A vaccination model for COVID-19 with asymptomatic transmission, waning immunity, and outbreak calibration

Abstract

The COVID-19 pandemic has posed unprecedented challenges to global public health, revealing critical gaps in our understanding of transmission dynamics, particularly the roles of asymptomatic spread and waning immunity. To address these gaps, we develop a mathematical model of COVID-19 transmission that incorporates asymptomatic and symptomatic pathways, vaccination dynamics, and the waning of both vaccine-induced and naturally acquired immunity. The basic reproduction number ${\mathcal{R}}_0$ is derived and decomposed into three transmission pathways: direct asymptomatic transmission, direct symptomatic transmission, and asymptomatic-to-symptomatic progression. This decomposition enables quantification of the contribution of asymptomatic spread to disease persistence. Numerical simulations show that vaccination coverage and vaccine durability jointly shape long-term epidemic burden, while natural immunity waning is critical for endemic persistence. Calibration of the model to an outbreak in Nanjing, China, yields excellent fit (R² = 0.9617) and reveals that asymptomatic transmission accounts for approximately 53.7% of the modeled transmission potential. The integrated framework provides both mathematical rigor for analyzing disease dynamics and practical insights for evidence-based pandemic response, highlighting the need for adaptive control strategies that address asymptomatic transmission and waning immunity.

Highlights

• •A COVID-19 model with asymptomatic transmission, vaccination, and waning immunity is developed.
• •The basic reproduction number ${\mathcal{R}}_0$ is decomposed into three transmission pathways.
• •Natural immunity waning is critical for endemic persistence.
• •Calibration to Nanjing outbreak data yields excellent fit (R² = 0.962).
• •Asymptomatic transmission accounts for approximately 54% of modeled transmission.

1. Introduction

The COVID-19 pandemic has highlighted the indispensable role of mathematical modeling in understanding and controlling emerging infectious diseases (Lin et al., 2020). With over 700 million confirmed cases and 7 million deaths worldwide as of 2023 (World Health Organization, 2023), this global health crisis has revealed transmission dynamics that challenge conventional epidemiological frameworks. A particularly critical feature is the substantial contribution of asymptomatic and pre-symptomatic transmission, estimated to account for 40-45% of all infections (Oran & Topol, 2020), with individuals often transmitting the virus 1-3 days before symptom onset (He et al., 2020). This silent transmission pathway complicates surveillance and control efforts, as standard detection methods primarily identify symptomatic cases. Mathematical models that fail to distinguish between asymptomatic and symptomatic transmission risk substantially underestimating the effective reproduction number and overestimating the effectiveness of symptom-based interventions (Li et al., 2020).

The complex immune response to SARS-CoV-2 introduces additional modeling challenges. Both natural infection and vaccine-induced immunity exhibit significant waning over time, with protection against infection declining substantially within 4-6 months for vaccine-induced immunity (Goldberg et al., 2021) and within 6-12 months for naturally acquired immunity (Chemaitelly et al., 2021). This immunological reality implies that recovered individuals may become susceptible again, transforming COVID-19 into an endemic disease requiring sustained vaccination efforts (Foutel-Rodier et al., 2025; Gutierrez & Gog, 2026). Traditional SIR-type models that assume permanent immunity after recovery are therefore inadequate for describing COVID-19 dynamics. Our model addresses this by incorporating waning of both vaccine-induced and naturally acquired immunity, providing a more realistic framework for long-term analysis.

Previous mathematical studies have advanced our understanding of specific aspects of COVID-19 transmission (Ferguson et al., 2020; Li et al., 2020; Musa et al., 2022; Zhao et al., 2020a, 2020b; Zou et al., 2025). Ferguson et al. (Ferguson et al., 2020) demonstrated in their influential Imperial College model that 30-50% of transmission occurred from pre-symptomatic individuals, while Li et al. (Li et al., 2020) estimated that neglecting asymptomatic transmission could lead to underestimating the basic reproduction number by a factor of 2.5-3.5. However, many models treat asymptomatic individuals as a homogeneous group with fixed transmission rates, overlooking dynamic interactions between symptomatic and asymptomatic classes. Moreover, establishing stability for high-dimensional epidemiological models incorporating multiple infection classes and nonlinear population dynamics remains a significant mathematical challenge.

In this paper, we develop and analyze a mathematical model that addresses several important gaps in the COVID-19 modeling literature. Our model explicitly incorporates dual-pathway transmission with distinct asymptomatic and symptomatic compartments characterized by different transmission rates, recovery rates, and disease-induced mortality. We introduce vaccination dynamics accounting for waning immunity and ongoing vaccination campaigns, reflecting the non-permanent protection observed with COVID-19 vaccines. Critically, we also account for waning of naturally acquired immunity, allowing recovered individuals to return to the susceptible class. Through a combination of analytical methods and numerical simulations, we systematically investigate the model's dynamical behavior and evaluate the effectiveness of vaccination strategies.

The remainder of this paper is organized as follows. Section 2 presents the mathematical model formulation, including detailed descriptions of all model compartments, parameters, and governing differential equations. Section 3 focuses on the existence of equilibria and the derivation of the basic reproduction number ${\mathcal{R}}_0$ using the next-generation matrix method (van den Driessche & Watmough, 2002). Section 4 investigates the stability of equilibria and the persistence of the system. Section 5 presents numerical simulations, beginning with theoretical simulations using literature-based parameters to illustrate fundamental dynamics (threshold behavior, natural immunity waning, vaccination effects, and sensitivity analysis), followed by model calibration to empirical data from Nanjing. Section 6 concludes the paper with a summary of key findings, discussion of epidemiological implications, and directions for future research.

2. Model formulation

2.1. Model formulation and assumptions

The model is based on the following assumptions:

• (A1)Population structure. The population is partitioned into six disjoint compartments based on disease and vaccination status:

$$S(t),E(t),I_A(t),I_S(t),V(t),R(t),$$

where S denotes susceptible individuals, E denotes exposed (latently infected) individuals, I_A denotes asymptomatic infectious individuals, I_S denotes symptomatic infectious individuals, V denotes vaccinated individuals, and R denotes recovered individuals. The total population is N(t) = S(t) + E(t) + I_A(t) + I_S(t) + V(t) + R(t).

• (A2)Demography. The population is open with constant recruitment rate Λ > 0 into the susceptible class. Individuals in all compartments are subject to natural mortality at per capita rate d > 0.
• (A3)Transmission. Disease transmission occurs through direct contact. The force of infection is

$$\lambda (t)=\frac{{\beta }_1{I}_A(t)+{\beta }_2{I}_S(t)}{N(t)},$$

where β₁, β₂ > 0 are the transmission rates from asymptomatic and symptomatic individuals, respectively. The standard incidence form λ(t)S(t) accounts for frequency-dependent transmission.

• (A4)Disease progression. Exposed individuals become infectious at rate ω > 0. A proportion δ ∈ (0, 1) develop symptoms (entering I_S), while the complementary proportion 1 − δ remain asymptomatic (entering I_A). Asymptomatic individuals may subsequently develop symptoms at rate ξ > 0.
• (A5)Recovery and waning immunity. Asymptomatic and symptomatic individuals recover at rates η₁ > 0 and η₂ > 0, respectively. Naturally acquired immunity wanes at rate σ > 0, returning recovered individuals to the susceptible class.
• (A6)Vaccination. Susceptible individuals are vaccinated at rate μ > 0 and enter the vaccinated class. Vaccine-induced immunity wanes at rate α > 0, returning vaccinated individuals to susceptibility. Perfect vaccine efficacy is assumed while immunity persists.
• (A7)Disease-induced mortality. Symptomatic infected individuals experience additional mortality at rate m > 0.

2.2. Governing equations

The dynamics of the system are described by the following system of ordinary differential equations:

$$\left\{\begin{array}{ll}& \frac{\mathrm{d}S}{\mathrm{d}t}=\mathrm{\Lambda }+\alpha V+\sigma R-\frac{{\beta }_{1}S{I}_A}{N}-\frac{{\beta }_{2}S{I}_S}{N}-(d+\mu )S,\\ & \frac{\mathrm{d}E}{\mathrm{d}t}=\frac{{\beta }_{1}S{I}_A}{N}+\frac{{\beta }_{2}S{I}_S}{N}-(d+\omega )E,\\ & \frac{\mathrm{d}{I}_A}{\mathrm{d}t}=(1-\delta )\omega E-(d+{\eta }_1+\xi )I_A,\\ & \frac{\mathrm{d}{I}_S}{\mathrm{d}t}=\delta \omega E+\xi I_A-(d+{\eta }_2+m)I_S,\\ & \frac{\mathrm{d}V}{\mathrm{d}t}=\mu S-(\alpha +d)V,\\ & \frac{\mathrm{d}R}{\mathrm{d}t}={\eta }_1{I}_A+{\eta }_2{I}_S-(d+\sigma )R.\end{array}\right)$$ (2.1)

Summing equation (2.1) yields the total population dynamics:

$$\frac{\mathrm{d}N}{\mathrm{d}t}=\mathrm{\Lambda }-dN-m{I}_S.$$ (2.2)

In the absence of infection, i.e., when E = I_A = I_S = 0, equation (2.2) reduces to $\frac{\mathrm{d}N}{\mathrm{d}t}=\mathrm{\Lambda }-dN$, so N(t) → Λ/d as t → ∞. The biologically feasible region

$$\mathrm{\Omega }=\left\{(S,E,I_A,I_S,V,R)\in {\mathbb{R}}_{+}^6:S+E+I_A+I_S+V+R\le \frac{\mathrm{\Lambda }}{d}\right\}$$

is positively invariant under the flow of system (2.1). That is, if (S(0), …, R(0)) ∈ Ω, then (S(t), …, R(t)) ∈ Ω for all t ≥ 0.

