 1.6791-0.3\cdot 1.1999\approx 0.0568$

$\text{d}R/\text{d}t=0.3\cdot 1.1999=0.3599$

Updating the compartments ( Table 2 , Figure 2 ):

$S\left(5\right)=994.8752+5\cdot \left(-0.3581\right)=993.0847$

$E\left(5\right)=1.6791+5\cdot 0.0223=1.7906$

$I\left(5\right)=1.1999+5\cdot 0.0568=1.4839$

$R\left(5\right)=3.1345+5\cdot 0.3599=4.9340$

To visualize the progression of the SEIR model over time using the Euler method with a variable step size $h=t+1$, we can sketch a line plot showing the number of individuals in each compartment (Susceptible, Exposed, Infected, Recovered) from time $t=0$ to $t=5$.

The plot would show the Susceptible curve declining as individuals become exposed. The Exposed curve would rise initially as more individuals become exposed, then decline as they transition to the Infected compartment. The Infected curve would show a slight fluctuation before increasing, reflecting the dynamics of disease progression. The Recovered curve would rise steadily, indicating the accumulation of recovered individuals over time.

This visualization helps in understanding the dynamics of disease spread and the impact of parameters like transmission rate (β), incubation rate (σ), and recovery rate (γ) on the population compartments over time.

If you need assistance in creating this plot using software tools like Excel, Python (Matplotlib), or any other platform, feel free to ask!

We will utilize matrix representation to analyze the linear form of the SEIR model.

Structure: The SEIR model divides the population into four compartments:

Susceptible (S): Individuals who can contract the disease.

Exposed (E): Individuals who have been infected but are not yet infectious.

Infectious (I): Individuals capable of transmitting the disease.

Recovered (R): Individuals who have recovered and are assumed to have immunity.

The transitions between these compartments are governed by differential equations that model the rates of infection, incubation, and recovery.

The SEIR model and its extensions are instrumental in:

Predicting Disease Dynamics: Forecasting the progression of an epidemic under various scenarios.

Evaluating Intervention Strategies: Assessing the potential impact of public health measures like lockdowns and vaccination campaigns.

Informing Policy Decisions: Providing data-driven insights to guide governmental responses to epidemics.

By incorporating additional compartments and varying parameters, modern SEIR models offer a more nuanced understanding of disease spread, enabling more effective control and mitigation strategies.

Recent advancements in epidemiological modeling have led to significant extensions of the classical SEIR (Susceptible-Exposed-Infectious-Recovered) framework, enhancing its ability to capture the complexities of real-world epidemics. These modern adaptations incorporate various factors such as population heterogeneity, behavioral responses, vaccination strategies, and spatial dynamics to improve predictive accuracy and inform public health interventions.

While foundational compartmental models like SEIR provide a crucial starting point for understanding infectious disease dynamics, their simplified nature often necessitates adaptation to accurately reflect the complexities of real-world outbreaks. Emphasizing application and refinement, recent research has focused on enhancing these models. This involves tailoring the SEIR framework to incorporate more realistic biological and social factors, thereby improving their utility for analyzing specific epidemics, evaluating interventions, and informing public health strategies.

Recent research has expanded the SEIR model to better capture the complexities of real-world epidemics [4] .

Utilize fractional calculus to model memory effects in disease transmission, providing a more accurate representation of diseases like COVID-19 [5] .

Segment the population by age groups to assess the impact of interventions like vaccination strategies and mobility restrictions [6] .

Adjust parameters over time to reflect changes in public health policies and population behavior [7]

The 4 × 4 matrix representation of the linear SEIR model is particularly useful for:

1) Stability Analysis: Determining the stability of the disease-free equilibrium by analyzing the eigenvalues of the system matrix.

2) Threshold Conditions: Identifying critical thresholds, such as $R_0=1$, which delineate whether an infectious disease will die out or lead to an epidemic [8] [9] .

3) Policy Planning: Informing public health interventions by understanding how changes in parameters (e.g., transmission rate β, recovery rate γ) affect disease dynamics.

This linear approach provides a foundational framework for more complex models and is essential for early-stage epidemic analysis.

Determining the stability of the disease-free equilibrium by analyzing the eigenvalues of the system matrix.

Identifying critical thresholds, such as $R_0=1$, which delineate whether an infectious disease will die out or lead to an epidemic [8] [9] .

Informing public health interventions by understanding how changes in parameters (e.g., transmission rate β, recovery rate γ) affect disease dynamics.

This linear approach provides a foundational framework for more complex models and is essential for early-stage epidemic analysis.

In the SEIR model, the population is divided into four compartments:

S: Susceptible individuals

E: Exposed individuals (infected but not yet infectious)

I: Infectious individuals

R: Recovered individuals

First: Susceptible to Exposed (S → E): Susceptible people become exposed when they come into contact with an infectious individual and the rate of transmission is dictated by β.

Second: Exposed to Infectious (E → I): After the incubation period when the exposed individuals become infectious and this transition occurs at a rate σ. If the average incubation period for an infection is 5 days, $\sigma =1/5$ which means that 20% of the exposed individuals might be infected each day.

Third: Infectious to Recovered (I → R): The recovery rate γ. e.g. if an infection with an average infectious period of 10 days, $\gamma =1/10$, meaning 10% of the infectious population will be recovered each day.

The SEIR model is mathematically described utilizing a set of ordinary differential equations (ODEs), which describe how each factor changes over time (3, 4, 5, 6). These equations are the backbone of the model, defining how the disease spreads through the population.

The equations are:

$\frac{\text{d}S}{\text{d}t}=-\frac{\beta SI}{N}$

$\frac{\text{d}E}{\text{d}t}=\frac{\beta SI}{N}-\sigma E$

$\frac{\text{d}I}{\text{d}t}=\sigma E-\nu I$

$\frac{\text{d}R}{\text{d}t}=\gamma I$

$\frac{\text{d}S}{\text{d}t}$: The rate of susceptible population decreases. The more infectious individuals, the faster the susceptible population shrinks.

$\frac{\text{d}E}{\text{d}t}$: The rate of exposed population changes. This depends on how many susceptible individuals become exposed at first term, and how many exposed individuals progress to becoming infectious at second term.

$\frac{\text{d}I}{\text{d}t}$: The rate of infectious population change. Exposed individuals become infectious at a rate σ, while infectious individuals recover at a rate γ.

$\frac{\text{d}R}{\text{d}t}$: The rate at which the recovered population increases.

The dynamics of these compartments can be described by the following system of differential equations:

$\frac{\text{d}S}{\text{d}t}=-\beta S\left(t\right)I\left(t\right),$

$\frac{\text{d}E}{\text{d}t}=\beta S\left(t\right)I\left(t\right)-\sigma E\left(t\right),$

$\frac{\text{d}I}{\text{d}t}=\sigma E\left(t\right)-\gamma I\left(t\right)$

$\frac{\text{d}R}{\text{d}t}=\gamma I\left(t\right).$

To linearize this system around the disease-free equilibrium (DFE), where $S=S^{*}$, $E=0$, $I=0$, and $R=0$, it can be assumed that $S\approx S^{*}$ remains constant. This approximation simplifies the nonlinear terms, allowing us to express the system in matrix form:

$\frac{\text{d}}{\text{d}t}\left(\begin{array}{c}S\\ E\\ I\\ R\end{array}\right)=\left(\begin{array}{cccc}0& 0& -\beta s^{*}& 0\\ 0& -\sigma & \beta s^{*}& 0\\ 0& -\sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right)\left(\begin{array}{c}S\\ E\\ I\\ R\end{array}\right)$

This 4 × 4 matrix captures the linearized dynamics of the SEIR model near the disease-free equilibrium (DFE).