2.3. Proportional variables

To facilitate the stability analysis that follows, we reformulate the system in terms of proportions. This transformation replaces the rational transmission terms $\frac{\beta SI}{N}$ with bilinear products, thereby simplifying the computation of equilibria and their Jacobians.

Define

$$s=\frac{S}{N},e=\frac{E}{N},a=\frac{I_A}{N},i=\frac{I_S}{N},v=\frac{V}{N},r=\frac{R}{N},z=\frac{1}{N}.$$

These variables satisfy the conservation relation

$$s+e+a+i+v+r=1.$$ (2.3)

For any $x≔\frac{X}{N}\in \{s,e,a,i,v,r\}$, we have $\dot{x}=\frac{\dot{X}}{N}-x\frac{\dot{N}}{N}$. Using (2.2) and the original equation (2.1), a straightforward calculation yields the equivalent system:

$$\left\{\begin{array}{ll}& \frac{\mathrm{d}s}{\mathrm{d}t}=\mathrm{\Lambda }z(1-s)+\alpha v+\sigma r-{\beta }_{1}sa-{\beta }_{2}si-\mu s+mis,\\ & \frac{\mathrm{d}e}{\mathrm{d}t}={\beta }_{1}sa+{\beta }_{2}si-\omega e-\mathrm{\Lambda }ez+mie,\\ & \frac{\mathrm{d}a}{\mathrm{d}t}=(1-\delta )\omega e-({\eta }_1+\xi )a-\mathrm{\Lambda }az+mia,\\ & \frac{\mathrm{d}i}{\mathrm{d}t}=\delta \omega e+\xi a-({\eta }_2+m)i-\mathrm{\Lambda }iz+m{i}^2,\\ & \frac{\mathrm{d}v}{\mathrm{d}t}=\mu s-\alpha v-\mathrm{\Lambda }vz+miv,\\ & \frac{\mathrm{d}r}{\mathrm{d}t}={\eta }_{1}a+{\eta }_{2}i-\sigma r-\mathrm{\Lambda }rz+mir,\\ & \frac{\mathrm{d}z}{\mathrm{d}t}=-\mathrm{\Lambda }{z}^2+dz+miz.\end{array}\right)$$ (2.4)

The feasible region in the new variables is

$$\tilde{\mathrm{\Omega }}=\left\{(s,e,a,i,v,r,z)\in {\mathbb{R}}_{+}^6\times (0,\infty ):s+e+a+i+v+r=1\right\}.$$

The transformation is invertible for N > 0; hence system (2.4) is equivalent to the original system (2.1) on Ω.

3. Existence of equilibria and the basic reproduction number

3.1. Disease-free equilibrium (DFE)

Theorem 3.1

(Existence and uniqueness of the DFE). The system (2.4) admits a unique DFE

$$P_0=(s^0,e^0,a^0,i^0,v^0,r^0,z^0)\in \tilde{\mathrm{\Omega }},$$

given explicitly by

$$s^0=\frac{\alpha +d}{\alpha +d+\mu },v^0=\frac{\mu }{\alpha +d+\mu },z^0=\frac{d}{\mathrm{\Lambda }},e^0=a^0=i^0=r^0=0.$$

Moreover, the conservation relation s⁰ + v⁰ = 1 holds.

Proof. At the DFE, e = a = i = 0. From the sixth equation of system (2.4), $\frac{\mathrm{d}r}{\mathrm{d}t}=-(\sigma +\mathrm{\Lambda }z)r$, so r = 0 is the only nonnegative equilibrium. From the seventh equation of system (2.4), with i = 0, we have $\frac{\mathrm{d}z}{\mathrm{d}t}=z(d-\mathrm{\Lambda }z)$, which gives z = 0 or $z=\frac{d}{\mathrm{\Lambda }}$. Since z > 0, we obtain $z^0=\frac{d}{\mathrm{\Lambda }}$. Substituting into the first and the fifth equation of system (2.4) with e = a = i = r = 0 yields

$$0=\mathrm{\Lambda }{z}^0(1-s)+\alpha v-\mu s,0=\mu s-(\alpha +d)v,$$

where we have used Λz⁰ = d. Solving this linear system gives s⁰ and v⁰. The conservation relation s⁰ + v⁰ = 1 follows directly.

3.2. Basic reproduction number

The basic reproduction number ${\mathcal{R}}_0$ is computed using the next-generation matrix method (van den Driessche & Watmough, 2002) applied to the proportional system (2.4). The infected-state variables are (e, a, i). Let x = (e,a,i)^⊤. The infection subsystem can be written as

$$\dot{x}=\mathcal{F}(x)-\mathcal{V}(x),$$

where the new infection terms $\mathcal{F}$ and the transition terms $\mathcal{V}$ are defined as

$$\mathcal{F}=\left(\begin{array}{l}{\beta }_{1}sa+{\beta }_{2}si\\ 0\\ 0\end{array}\right),\mathcal{V}=\left(\begin{array}{l}\omega e+\mathrm{\Lambda }ez-mie\\ -(1-\delta )\omega e+({\eta }_1+\xi )a+\mathrm{\Lambda }az-mia\\ -\delta \omega e-\xi a+({\eta }_2+m)i+\mathrm{\Lambda }iz-m{i}^2\end{array}\right).$$

The Jacobian matrices of $\mathcal{F}$ and $\mathcal{V}$ at the DFE P₀ are given by

$$F=\left(\begin{array}{lll}0& {\beta }_1{s}^0& {\beta }_2{s}^0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),V=\left(\begin{array}{lll}d+\omega & 0& 0\\ -(1-\delta )\omega & d+{\eta }_1+\xi & 0\\ -\delta \omega & -\xi & d+{\eta }_2+m\end{array}\right).$$

The basic reproduction number ${\mathcal{R}}_0$ is the spectral radius of FV⁻¹. A direct computation yields

$${\mathcal{R}}_0=s^0\cdot \frac{\omega }{d+\omega }\left[\frac{(1-\delta ){\beta }_1}{d+{\eta }_1+\xi }+\frac{\delta {\beta }_2}{d+{\eta }_2+m}+\frac{\xi (1-\delta ){\beta }_2}{(d+{\eta }_1+\xi )(d+{\eta }_2+m)}\right].$$

Substituting $s^0=\frac{\alpha +d}{\alpha +d+\mu }$, we obtain the explicit formula

$${\mathcal{R}}_0=\frac{\alpha +d}{\alpha +d+\mu }\cdot \frac{\omega }{d+\omega }\left[\frac{(1-\delta ){\beta }_1}{d+{\eta }_1+\xi }+\frac{\delta {\beta }_2}{d+{\eta }_2+m}+\frac{\xi (1-\delta ){\beta }_2}{(d+{\eta }_1+\xi )(d+{\eta }_2+m)}\right].$$

Remark 1

(Decomposition and interpretation of ${\mathcal{R}}_0$). The basic reproduction number can be decomposed into three distinct transmission pathways:

$${\mathcal{R}}_0=s^0\left({\mathcal{R}}_0^{\text{asym}}+{\mathcal{R}}_0^{\text{sym}}+{\mathcal{R}}_0^{\text{asym}2\text{sym}}\right),$$

where

$$\begin{array}{ll}& {\mathcal{R}}_0^{\text{asym}}=\frac{(1-\delta )\omega {\beta }_1}{(d+\omega )(d+{\eta }_1+\xi )},\\ & {\mathcal{R}}_0^{\text{sym}}=\frac{\delta \omega {\beta }_2}{(d+\omega )(d+{\eta }_2+m)},\\ & {\mathcal{R}}_0^{\text{asym}2\text{sym}}=\frac{\xi (1-\delta )\omega {\beta }_2}{(d+\omega )(d+{\eta }_1+\xi )(d+{\eta }_2+m)}.\end{array}$$

The factor $s^0=\frac{\alpha +d}{\alpha +d+\mu }$ is the proportion of susceptible individuals at the DFE. It is placed outside the decomposition to separate the effect of vaccination from the intrinsic transmission pathways. Specifically, s⁰ decreases as the vaccination rate μ increases, reflecting the fact that vaccination reduces the effective susceptible population available for initial disease invasion. The three terms inside the parentheses represent the intrinsic transmission potential per susceptible individual, independent of vaccination coverage. Thus, multiplying by s⁰ adjusts the intrinsic transmission potential to account for the fraction of the population that is actually susceptible at the DFE. The classical basic reproduction number in a fully susceptible population (without vaccination) would be ${\mathcal{R}}_0^{\text{full}}=\frac{{\mathcal{R}}_0}{s^0}$.