We define the state of the population at time t as a column vector:

$X\left(t\right)=\left[\begin{array}{c}\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\end{array}\\ R\left(t\right)\end{array}\right]$

We represent the system of linear differential equations in matrix form:

$\left[\begin{array}{c}{S}^{\prime }\left(t\right)\\ E^{\prime }\left(t\right)\\ I^{\prime }\left(t\right)\\ R^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\left[\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\end{array}\right]$

$X^{\prime }\left(t\right)=\left[\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\cdot X\left(t\right)$

where:

$\beta$ is the infection rate (e.g., 0.3)

$\sigma$ is the exposed rate (e.g., 0.19)

$\gamma$ is the recovery rate (e.g., 0.1)

$S\left(t\right)$, $E\left(t\right),$ $I\left(t\right)$, and $R\left(t\right)$ are the number of susceptible, exposed, infected, and recovered individuals at time t.

$X^{\prime }\left(t\right)$ gives the rate of change of each compartment

Example (Initial State):

Let’s suppose at day 0:

$X\left(0\right)=\left[\begin{array}{c}999\\ 0\\ 1\\ 0\end{array}\right]$

Using Euler’s method:

$X\left(t+1\right)\approx X\left(t\right)+\Delta t\cdot X^{\prime }\left(t\right)$

This matrix-based approach simplifies computation, especially when paired with numerical solvers or computational tools.

The SEIR model extends the classical SIR model by adding the Exposed compartment, which represents individuals who have been infected but are not yet infectious. The model is governed by the following model of differential equations:

$S^{\prime }\left(t\right)=-\frac{\beta S\left(t\right)I\left(t\right)}{N}$

$E^{\prime }\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N}-\sigma E\left(t\right)$

$I^{\prime }\left(t\right)=\sigma E\left(t\right)-\gamma I\left(t\right)$

$R^{\prime }\left(t\right)=\gamma I\left(t\right)$

Where the SEIR process ( $S\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),t\ge 0$) divides a population of size N undergoing an epidemic into three classes called “susceptible, exposed, infectives, and removed” [2] - [6] .

$\beta$ (beta): Transmission rate

$\sigma$ (sigma): Rate at which exposed individuals become infectious (i.e., 1/σ is the average incubation period)

$\gamma$ (gamma): Recovery rate

N: Total population (assumed constant)

Step 1: Define the SEIR Model Parameters

We’ll assume the following:

N = 1000 (total population)

Initial values:

Parameters:

Time step: $\Delta t=1\text{day}$.

In traditional SEIR models, the population is divided into four compartments: susceptible (S), E—representing individuals are exposed but are not yet infectious, infected (I), and recovered (R). The transitions between these compartments are governed by differential equations. Matrix-SEIR models generalize this approach by introducing matrices to capture the interactions between multiple subgroups within the population. This allows for a more nuanced representation of disease dynamics, especially in heterogeneous populations.

The term “SYR” was proposed to emphasize the structural similarity to the classic SIR model while highlighting the use of matrix operations to handle complex interactions among subgroups.

To extend the SIR framework and incorporate the exposed (latent) stage of infection, the SEIR model introduces a fourth compartment—E(t)—representing individuals who have been exposed to the disease but are not yet infectious. This modification is particularly important for diseases with an incubation period.

For a more realistic and accurate representation of disease dynamics, the system is modeled using non-linear differential equations, where the infection term depends on the product of susceptible and infected individuals, rather than being approximated linearly.

The SEIR model can be expressed in matrix form as follows:

$\left[\begin{array}{c}{S}^{\prime }\left(t\right)\\ E^{\prime }\left(t\right)\\ I^{\prime }\left(t\right)\\ R^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cccc}-\beta I\left(t\right)/N& 0& 0& 0\\ \beta I\left(t\right)/N& -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\left[\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\end{array}\right]$

In this formulation:

$\beta$ (beta) is the transmission rate,

$\sigma$ (sigma) is the rate at which exposed individuals become infectious (i.e., the inverse of the incubation period),

$\gamma$ (gamma) is the recovery rate,

$N$ is the total population.

This matrix representation simplifies the numerical solution of the model and provides a clear structural view of transitions between compartments. It is especially useful for simulations using numerical methods such as Euler’s method or Runge–Kutta techniques.