3.3. Endemic equilibrium (EE)

Theorem 3.2

(Existence and uniqueness of the EE). If ${\mathcal{R}}_0>1$, then the proportional system (2.4) admits a unique EE

$$P^{\ast }=(s^{\ast },e^{\ast },a^{\ast },i^{\ast },v^{\ast },r^{\ast },z^{\ast })\in \tilde{\mathrm{\Omega }}$$

satisfying s∗ + e∗ + a∗ + i∗ + v∗ + r∗ = 1. And

$$\begin{array}{l}{s}^{\ast }=\frac{1}{{\mathcal{R}}_0}\cdot \frac{\alpha +d}{\alpha +d+\mu },i^{\ast }=\frac{c_2\left(1-\frac{1}{{\mathcal{R}}_0}\right)}{C},e^{\ast }=\frac{1-\frac{1}{{\mathcal{R}}_0}}{C},\\ a^{\ast }=c_1{e}^{\ast },v^{\ast }=\frac{\mu }{\alpha +d}{s}^{\ast },r^{\ast }=c_3{e}^{\ast },z^{\ast }=\frac{d+m{i}^{\ast }}{\mathrm{\Lambda }},\end{array}$$

where C = 1 + c₁ + c₂ + c₃ and

$$c_1=\frac{(1-\delta )\omega }{d+{\eta }_1+\xi },c_2=\frac{\omega \left[\delta (d+{\eta }_1)+\xi \right]}{(d+{\eta }_1+\xi )(d+{\eta }_2+m)},c_3=\frac{{\eta }_1{c}_1+{\eta }_2{c}_2}{d+\sigma }.$$

Proof. Any positive equilibrium P∗ satisfies the following system

$$\left\{\begin{array}{ll}& \mathrm{\Lambda }{z}^{\ast }(1-s^{\ast })+\alpha v^{\ast }+\sigma r^{\ast }-{\beta }_1{s}^{\ast }{a}^{\ast }-{\beta }_2{s}^{\ast }{i}^{\ast }-\mu s^{\ast }+m{i}^{\ast }{s}^{\ast }=0,\\ & {\beta }_1{s}^{\ast }{a}^{\ast }+{\beta }_2{s}^{\ast }{i}^{\ast }-\omega e^{\ast }-\mathrm{\Lambda }{e}^{\ast }{z}^{\ast }+m{i}^{\ast }{e}^{\ast }=0,\\ & (1-\delta )\omega e^{\ast }-({\eta }_1+\xi )a^{\ast }-\mathrm{\Lambda }{a}^{\ast }{z}^{\ast }+m{i}^{\ast }{a}^{\ast }=0,\\ & \delta \omega e^{\ast }+\xi a^{\ast }-({\eta }_2+m)i^{\ast }-\mathrm{\Lambda }{i}^{\ast }{z}^{\ast }+m{(i^{\ast })}^2=0,\\ & \mu s^{\ast }-\alpha v^{\ast }-\mathrm{\Lambda }{v}^{\ast }{z}^{\ast }+m{i}^{\ast }{v}^{\ast }=0,\\ & {\eta }_1{a}^{\ast }+{\eta }_2{i}^{\ast }-\sigma r^{\ast }-\mathrm{\Lambda }{r}^{\ast }{z}^{\ast }+m{i}^{\ast }{r}^{\ast }=0,\\ & -\mathrm{\Lambda }{(z^{\ast })}^2+d{z}^{\ast }+m{i}^{\ast }{z}^{\ast }=0.\end{array}\right)$$ (3.5)

From the seventh equation of system (3.5), we obtain

$$z^{\ast }=\frac{d+m{i}^{\ast }}{\mathrm{\Lambda }}.$$

Substituting this into the third, fourth, and sixth equations yields

$$a^{\ast }=c_1{e}^{\ast },i^{\ast }=c_2{e}^{\ast },r^{\ast }=c_3{e}^{\ast }$$

where

$$c_1=\frac{(1-\delta )\omega }{d+{\eta }_1+\xi },c_2=\frac{\omega \left[\delta (d+{\eta }_1)+\xi \right]}{(d+{\eta }_1+\xi )(d+{\eta }_2+m)},c_3=\frac{{\eta }_1{c}_1+{\eta }_2{c}_2}{d+\sigma }.$$

From the second equation, it follows that

$$s^{\ast }=\frac{d+\omega }{{\beta }_1{c}_1+{\beta }_2{c}_2}.$$

Note that the basic reproduction number ${\mathcal{R}}_0$ can be expressed as

$${\mathcal{R}}_0=\frac{s^0({\beta }_1{c}_1+{\beta }_2{c}_2)}{d+\omega },$$

we obtain

$$s^{\ast }=\frac{s^0}{{\mathcal{R}}_0}=\frac{1}{{\mathcal{R}}_0}\cdot \frac{\alpha +d}{\alpha +d+\mu }.$$

From the fifth equation, we have

$$v^{\ast }=\frac{\mu }{\alpha +d}{s}^{\ast }.$$

Using the normalization condition

$$s^{\ast }+e^{\ast }+a^{\ast }+i^{\ast }+v^{\ast }+r^{\ast }=1,$$

we derive

$$e^{\ast }=\frac{1-\frac{1}{{\mathcal{R}}_0}}{C},i^{\ast }=c_2{e}^{\ast }.$$

Since the sum of the right-hand sides of the first six equations in model (2.4) is identically zero on the hyperplane s + e + a + i + v + r = 1, the first equilibrium equation is automatically satisfied. Thus, when ${\mathcal{R}}_0>1$, the system admits a unique EE P∗.

4. Stability and persistence analysis

4.1. Local stability of the DFE

Theorem 4.1

(Local stability of the DFE). For the proportional system (2.4), the DFE P₀ = (s⁰, 0, 0, 0, v⁰, 0, z⁰) is locally asymptotically stable if ${\mathcal{R}}_0<1$, and unstable if ${\mathcal{R}}_0>1$.

Proof. We analyze the Jacobian matrix of the proportional system (2.4) at P₀. Ordering the variables as (e, a, i, s, v, r, z), the Jacobian matrix can be written in the following block triangular form:

$$J(P_0)=\left(\begin{array}{lll}{J}_{\text{inf}}& 0& 0\\ J_{21}& J_{22}& 0\\ J_{31}& 0& J_{33}\end{array}\right),$$

where each block is specified below.

Linearizing the subsystem corresponding to e, a, i at P₀, and noting that Λz⁰ = d, yields

$$J_{\text{inf}}=\left(\begin{array}{lll}-\omega -d& {\beta }_1{s}^0& {\beta }_2{s}^0\\ (1-\delta )\omega & -({\eta }_1+\xi )-d& 0\\ \delta \omega & \xi & -({\eta }_2+m)-d\end{array}\right).$$

The block corresponding to the variables (s, v, r) is given by:

$$J_{22}=\left(\begin{array}{lll}-d-\mu & \alpha & \sigma \\ \mu & -(\alpha +d)& 0\\ 0& 0& -d-\sigma \end{array}\right),$$

and the scalar block associated with z is:

$$J_{33}=-d.$$

The off-diagonal blocks J₂₁ and J₃₁ are generally nonzero; however, due to the block triangular structure of J(P₀), they do not affect the eigenvalues.

The characteristic polynomial of J₂₂ is

$$(\lambda +d)(\lambda +d+\sigma )(\lambda +d+\alpha +\mu )=0,$$

which yields three eigenvalues λ₁ = −d < 0, λ₂ = −(d + σ) < 0, λ₃ = −(d + α + μ) < 0. Hence, all eigenvalues of J₂₂ are negative. Furthermore, the eigenvalue of J₃₃ is λ = −d < 0.

Next, observe that J_inf = F − V, where F and V are the next-generation matrices defined in Section 3.2:

$$F=\left(\begin{array}{lll}0& {\beta }_1{s}^0& {\beta }_2{s}^0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),V=\left(\begin{array}{lll}d+\omega & 0& 0\\ -(1-\delta )\omega & d+{\eta }_1+\xi & 0\\ -\delta \omega & -\xi & d+{\eta }_2+m\end{array}\right).$$

By the next-generation matrix theory (van den Driessche & Watmough, 2002), all eigenvalues of F − V have negative real parts if and only if ${\mathcal{R}}_0<1$, whereas at least one eigenvalue has a positive real part if ${\mathcal{R}}_0>1$.

Since J(P₀) is block triangular, its eigenvalues consist of those of J_inf, J₂₂, and J₃₃. As all eigenvalues of J₂₂ and J₃₃ are negative, it follows that if ${\mathcal{R}}_0<1$, all eigenvalues of J(P₀) have negative real parts, and thus P₀ is locally asymptotically stable; if ${\mathcal{R}}_0>1$, then J_inf admits an eigenvalue with positive real part, and hence P₀ is unstable.

4.2. Uniform persistence

Theorem 4.2

(Uniform persistence). If ${\mathcal{R}}_0>1$, then the proportional system (2.4) is uniformly persistent with respect to the boundary $\partial \tilde{\mathrm{\Omega }}$. More precisely, there exists ϵ > 0 such that for any solution with initial condition in the interior $\tilde{\mathrm{\Omega }}°$ of $\tilde{\mathrm{\Omega }}$,

$${\mathrm{lim\; inf}}_{t\to \infty }e(t)\ge ϵ,{\mathrm{lim\; inf}}_{t\to \infty }a(t)\ge ϵ,{\mathrm{lim\; inf}}_{t\to \infty }i(t)\ge ϵ.$$

Proof. Let $\tilde{\mathrm{\Omega }}$ be the feasible region defined in Section 2. Denote by $\tilde{\mathrm{\Omega }}°=\{(s,e,a,i,v,r,z)\in \tilde{\mathrm{\Omega }}:e>0,a>0,i>0\}$ the interior, and let $\partial \tilde{\mathrm{\Omega }}=\tilde{\mathrm{\Omega }}\setminus \tilde{\mathrm{\Omega }}°$ be the boundary. Let Φ_t be the semiflow generated by the proportional system (2.4).

Define the maximal compact invariant set in $\partial \tilde{\mathrm{\Omega }}$ as

$$M_{\partial }=\{x\in \partial \tilde{\mathrm{\Omega }}:{\mathrm{\Phi }}_t(x)\in \partial \tilde{\mathrm{\Omega }}\text{for all}t\ge 0\}.$$

We first show that any trajectory contained in M_∂ must satisfy e(t) = a(t) = i(t) = 0 for all t ≥ 0. Indeed, suppose x ∈ M_∂ and consider its orbit {Φ_t(x): t ≥ 0}. Since the orbit remains in $\partial \tilde{\mathrm{\Omega }}$ for all t, at least one of e(t), a(t), i(t) is zero for each t. If at some time e(t) = 0 while a(t) > 0 or i(t) > 0, then either s(t) > 0, in which case $\frac{\mathrm{d}e}{\mathrm{d}t}={\beta }_{1}sa+{\beta }_{2}si>0,$ or s(t) = 0, in which case $\frac{\mathrm{d}s}{\mathrm{d}t}=\mathrm{\Lambda }z+\alpha v+\sigma r>0$ since z > 0, so that s becomes positive immediately and hence $\frac{\mathrm{d}e}{\mathrm{d}t}>0$. In either case, this contradicts the invariance of $\partial \tilde{\mathrm{\Omega }}$. Hence, whenever e(t) = 0, one must have a(t) = i(t) = 0.

Similarly, if a(t) = 0 for some t with e(t) > 0, then $\frac{\mathrm{d}a}{\mathrm{d}t}=(1-\delta )\omega e(t)>0,$ forcing a to become positive. The same argument applies to i(t). Consequently, any trajectory in M_∂ must satisfy e(t) = a(t) = i(t) = 0 for all t ≥ 0. Therefore, $M_{\partial }\subset \{(s,0,0,0,v,r,z)\in \tilde{\mathrm{\Omega }}\}.$

On this invariant subset, the infected compartments vanish identically, and the system reduces to the disease-free subsystem. By the stability of the DFE on this subsystem, every trajectory in M_∂ converges to P₀. Hence, M_∂ = {P₀}.

When ${\mathcal{R}}_0>1$, Theorem 4.1 implies that P₀ is unstable. The restriction of the Jacobian at P₀ to the infected compartments is

$$J_{\text{inf}}=\left(\begin{array}{lll}-\omega -d& {\beta }_1{s}^0& {\beta }_2{s}^0\\ (1-\delta )\omega & -({\eta }_1+\xi )-d& 0\\ \delta \omega & \xi & -({\eta }_2+m)-d\end{array}\right),$$

which is an irreducible Metzler matrix. Moreover, ${\mathcal{R}}_0>1$ implies that the spectral bound of J_inf is positive. Therefore, J_inf possesses a positive eigenvalue with a corresponding positive eigenvector. Hence, P₀ is repelling in the infected directions. More precisely, since the infected subsystem is cooperative near P₀ and its linearization has positive spectral bound, the standard comparison and linearization argument for irreducible Metzler systems implies that no solution initiating in $\tilde{\mathrm{\Omega }}°$ can have P₀ as its omega-limit point.

Since $\tilde{\mathrm{\Omega }}$ is positively invariant and the semiflow Φ_t is point-dissipative, and since the boundary dynamics admits the isolated acyclic covering {P₀}, all hypotheses of the persistence theorem (Theorem 4.1 of (Thieme, 1993), or equivalently Theorem 1.3.1 of (Zhao, 2003)) are satisfied. Therefore, Φ_t is uniformly persistent. Hence, there exists ϵ > 0 such that for any initial condition in $\tilde{\mathrm{\Omega }}°$,

$${\mathrm{lim\; inf}}_{t\to \infty }e(t)\ge ϵ,{\mathrm{lim\; inf}}_{t\to \infty }a(t)\ge ϵ,{\mathrm{lim\; inf}}_{t\to \infty }i(t)\ge ϵ.$$

4.3. Local stability of the EE

The Schur complement and diagonal dominance are useful matrix-theoretic tools for analyzing the spectral properties of structured Jacobian matrices; see, e.g. (Berman & Plemmons, 1994), for relevant matrix-theoretic foundations. Related Schur-complement and diagonal-dominance arguments have also been employed in the analysis of epidemic models (Gromov et al., 2019; Wang et al., 2019).

In this subsection, we analyze the local asymptotic stability of the EE P∗ under the condition ${\mathcal{R}}_0>1$. Unlike the disease-free case, where the Jacobian matrix admits a straightforward block triangular structure, the EE requires a more delicate analysis due to the coupling between infected and non-infected compartments. Our analysis proceeds as follows. We first derive the exact Jacobian matrix at P∗ in proportional variables, and then apply a combination of Schur complement reduction and a strict column diagonal dominance argument to show that all eigenvalues lie strictly in the open left half-plane.

We first present the exact Jacobian matrix at P∗. Ordering the variables as (e, a, i, s, v, r, z), a direct linearization of system (2.4) at the EE P∗ yields the following Jacobian matrix (obtained by symbolic differentiation and verified at the equilibrium point):

$$J(P^{\ast })=\left[\begin{array}{lllllll}-d-\omega & {\beta }_1{s}^{\ast }& e^{\ast }m+s^{\ast }{\beta }_2& a^{\ast }{\beta }_1+i^{\ast }{\beta }_2& 0& 0& -\mathrm{\Lambda }{e}^{\ast }\\ (1-\delta )\omega & -d-\xi -{\eta }_1& m{a}^{\ast }& 0& 0& 0& -\mathrm{\Lambda }{a}^{\ast }\\ \delta \omega & \xi & i^{\ast }m-d-m-{\eta }_2& 0& 0& 0& -\mathrm{\Lambda }{i}^{\ast }\\ 0& -{\beta }_1{s}^{\ast }& m{s}^{\ast }-s^{\ast }{\beta }_2& -a^{\ast }{\beta }_1-i^{\ast }{\beta }_2-d-\mu & \alpha & \sigma & \mathrm{\Lambda }(1-s^{\ast })\\ 0& 0& m{v}^{\ast }& \mu & -\alpha -d& 0& -\mathrm{\Lambda }{v}^{\ast }\\ 0& {\eta }_1& m{r}^{\ast }+{\eta }_2& 0& 0& -d-\sigma & -\mathrm{\Lambda }{r}^{\ast }\\ 0& 0& \frac{m(d+m{i}^{\ast })}{\mathrm{\Lambda }}& 0& 0& 0& -d-m{i}^{\ast }\end{array}\right].$$

For notational convenience we introduce the following shorthands (all evaluated at P∗):

$$l≔{\beta }_1{a}^{\ast }+{\beta }_2{i}^{\ast },\rho ≔\mathrm{\Lambda }{z}^{\ast }=d+m{i}^{\ast }.$$

The matrix can then be written in block form as

$$J(P^{\ast })=\left(\begin{array}{ll}{J}_{11}& L\\ M& K\end{array}\right),$$

with the blocks defined as follows (the variable order is (e, a, i) for the first three rows/columns and (s, v, r, z) for the last four).

$$\begin{array}{cc}{J}_{11}=\left(\begin{array}{lll}\hfill -d-\omega \hfill & \hfill {\beta }_1{s}^{\ast }\hfill & \hfill e^{\ast }m+{\beta }_2{s}^{\ast }\hfill \\ \hfill (1-\delta )\omega \hfill & \hfill -d-\xi -{\eta }_1\hfill & \hfill m{a}^{\ast }\hfill \\ \hfill \delta \omega \hfill & \hfill \xi \hfill & \hfill i^{\ast }m-d-m-{\eta }_2\hfill \end{array}\right),\hfill & L=\left(\begin{array}{llll}\hfill l\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\mathrm{\Lambda }{e}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\mathrm{\Lambda }{a}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\mathrm{\Lambda }{i}^{\ast }\hfill \end{array}\right),\hfill \\ M=\left(\begin{array}{lll}\hfill 0\hfill & \hfill -{\beta }_1{s}^{\ast }\hfill & \hfill m{s}^{\ast }-{\beta }_2{s}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m{v}^{\ast }\hfill \\ \hfill 0\hfill & \hfill {\eta }_1\hfill & \hfill m{r}^{\ast }+{\eta }_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{m\rho }{\mathrm{\Lambda }}\hfill \end{array}\right),\hfill & K=\left(\begin{array}{llll}\hfill -l-d-\mu \hfill & \hfill \alpha \hfill & \hfill \sigma \hfill & \hfill \mathrm{\Lambda }(1-s^{\ast })\hfill \\ \hfill \mu \hfill & \hfill -\alpha -d\hfill & \hfill 0\hfill & \hfill -\mathrm{\Lambda }{v}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d-\sigma \hfill & \hfill -\mathrm{\Lambda }{r}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\rho \hfill \end{array}\right).\hfill \end{array}$$

Theorem 4.3

(Local stability of the EE). Assume that ${\mathcal{R}}_0>1$. Let

$${\mathcal{C}}_2≔{\sup }_{\text{Re}\lambda \ge 0}|S_{12}(\lambda )|,{\mathcal{C}}_3≔{\sup }_{\text{Re}\lambda \ge 0}\left(|S_{13}(\lambda )|+|S_{23}(\lambda )|\right),$$

where S₁₂(λ), S₁₃(λ), S₂₃(λ) are the off-diagonal entries of the Schur complement matrix S(λ) defined in the proof. If

$${\mathcal{C}}_2<d+{\eta }_1\text{and}{\mathcal{C}}_3<\mathcal{L},$$

where $\mathcal{L}={\inf }_{\text{Re}\lambda \ge 0}|S_{33}(\lambda )|$, then the EE P∗ is locally asymptotically stable.