For linear approximation (or in simplified teaching/simulation contexts), we can represent this in matrix form as:

$\left[\begin{array}{c}{S}^{\prime }\left(t\right)\\ E^{\prime }\left(t\right)\\ I^{\prime }\left(t\right)\\ R^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\left[\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\end{array}\right]$

Notes:

This form assumes constant coefficients, and is a linearization that ignores the full nonlinearity $\beta SI/N$.

For more accurate simulations, the nonlinear form is preferred and often solved using numerical methods like Euler or Runge-Kutta.

Application of Matrices in Pharmaceutical Science: Matrices play a significant role in various domains of pharmaceutical science, particularly in the modeling and analysis of dynamic systems. Their structured and systematic nature makes them highly suitable for solving complex problems related to drug behavior and interactions within the human body.

The SEIR (Susceptible, Exposed, Infected, Recovered) model is a well-known epidemiological model used to describe how infectious diseases [9] - [12] spread through populations. In this model, matrices can be used to represent the transitions between different states of individuals in a population.

Susceptible (S): Individuals who can catch the disease.

Exposed (E): individuals who have been infected but are not yet infectious themselves.

Infected (I): Individuals who have the disease and can spread it.

Recovered (R): Individuals who have recovered and are immune.

The transition from one state to another can be described by a set of differential equations, which can be solved using matrices for large-scale or complex populations.

When a susceptible individual comes into contact with an infectious person, they move into the exposed compartment.

Exposed individuals are infected but not yet capable of transmitting the disease. This period is crucial for diseases with significant incubation times, such as COVID-19.

After the incubation period, exposed individuals become infectious and move into the infectious compartment.

To simplify and systematize the analysis of the SEIR (Susceptible–Exposed–Infected–Recovered) model, the system of differential equations is expressed in matrix form. This approach allows the dynamics of the model to be represented as the product of a coefficient matrix and a state vector. Specifically, the model equations are written as:

$\left[\begin{array}{c}{S}^{\prime }\left(t\right)\\ E^{\prime }\left(t\right)\\ I^{\prime }\left(t\right)\\ R^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\left[\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\end{array}\right]$

Here, the first matrix contains the constants representing infection and recovery rates, while the column vector holds the time-dependent variables. This linear matrix form provides a compact structure for applying numerical methods such as Euler’s method—it is one way mathematicians model differential equations that cannot be solved. Euler’s method treats each step of a differential equation as a linear equation, and facilitates computational simulations. The attached file illustrates this formulation in detail and shows how the interactions among susceptible, infected, and recovered individuals are governed by the matrix multiplication.

For example, the system can be represented as:

$\left[\begin{array}{c}{S}^{\prime }\left(t\right)\\ E^{\prime }\left(t\right)\\ I^{\prime }\left(t\right)\\ R^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cccc}-\beta & 0& 0& 0\\ \beta & -\sigma & 0& 0\\ 0& \sigma & -\gamma & 0\\ 0& 0& \gamma & 0\end{array}\right]\left[\begin{array}{c}S\left(t\right)\\ E\left(t\right)\\ I\left(t\right)\\ R\left(t\right)\end{array}\right]$

$S_{n+1}=S_n-\beta S_n$

$E_{n+1}=E_n+\beta S_n-\sigma E_n$

$I_{n+1}=I_n+\sigma E_n-\gamma I_n$

$R_{n+1}=R_n+\gamma I_n$

where:

$\beta$ (beta) is the infection rate (e.g., 0.03),

$\sigma$ (sigma) is the transition rate (e.g., 0.2),

$\gamma$ (gamma) is the recovery rate (e.g., 0.1).

1) Definition: σ is the transition rate from the Exposed (E) compartment to the Infectious (I) compartment.

2) Interpretation: If the average incubation period (the time between exposure to the pathogen and becoming infectious) is D days, then:

$\sigma =\frac{1}{D}$

For example, if the average incubation period is 5 days, then σ = 1/5 = 0.2 per day. This means that each day, 20% of the exposed individuals become infectious.