Proof. Ordering the variables as (e, a, i, s, v, r, z), the Jacobian matrix at P∗ has the block form

$$J(P^{\ast })=\left(\begin{array}{ll}{J}_{11}& L\\ M& K\end{array}\right),$$

where $J_{11}\in {\mathbb{R}}^{3\times 3}$, $L\in {\mathbb{R}}^{3\times 4}$, $M\in {\mathbb{R}}^{4\times 3}$, and $K\in {\mathbb{R}}^{4\times 4}$ are defined previously. For any λ satisfying Re λ ≥ 0, Appendix A.1 shows that K − λI₄ is invertible. Hence, by the Schur complement formula,

$$\det (J(P^{\ast })-\lambda I_7)=\det (K-\lambda I_4)\det (S(\lambda )),$$

where

$$S(\lambda )=J_{11}-\lambda I_3-L{(K-\lambda I_4)}^{-1}M.$$

Therefore, it suffices to show that

$$\det S(\lambda )\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

From Appendices A.2-A.3, the matrix S(λ) takes the form

$$S(\lambda )=\left(\begin{array}{lll}{S}_{11}(\lambda )& S_{12}(\lambda )& S_{13}(\lambda )\\ S_{21}(\lambda )& S_{22}(\lambda )& S_{23}(\lambda )\\ S_{31}(\lambda )& S_{32}(\lambda )& S_{33}(\lambda )\end{array}\right),$$

with

$$\begin{array}{c}{S}_{11}(\lambda )=-d-\omega -\lambda ,S_{21}(\lambda )=(1-\delta )\omega ,S_{31}(\lambda )=\delta \omega ,S_{22}(\lambda )=-d-\xi -{\eta }_1-\lambda ,\hfill \\ S_{32}(\lambda )=\xi ,S_{23}(\lambda )=\frac{m{a}^{\ast }\lambda }{\rho +\lambda },S_{33}(\lambda )=-(d+m+{\eta }_2+\lambda )+\frac{i^{\ast }m\lambda }{\rho +\lambda }.\hfill \end{array}$$

The entries S₁₂(λ) and S₁₃(λ) are rational functions of λ, analytic on the open right half-plane and continuous and bounded on the closed right half-plane; their explicit forms are unnecessary for the following estimates. We now analyze S₃₃(λ). Substituting

$$\rho =\mathrm{\Lambda }{z}^{\ast }=d+m{i}^{\ast },$$

we rewrite

$$S_{33}(\lambda )=-(d+m+{\eta }_2+\lambda )+\frac{i^{\ast }m\lambda }{\rho +\lambda }=\frac{\mathcal{N}(\lambda )}{\rho +\lambda },$$

where

$$\mathcal{N}(\lambda )=-(d+m+{\eta }_2)\rho -\left(d+m+{\eta }_2+\rho -i^{\ast }m\right)\lambda -{\lambda }^2.$$

Set

$$A=d+m+{\eta }_2>0,B=\rho >0,C=i^{\ast }m>0.$$

Then

$$\mathcal{N}(\lambda )=-AB-(A+B-C)\lambda -{\lambda }^2.$$

Thus the equation $\mathcal{N}(\lambda )=0$ is equivalent to

$${\lambda }^2+(A+B-C)\lambda +AB=0.$$

Since

$$A+B-C=(d+m+{\eta }_2)+\rho -i^{\ast }m=(d+m+{\eta }_2)+(d+m{i}^{\ast })-i^{\ast }m=2d+m+{\eta }_2>0,$$

and

$$AB>0,$$

the quadratic equation

$${\lambda }^2+(A+B-C)\lambda +AB=0$$

has coefficients satisfying

$$A+B-C>0,AB>0.$$

Its discriminant is

$$\mathrm{\Delta }={(A+B-C)}^2-4AB.$$

If Δ < 0, then the two roots form a complex conjugate pair with real part

$$-\frac{A+B-C}{2}<0.$$

If Δ ≥ 0, then the roots are real. In this case, since the sum of the roots is

$$-(A+B-C)<0$$

and the product is

$$AB>0,$$

both roots must be strictly negative. Therefore, in all cases, every root lies strictly in the open left half-plane. Equivalently,

$$\mathcal{N}(\lambda )\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

Moreover, as |λ| → ∞ with Re λ ≥ 0,

$$S_{33}(\lambda )=-\lambda +O(1),$$

and hence

$$|S_{33}(\lambda )|\to \infty .$$

Therefore, since S₃₃(λ) is continuous and nonzero on the closed right half-plane and diverges at infinity there, it is bounded away from zero. Consequently, there exists a constant

$$\mathcal{L}≔{\inf }_{\text{Re}\lambda \ge 0}|S_{33}(\lambda )|>0$$

such that

$$|S_{33}(\lambda )|\ge \mathcal{L},\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

Now for every λ satisfying Re λ ≥ 0, we have

$$\begin{array}{llll}& |S_{11}(\lambda )|\ge d+\omega ,& & |S_{21}(\lambda )|+|S_{31}(\lambda )|=\omega ,\\ & |S_{22}(\lambda )|\ge d+\xi +{\eta }_1,& & |S_{32}(\lambda )|=\xi ,\\ & |S_{12}(\lambda )|\le {\mathcal{C}}_2,& & |S_{13}(\lambda )|+|S_{23}(\lambda )|\le {\mathcal{C}}_3.\end{array}$$

If

$${\mathcal{C}}_2<d+{\eta }_1,{\mathcal{C}}_3<\mathcal{L},$$

then for every λ with Re λ ≥ 0,

$$|S_{11}(\lambda )|\ge d+\omega >\omega =|S_{21}(\lambda )|+|S_{31}(\lambda )|,$$

$$|S_{22}(\lambda )|\ge d+\xi +{\eta }_1>{\mathcal{C}}_2+\xi \ge |S_{12}(\lambda )|+|S_{32}(\lambda )|,$$

and

$$|S_{33}(\lambda )|\ge \mathcal{L}>{\mathcal{C}}_3\ge |S_{13}(\lambda )|+|S_{23}(\lambda )|.$$

Thus every column of S(λ) is strictly diagonally dominant for all Re λ ≥ 0. By the strict column diagonal dominance criterion,

$$\det S(\lambda )\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

Since Appendix A.1 shows that

$$\det (K-\lambda I_4)\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0,$$

the Schur determinant formula yields

$$\det (J(P^{\ast })-\lambda I_7)=\det (K-\lambda I_4)\det S(\lambda )\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

Therefore, there exists no eigenvalue of J(P∗) in the closed right half-plane. Hence all eigenvalues of J(P∗) lie strictly in the open left half-plane.

Remark 2

The key idea is to exploit the block structure of the Jacobian matrix and use the Schur complement decomposition to reduce the local stability analysis to the nonsingularity of S(λ) on the closed right half-plane. By deriving explicit lower and upper bounds for the diagonal and off-diagonal entries of S(λ), and establishing strict column diagonal dominance under the conditions ${\mathcal{C}}_2<d+{\eta }_1,{\mathcal{C}}_3<\mathcal{L},$ where $\mathcal{L}≔{\inf }_{\text{Re}\lambda \ge 0}|S_{33}(\lambda )|>0,$ one obtains $\det (S(\lambda ))\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0,$ which implies that all eigenvalues of J(P∗) lie strictly in the open left half-plane.

5. Numerical results

For notational convenience, we refer to the model system (2.1) as the SEIAVSR model, where the acronym denotes the six compartments: Susceptible, Exposed, Infectious Asymptomatic, Infectious Symptomatic, Vaccinated, and Recovered.

We emphasize that the original system (2.1) and the proportional system (2.4) are mathematically equivalent via the transformation

$$s=\frac{S}{N},e=\frac{E}{N},a=\frac{I_A}{N},i=\frac{I_S}{N},v=\frac{V}{N},r=\frac{R}{N},z=\frac{1}{N}$$

(see Section 2). All parameters retain identical values and interpretations across both formulations. Therefore, the theoretical results derived from the proportional system, including the existence and stability of equilibria and the threshold condition based on ${\mathcal{R}}_0$, apply directly to the original system. All numerical simulations reported in this section are performed on the original system (2.1).

The section comprises two complementary parts. First, theoretical simulations using literature-based parameters (Table 1) illustrate fundamental dynamical behaviors and validate the analytical results established in Sections 3, 4. Second, we calibrate the model to daily case data from Nanjing, China, obtaining estimates of transmission parameters under outbreak conditions.

Table 1.

Baseline theoretical parameter values for system (2.1). These parameters represent a pre-vaccination, pre-variant scenario without interventions.

Parameter	Symbol	Value	Unit	Description and Epidemiological Basis
Recruitment rate	Λ	1000/365	individuals day−1	Daily recruitment (annual influx of 1000 individuals)
Natural mortality rate	d	3.913 × 10−5	day−1	Corresponds to 70-year life expectancy: d = 1/(70 × 365)
Transmission rate (asymptomatic)	β1	0.15	day−1	Asymptomatic transmission, 60% of symptomatic rate (He et al., 2020)
Transmission rate (symptomatic)	β2	0.25	day−1	Symptomatic transmission, yields ${\mathcal{R}}_0\approx 2.483$
Incubation rate	ω	1/5.2	day−1	Mean latent period 5.2 days (Li et al., 2020)
Symptom development proportion	δ	0.6	-	60% develop symptoms (Oran & Topol, 2020)
Recovery rate (asymptomatic)	η1	1/10	day−1	10-day infectious period for asymptomatic individuals
Recovery rate (symptomatic)	η2	1/14	day−1	14-day illness duration for symptomatic cases
Asymptomatic to symptomatic rate	ξ	1/20	day−1	Mean progression from asymptomatic to symptomatic infection over approximately 20 days
Disease mortality rate	m	1.071 × 10−3	day−1	1.5% case fatality over 14 days: m = 0.015/14
Vaccination rate	μ	0.001	day−1	Base vaccination rate (0.1% per day, 36.5% per year)
Vaccine waning rate	α	1/180	day−1	Immunity wanes after 6 months (Goldberg et al., 2021)
Natural immunity waning rate	σ	1/360	day−1	Natural immunity wanes after approximately 1 year (Chemaitelly et al., 2021)
Initial population	N0	Λ/d ≈ 70, 000	individuals	Total initial population

Note on parameter scaling: The proportional transformation in Section 2.3 rescales the transmission rates. The baseline values in this table are suitable for direct population-level simulation. The calibrated values in Section 5.2 are estimated within the proportional framework and differ in magnitude for mathematical consistency; they should not be directly compared numerically.

5.1. Theoretical simulations with baseline parameters

5.1.1. Baseline parameterization

Table 1 summarizes the baseline parameters for theoretical simulations. These values reflect the transmission characteristics of the original SARS-CoV-2 strain during the early pandemic phase (2020-2021) prior to widespread vaccination or immune evasion by variants. The demographic parameters are calibrated to a population of approximately 70,000 individuals, corresponding to the equilibrium population size $\frac{\mathrm{\Lambda }}{d}$.

Using these values, the basic reproduction number is ${\mathcal{R}}_0\approx 2.483>1$, Theorem 3.2 guarantees the existence of a unique EE, which numerical simulations confirm is locally asymptotically stable under the baseline parameters.

The initial conditions are configured to represent an early epidemic phase with approximately 70 initial infections (0.1% of the population):

Here N₀ = Λ/d ≈ 70, 000 is the equilibrium population size.

$$\begin{array}{llllll}& S(0)=0.999N_0\approx 69,930,& & E(0)=0.0003N_0\approx 21,& & I_A(0)=0.0004N_0\approx 28,\\ & I_S(0)=0.0003N_0\approx 21,& & V(0)=0,& & R(0)=0.\end{array}$$

The simulation covers 2000 days (approximately 5.5 years) to capture both the initial outbreak phase and the transition to EE.

5.1.2. Threshold dynamics

Fig. 1 illustrates the threshold behavior of system (2.1) under two transmission scenarios. In panel (a), reduced transmission rates (β₁ = 0.05, β₂ = 0.08) yield ${\mathcal{R}}_0=0.799<1$. Starting from 49 initial infected cases, the infection burden decreases continuously over time and eventually approaches zero, indicating disease extinction. In panel (b), the baseline transmission rates (β₁ = 0.15, β₂ = 0.25) produce ${\mathcal{R}}_0=2.483>1$. Under this regime, total infections increase rapidly to a major outbreak peak of approximately 12,240 cases around day 100, followed by damped oscillatory decline, and ultimately approach a sustained positive endemic level of approximately 1380 cases by day 2000. These numerical results clearly demonstrate the threshold role of ${\mathcal{R}}_0$: when ${\mathcal{R}}_0<1$, the disease cannot sustain transmission and eventually dies out, whereas when ${\mathcal{R}}_0>1$, the disease remains persistent over the long term.

Remark 3

(On the sufficient conditions for EE stability). Theorem 4.3 establishes local stability of the EE under the explicit sufficient conditions ${\mathcal{C}}_2<d+{\eta }_1$ and ${\mathcal{C}}_3<\mathcal{L}$, in addition to ${\mathcal{R}}_0>1$. These conditions arise from the column diagonal dominance argument used in the proof and are sufficient but may not be necessary. In all numerical simulations conducted across a wide range of biologically plausible parameter values, including those reported in Subsections 5.1.1-5.1.3 and the calibrated parameter set in Section 5.2, we consistently observed convergence to the EE whenever ${\mathcal{R}}_0>1$, without the need to verify the additional conditions explicitly. This suggests that the EE may be locally asymptotically stable under the sole condition ${\mathcal{R}}_0>1$ for this model. However, a rigorous proof of this stronger statement would require a more delicate analysis beyond the present Schur complement framework, and we therefore leave it as a conjecture for future investigation.

Fig. 1.

Threshold dynamics of system (2.1). (a) ${\mathcal{R}}_0=0.799<1$: disease extinction. (b) ${\mathcal{R}}_0=2.483>1$: endemic persistence. Here “total infected cases” refers to the sum I_A + I_S of asymptomatic and symptomatic infectious individuals. The contributions of each compartment are displayed separately to illustrate the relative roles of asymptomatic and symptomatic transmission.

5.1.3. Impact of waning natural immunity

Fig. 2 demonstrates the substantial influence of natural immunity duration on long-term epidemic outcomes. As the natural immunity waning rate increases from σ = 1/720 (average protection of 2 years) to σ = 1/90 (average protection of 3 months), the long-term endemic infection level rises dramatically from approximately 785 to 4475 cases, corresponding to approximately a 5.7-fold increase. Under the baseline 1-year natural immunity scenario (σ = 1/360), the endemic level is approximately 1390 cases. When immunity duration shortens to 6 months (σ = 1/180), the endemic burden increases to approximately 2565 cases, while rapid waning over 3 months (σ = 1/90) generates the highest sustained burden.

Fig. 2.

Effect of the natural immunity waning rate σ on epidemic dynamics. Larger σ (faster loss of naturally acquired immunity) substantially increases long-term endemic prevalence.

This phenomenon occurs because faster waning accelerates susceptible replenishment, increasing the steady-state prevalence. From the EE expressions in Section 3, the infection level satisfies $i^{\ast }\propto (1-\frac{1}{{\mathcal{R}}_0})\cdot \frac{c_2}{C}$, where c₂ and ${\mathcal{R}}_0$ are independent of σ, while C depends on σ through c₃ = (η₁c₁ + η₂c₂)/(d + σ). As σ increases, c₃ and hence C decrease, leading to an increase in i∗. These results indicate that in settings where natural immunity wanes rapidly, sustained vaccination or booster strategies become increasingly important for compensating for declining population-level protection.

5.1.4. Vaccination effects

Fig. 3 illustrates the combined roles of vaccination coverage and vaccine durability in shaping long-term epidemic outcomes. Panel (a) examines the effect of varying the vaccination rate μ while holding the natural immunity waning rate fixed at σ = 1/360 and the baseline vaccine waning rate fixed at α = 1/180. In the absence of vaccination (μ = 0), the long-term endemic prevalence is approximately 1523 cases. Introducing low vaccination coverage (μ = 0.0005, or 0.05% per day) reduces the endemic burden to approximately 1457 cases (4.3% reduction). Under the baseline vaccination rate (μ = 0.001, or 0.1% per day), endemic prevalence further declines to approximately 1390 cases (8.7% reduction), while higher vaccination coverage (μ = 0.0015, or 0.15% per day) lowers the endemic level to approximately 1322 cases (13.2% reduction). The diminishing returns reflect the nonlinear relationship between μ and the susceptible fraction $s^{\ast }=(\alpha +d)/(\alpha +d+\mu )\cdot 1/{\mathcal{R}}_0$.

Fig. 3.

Vaccination effects on epidemic dynamics. (a) Effect of vaccination rate μ under fixed natural immunity waning and baseline vaccine waning conditions. (b) Effect of vaccine-induced immunity waning rate α under fixed vaccination coverage.

Panel (b) investigates the effect of vaccine-induced immunity duration by varying the vaccine waning rate α while fixing μ = 0.001day⁻¹. Longer-lasting vaccine protection (α = 1/720, corresponding to approximately 2 years of protection) yields the lowest endemic prevalence of approximately 994 cases. When vaccine protection lasts about 1 year (α = 1/360), the endemic level increases to approximately 1255 cases. Under the baseline 6-month vaccine protection scenario (α = 1/180), endemic prevalence reaches approximately 1390 cases, while rapid waning over 3 months (α = 1/90) further increases the endemic burden to approximately 1458 cases. These findings demonstrate that long-term vaccination effectiveness depends not only on vaccination coverage but also critically on the durability of vaccine-induced immunity. Under fixed vaccination coverage, longer-lasting vaccine protection substantially enhances sustained epidemic suppression, whereas rapid vaccine waning can markedly weaken long-term population-level benefits.

5.1.5. Sensitivity analysis

Fig. 4 presents the elasticity-based sensitivity analysis of the basic reproduction number ${\mathcal{R}}_0$ under baseline conditions. Among all parameters, the symptomatic transmission rate β₂ exerts the strongest positive influence on ${\mathcal{R}}_0$ (sensitivity index 0.863), indicating that transmission from symptomatic individuals is the dominant driver of epidemic invasion. In contrast, the symptomatic recovery rate η₂ has the strongest negative influence (index −0.858), implying that accelerating symptomatic recovery is one of the most effective mechanisms for reducing transmission potential. The symptom development proportion δ also has a substantial positive influence (index 0.266), reflecting that shifting infections toward the more transmissible symptomatic class increases epidemic potential.

Fig. 4.

Elasticity-based sensitivity analysis of ${\mathcal{R}}_0$ with respect to system parameters under baseline conditions. Sensitivity indices quantify the proportional response of ${\mathcal{R}}_0$ to a 10% perturbation in each parameter. Note that σ has zero sensitivity because recovered individuals are absent at the DFE.

The asymptomatic transmission rate β₁ has a moderate positive effect (index 0.137), while the asymptomatic recovery rate η₁ shows a moderate negative effect (index −0.197). Vaccination rate μ exhibits a negative sensitivity index (index −0.152), confirming that increased vaccination coverage directly suppresses the invasion potential of the disease. Conversely, the vaccine waning rate α has a positive sensitivity index of approximately equal magnitude (index 0.152), indicating that faster loss of vaccine-induced immunity counteracts vaccination benefits by increasing susceptibility. The asymptomatic-to-symptomatic progression rate ξ has a relatively weak positive influence (index 0.059), whereas disease-induced mortality m contributes only marginally (index −0.013).

Notably, the natural immunity waning rate σ has an approximately zero sensitivity index because ${\mathcal{R}}_0$ is derived from the DFE, where the recovered compartment is absent and σ does not directly enter the invasion threshold. However, as demonstrated in Fig. 2, although σ does not affect ${\mathcal{R}}_0$, it plays a crucial role in determining long-term endemic burden once sustained transmission is established. Overall, these results suggest that reducing symptomatic transmission, accelerating symptomatic recovery, expanding vaccination coverage, and prolonging vaccine durability are among the most influential strategies for lowering epidemic invasion potential.