3) Role in the SEIR Model: σ determines how quickly the disease progresses from the exposed stage to the infectious stage. A higher σ implies a shorter incubation period, leading to a faster spread of the disease.

σ is the transition rate from the Exposed (E) compartment to the Infectious (I) compartment.

If the average incubation period (the time between exposure to the pathogen and becoming infectious) is D days, then:

$\sigma =\frac{1}{D}$

For example, if the average incubation period is 5 days, then σ = 1/5 = 0.2 per day. This means that each day, 20% of the exposed individuals become infectious.

σ determines how quickly the disease progresses from the exposed stage to the infectious stage. A higher σ implies a shorter incubation period, leading to a faster spread of the disease.

Example

Consider a disease with an average incubation period of 4 days. Then:

$\sigma =\frac{1}{4}=0.25$ per day

This indicates that 25% of the exposed individuals become infectious each day.

Understanding σ is crucial for accurately modeling the dynamics of infectious diseases, especially those with significant incubation periods.

This matrix equation models how individuals move between susceptible, exposed, infected, and recovered states over time.

Computing each day manually here using the Euler method and the linear matrix model. Let’s do it for a few days:

Parameters:

$\beta =0.03$

$\sigma =0.2$

$\gamma =0.1$

Initial Conditions:

$S_0=999$

$E_0=0$

$I_0=1$

$R_0=0$

We’ll compute the values step by step:

Step 0:

$S_1=999-0.03\times 999=969.03$

$E_1=0+0.03\times 999-0.2\times 0=29.97$

$I_1=1+0.2\times 0-0.1\times 1=0.9$

$R_1=0+0.1\times 1=0.1$

Step 1:

$S_2=969.03-0.03\times 969.03=939$

$E_2=29.97+0.03\times 969.03-0.2\times 29.97=53.05$

$I_2=0.9+0.2\times 29.97-0.1\times 0.9=6.80$

$R_2=0.1+0.1\times 0.9=0.19$

Step 2:

$S_3=939.96-0.03\times 939.96=911$

$E_3=53.05+0.03\times 939.96-0.2\times 53.05=75.38$

$I_3=6.80+0.2\times 53.05-0.1\times 6.80=17.92$

$R_3=0.19+0.1\times 6.80=0.87$

Step 3:

$S_4=911.76-0.03\times 911.76=884$

$E_4=75.38+0.03\times 911.76-0.2\times 75.38=96$

$I_4=17.92+0.2\times 75.38-0.1\times 17.92=33.26$

$R_4=0.87+0.1\times 17.92=2.65$

Step 4:

$S_5=884.41-0.03\times 884.41=857.88$

$E_5=96.25+0.03\times 884.41-0.2\times 96.25=115.14$

$I_5=33.26+0.2\times 96.25-0.1\times 33.26=53.52$

$R_5=2.65+0.1\times 33.26=5.36$

This table illustrates the progression of the disease over the first five time steps using the discrete SEIR model.

Note: Values are rounded to two decimal places for clarity.

The SEIR model is governed by a system of nonlinear differential equations, with the infection rate depending on the product of susceptible and infected individuals, i.e., $\text{d}S/\text{d}t=-\beta S\frac{I}{N}$.

The total population $N=S+E+I+R$ remains constant over time, assuming no births or deaths.

Realism: By modeling the exposed phase, the SEIR model provides a more realistic depiction of disease progression, especially for illnesses with incubation periods.

Matrix Model

Linear Approximation: The matrix model simplifies the dynamics using linear equations, often neglecting the nonlinear interaction between compartments.

Simplified Calculations: While easier to compute and analyze, the matrix model may not conserve the total population and can oversimplify the disease dynamics.

Instructional Use: Due to its simplicity, the matrix model is often used for educational purposes or initial approximations but may lack accuracy in realistic scenarios.

In summary, while the matrix model offers computational simplicity, the SEIR model’s incorporation of the exposed compartment and nonlinear interactions provides a more accurate and realistic framework for modeling infectious diseases with incubation periods.