Notably, the incubation rate ω also exhibits a sensitivity index close to zero under the baseline parameter setting. This near-zero sensitivity can be understood from the factor ω/(d + ω) in ${\mathcal{R}}_0$. Indeed, the normalized sensitivity index of ${\mathcal{R}}_0$ with respect to ω is d/(d + ω), which becomes small when ω is large relative to the natural death rate d. Thus, under the baseline parameter setting, ${\mathcal{R}}_0$ is relatively insensitive to the exact duration of the latent period. These results indicate that the effect of the incubation rate on disease invasion becomes weak once progression from the exposed class is sufficiently rapid.

5.1.6. The structural role of natural immunity waning

Fig. 5 provides a supplementary mechanistic demonstration of the structural importance of natural immunity waning in shaping long-term epidemic behavior within the SEIAVSR framework. In the absence of natural immunity waning (σ = 0), recovered individuals retain permanent immunity, preventing replenishment of the susceptible population through immunity loss. Under this scenario, infections peak at approximately 13,600 cases during the initial outbreak but subsequently decline to extinction, with final prevalence approaching zero. By contrast, when natural immunity wanes at the baseline rate (σ = 1/360), immunity loss continuously returns recovered individuals to the susceptible class, thereby replenishing the susceptible pool and sustaining recurrent epidemic waves. In this case, infections peak at approximately 13,920 cases and ultimately persist at a positive long-term endemic level of approximately 1520 cases.

Fig. 5.

Structural necessity of natural immunity waning. Without waning (σ = 0), disease eventually extinguishes after the initial outbreak. With waning (σ = 1/360), susceptible replenishment through immunity loss sustains recurrent transmission and long-term endemic persistence.

These results demonstrate that incorporating natural immunity waning fundamentally alters long-term model behavior by transforming epidemic trajectories from eventual extinction to sustained endemic persistence under the baseline transmission regime. Thus, immunity waning is not merely a secondary quantitative refinement but a structurally important mechanism for accurately capturing recurrent transmission patterns and long-term endemic burden in diseases where post-infection immunity is temporary.

5.2. Model calibration to empirical outbreak data

5.2.1. Data and calibration setup

Having explored the theoretical behavior of the SEIAVSR model under literature-based parameters, we now turn to empirical validation using real-world outbreak data. Specifically, we calibrate the SEIAVSR model to daily reported COVID-19 cases from Nanjing, China, during an 18-day outbreak period (February 14 - March 3, 2022). The outbreak was caused by the Omicron BA.2 sublineage. The data exhibit a clear epidemic curve with a peak of 110 cases on day 9.

Crucial clarification: The parameters estimated in this calibration reflect effective transmission rates under the intervention policies in effect during the outbreak (contact tracing, quarantine, mask mandates, etc.). They should not be interpreted as intrinsic biological transmission rates. Consequently, the quantity we denote as ${\mathcal{R}}_0^{(\text{calibrated})}$ is actually an effective reproduction number under the prevailing control measures. Strictly speaking, the value below unity indicates that the implemented controls were sufficient to contain the outbreak.

5.2.2. Calibration algorithm

The calibration procedure employs a weighted least squares framework to estimate the parameter vector ${\mathrm{\Theta }}^{\ast }=({\beta }_1^{\ast },{\beta }_2^{\ast },{\xi }^{\ast },{\eta }_1^{\ast },{\eta }_2^{\ast },{\omega }^{\ast },r_E^{\ast },r_A^{\ast })$ by minimizing the discrepancy between model output and observed data. Here, $r_E=\frac{E_0}{I_0}$ and $r_A=\frac{I_A^0}{I_0}$ denote the initial exposed-to-symptomatic and asymptomatic-to-symptomatic ratios, respectively. To improve fitting performance near the epidemic peak, larger weights are assigned to observations around the peak day via the weight function

$$w_t=1+c\exp \left(-\frac{{(t-t_p)}^2}{2h^2}\right),$$

where we set c = 3 and h = 3 days in the implementation. Parameter estimation is implemented using a multi-start optimization strategy with the MATLAB function fmincon to reduce sensitivity to local minima. The full calibration procedure is summarized in Algorithm 1.

Based on Algorithm 1, we implement the calibration program in MATLAB and apply it to the Nanjing outbreak dataset to obtain the fitted parameter set Θ∗. Using these calibrated parameters, we simulate the model and reconstruct the corresponding daily new-case trajectory. Fig. 6 presents the resulting calibration performance against the empirical data.

Algorithm 1

Parameter calibration for the SEIAVSR system using outbreak data

Require: Data ${\{y_t\}}_{t=0}^T$, population N, initial symptomatic cases I0 = y0, peak day tp, weight parameters c = 3, h = 3

Ensure: Fitted parameter set Θ∗, fitted daily cases $\{{\hat{y}}_t\}$, model-based reproduction number ${\mathcal{R}}_0$, metrics (RMSE, R2)

1: Fixed parameters:δ, μ, α, m, d, Λ

2: Estimated parameters: Θ = (β1, β2, ξ, η1, η2, ω, rE, rA)

3: Bounds: $$\mathrm{l}\mathrm{b}=({\beta }_1^{\min },{\beta }_2^{\min },{\xi }^{\min },{\eta }_1^{\min },{\eta }_2^{\min },{\omega }^{\min },r_E^{\min },r_A^{\min }),$$ $$\mathrm{u}\mathrm{b}=({\beta }_1^{\max },{\beta }_2^{\max },{\xi }^{\max },{\eta }_1^{\max },{\eta }_2^{\max },{\omega }^{\max },r_E^{\max },r_A^{\max }).$$

4: Θ0 ← initial guess

5: Define objective functionJ(Θ):

6: E0 = I0rE, IA0 = I0rA, S0 = N − E0 − IA0 − I0, V0 = 0, R(0) = 0

7: s0 = S0/N, e0 = E0/N, a0 = IA0/N, i0 = I0/N, v0 = 0, r0 = 0, z0 = 1/N

8: [t, Y] ←ODE45(fSEIAVSR, [0, T], [s0, e0, a0, i0, v0, r0, z0], Θ)

9: ${\hat{y}}_t\leftarrow N(\delta \omega e(t)+\xi a(t)),t=0,1,\dots ,T$

10: $w_t\leftarrow 1+c\exp \left(-\frac{{(t-t_p)}^2}{2h^2}\right)$

11: $J(\mathrm{\Theta })\leftarrow \sqrt{\frac{1}{T+1}{\sum }_{t=0}^T{w}_t{({\hat{y}}_t-y_t)}^2}$

12: Multi-start optimization: best ←∞

13: fork = 1 to Mdo

14: Θs ←Θ0 if k = 1, otherwise Θs ←lb + rand(1, 8) ⊙ (ub − lb)

15: [Θk, rmsek] ←fmincon(J, Θs, lb, ub)

16: if rmsek < best then

17: best ←rmsek, Θ∗ ←Θk

18: end if

19: end for

20: Simulate the model with Θ∗ to obtain ${\{{\hat{y}}_t\}}_{t=0}^T$$r_t\leftarrow y_t-{\hat{y}}_t,{\sigma }_r\leftarrow \mathrm{s}\mathrm{t}\mathrm{d}(r)$

21: $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}\leftarrow \sqrt{\frac{1}{T+1}{\sum }_{t=0}^T{r}_t^2}$$R^2\leftarrow 1-\frac{{\sum }_{t=0}^T{r}_t^2}{{\sum }_{t=0}^T{(y_t-\overline{y})}^2}$

22: ${\mathcal{R}}_0^{\text{asym}}\leftarrow \frac{(1-\delta )\omega {\beta }_1}{(d+\omega )(d+{\eta }_1+\xi )}$${\mathcal{R}}_0^{\text{sym}}\leftarrow \frac{\delta \omega {\beta }_2}{(d+\omega )(d+{\eta }_2+m)}$

23: ${\mathcal{R}}_0^{\text{asym}2\text{sym}}\leftarrow \frac{\xi (1-\delta )\omega {\beta }_2}{(d+\omega )(d+{\eta }_1+\xi )(d+{\eta }_2+m)}$

24: $s^0\leftarrow \frac{\alpha +d}{\alpha +d+\mu }$${\mathcal{R}}_0\leftarrow s^0\left({\mathcal{R}}_0^{\text{asym}}+{\mathcal{R}}_0^{\text{sym}}+{\mathcal{R}}_0^{\text{asym}2\text{sym}}\right)$

25: ${\mathrm{C}\mathrm{I}}_{\text{upper}}\leftarrow {\hat{y}}_t+1.96{\sigma }_r,{\mathrm{C}\mathrm{I}}_{\text{lower}}\leftarrow \max ({\hat{y}}_t-1.96{\sigma }_r,0)$return${\mathrm{\Theta }}^{\ast },\{{\hat{y}}_t\},{\mathcal{R}}_0,(\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E},R^2)$

Fig. 6.

 Calibration of the SEIAVSR model to Nanjing COVID-19 daily cases. (a) Black circles represent observed daily cases, the red solid line denotes the fitted trajectory under calibrated parameters Θ∗, and the pink shaded region indicates the 95% confidence interval. (b) Residual analysis: blue dots represent residuals between observed and fitted values, the red dashed line denotes zero residual, and the green dashed lines indicate the ± 1.96σ bounds.

5.2.3. Calibration results

Fig. 6(a) demonstrates that the calibrated model closely captures the principal epidemiological features of the Nanjing outbreak. In particular, the observed epidemic peak occurs on day 9 with 110 reported cases, while the model predicts a peak of 111 cases on the same day, corresponding to a deviation of less than 1%. The estimated epidemic duration, measured by the half-maximum width (14 days), also closely matches the observed outbreak profile. Fig. 6(b) presents the residual analysis, showing that residuals remain generally centered around zero without strong systematic bias, and most observations fall within the ± 1.96σ bounds, supporting satisfactory calibration performance. Standard goodness-of-fit measures further confirm strong agreement between model and data, with RMSE = 6.01 cases/day and R² = 0.9617.

The calibrated parameter estimates are summarized as follows. The effective transmission rates under interventions are β₁ = 6.000day⁻¹ for asymptomatic individuals and β₂ = 7.692day⁻¹ for symptomatic individuals. The progression rate from exposed to infectious status is ω = 0.200day⁻¹, corresponding to an average latent period of $\frac{1}{\omega }=5.0$ days. The recovery rates are η₁ = 0.080day⁻¹ for asymptomatic cases (mean infectious period of 12.5 days) and η₂ = 0.390day⁻¹ for symptomatic cases (mean infectious period of 2.6 days). The asymptomatic-to-symptomatic progression rate is ξ = 0.329day⁻¹, corresponding to an average transition time of approximately 3.0 days. The symptom-development proportion is fixed at δ = 0.6.

5.2.4. Reproduction number decomposition

Under the calibrated proportional-system parameterization, the model-based reproduction metric is calculated as ${\mathcal{R}}_0=s^0\left({\mathcal{R}}_0^{\text{asym}}+{\mathcal{R}}_0^{\text{sym}}+{\mathcal{R}}_0^{\text{asym}2\text{sym}}\right)\approx 0.572$. Its three pathway-specific components are $s^0{\mathcal{R}}_0^{\text{asym}}=0.165$ for direct asymptomatic transmission, $s^0{\mathcal{R}}_0^{\text{sym}}=0.265$ for direct symptomatic transmission, and $s^0{\mathcal{R}}_0^{\text{asym}2\text{sym}}=0.142$ for the asymptomatic-to-symptomatic transition pathway. Asymptomatic-related contributions (direct plus transition-associated) total approximately 0.307(53.7%), while direct symptomatic transmission accounts for approximately 0.265(46.3%).

Important interpretation: This decomposition reflects transmission potential under the calibrated intervention scenario, not intrinsic biological potential. The finding that 53.7% of modeled transmission potential is attributable to asymptomatic pathways aligns with published estimates for Omicron BA.2 (40-60% of transmission from asymptomatic or presymptomatic individuals) and underscores the importance of asymptomatic surveillance (testing, contact tracing) even in the absence of symptoms.

5.2.5. Parameter interpretation and limitations

Several important qualifications accompany these calibration results:

• (1)Intervention context: The estimated ${\mathcal{R}}_0\approx 0.572<1$ is consistent with eventual outbreak containment under the implemented public health measures, not with uncontrolled spread. The rapid initial case increase reflects a high intrinsic transmission rate that was subsequently brought under control.
• (2)Parameter scaling: The large β₁, β₂ values (6-8 day⁻¹) arise from the proportional system formulation and should not be directly compared with the baseline theoretical values (0.15-0.25 day⁻¹) from Table 1, which assume population-level incidence in a large, fully susceptible population. These values are consistent with Omicron BA.2's high intrinsic transmissibility, though considerably reduced by interventions relative to an uncontrolled scenario.
• (3)Short symptomatic infectious period: The estimated symptomatic infectious period ${\eta }_2^{-1}=2.6$ days likely reflects rapid isolation of symptomatic cases under outbreak management rather than biological recovery. The model does not distinguish between recovery and isolation; both are captured by the exit rate from I_S.
• (4)Asymptomatic contribution: The estimated asymptomatic contribution (53.7%) depends on model structure, particularly the assumption that asymptomatic individuals can either recover directly or progress to symptomatic infection. If the progression rate ξ were higher, the relative contribution of direct asymptomatic transmission would decrease.

6. Conclusion and discussion

This study develops and analyzes a comprehensive SEIAVSR mathematical framework for COVID-19 transmission that explicitly incorporates asymptomatic and symptomatic infection pathways, vaccination with waning vaccine-induced immunity, and loss of naturally acquired immunity. By integrating rigorous theoretical analysis, mechanism-oriented numerical simulations, and empirical outbreak calibration, the study provides a unified mathematical-epidemiological framework for understanding both epidemic invasion and long-term endemic persistence.

Mathematically, we establish that the basic reproduction number ${\mathcal{R}}_0$ serves as a fundamental threshold governing disease invasion. When ${\mathcal{R}}_0<1$, the DFE is locally asymptotically stable (Theorem 4.1); when ${\mathcal{R}}_0>1$, the system is uniformly persistent (Theorem 4.2), a unique EE exists (Theorem 3.2), and the EE is locally asymptotically stable under explicit verifiable parametric conditions (Theorem 4.3). Notably, the local stability analysis of the EE employs a Schur complement reduction combined with a column diagonal dominance argument, an approach that is relatively uncommon in epidemiological modeling yet well suited to the high-dimensional structure of the present system. A pathway-specific decomposition of ${\mathcal{R}}_0$ quantifies the distinct contributions of direct asymptomatic transmission, direct symptomatic transmission, and asymptomatic-to-symptomatic progression, thereby providing a mechanistic basis for targeted intervention design. Due to the high dimensionality and intrinsic nonlinearity of the model, establishing global stability remains analytically challenging and is left for future investigation.

Mechanistically, the numerical simulations reveal that vaccination coverage and vaccine durability jointly shape long-term epidemic burden, with diminishing returns at higher coverage levels. Critically, natural immunity waning is not merely a quantitative modifier but a qualitative driver that replenishes susceptibility and transforms epidemic trajectories from post-outbreak extinction to recurrent endemic persistence; without waning (σ = 0), the disease eventually extinguishes even when ${\mathcal{R}}_0>1$. Sensitivity analysis further identifies symptomatic transmission and symptomatic recovery as the dominant determinants of invasion potential, whereas natural immunity waning has minimal influence on ${\mathcal{R}}_0$ itself but profoundly affects endemic burden after invasion.

Empirically, calibration to the Nanjing Omicron BA.2 outbreak demonstrates that the model successfully reproduces observed epidemic trajectories with strong goodness-of-fit (R² = 0.9617). The calibrated effective reproduction number $({\mathcal{R}}_0\approx 0.572)$ remains below unity under fitted intervention conditions, reflecting successful outbreak control rather than intrinsic transmissibility. Pathway decomposition reveals that asymptomatic-related transmission contributes approximately 53.7% of the modeled transmission potential, highlighting the substantial epidemiological importance of silent spread during early outbreak escalation. These findings are broadly consistent with the transmission characteristics of Omicron BA.2 and emphasize the importance of intervention strategies that extend beyond symptom-based containment.

From a public health perspective, the results suggest that sustainable epidemic management requires not only vaccination expansion but also maintenance of durable immunity, reinforcement of surveillance for asymptomatic spread, and interventions that reduce symptomatic transmission intensity. The framework also underscores that parameters governing long-term susceptibility replenishment may be as important for endemic management as classical threshold parameters are for initial outbreak prevention.

Several extensions may further enhance the applicability of this framework, including stochastic perturbations, age or contact structure, multiple co-circulating variants, behavioral adaptation, and broader validation using heterogeneous regional datasets. Such extensions would strengthen the model's capacity to inform future responses to evolving respiratory pathogens. Additionally, the current assumption of perfect vaccine efficacy while immunity persists could be relaxed to allow breakthrough infections in future work.

In summary, this work establishes a mathematically rigorous and epidemiologically interpretable framework that bridges threshold theory, mechanistic simulation, and empirical calibration. Beyond COVID-19, the proposed framework provides a generalizable platform for analyzing infectious diseases in which asymptomatic transmission, temporary immunity, and vaccination durability jointly determine long-term population-level outcomes.

CRediT authorship contribution statement

Yiping Tan: Writing – original draft, Software, Methodology, Formal analysis, Conceptualization. Yongli Cai: Methodology, Formal analysis. Weiming Wang: Writing – original draft, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Handling Editor: Dr Yijun Lou

Appendix A. Detailed calculations for the Schur complement stability analysis

This appendix presents the explicit block decomposition of the Jacobian matrix at the EE P∗, the construction of the associated Schur complement, and the verification of the sufficient column diagonal dominance conditions used in Theorem 4.3.

Appendix A.1. Invertibility of K − λI₄ in the closed right half-plane

For $\lambda \in \mathbb{C}$, write

$$K-\lambda I_4=\left(\begin{array}{ll}{K}_{11}& K_{12}\\ 0& K_{22}\end{array}\right),$$

where

$$K_{11}=\left(\begin{array}{lll}-l-d-\mu -\lambda & \alpha & \sigma \\ \mu & -\alpha -d-\lambda & 0\\ 0& 0& -d-\sigma -\lambda \end{array}\right),$$

$$K_{12}=\left(\begin{array}{l}\mathrm{\Lambda }(1-s^{\ast })\\ -\mathrm{\Lambda }{v}^{\ast }\\ -\mathrm{\Lambda }{r}^{\ast }\end{array}\right),K_{22}=-\rho -\lambda .$$

The upper-left 2 × 2 block of K₁₁ can be written as

$$\mathcal{B}(\lambda )=B_0-\lambda I_2,$$

where

$$B_0=\left(\begin{array}{ll}-l-d-\mu & \alpha \\ \mu & -\alpha -d\end{array}\right).$$

Since

$$\text{tr}(B_0)=-(l+\mu +\alpha +2d)<0,$$

and

$$\det (B_0)=(l+d+\mu )(\alpha +d)-\alpha \mu =(d+l)(\alpha +d)+\mu d>0,$$

all eigenvalues of B₀ lie in the open left half-plane. Therefore,

$$\det \mathcal{B}(\lambda )=\det (B_0-\lambda I_2)\ne 0,\text{Re}\lambda \ge 0.$$

Moreover,

$$d+\sigma +\lambda \ne 0,\rho +\lambda \ne 0,\text{Re}\lambda \ge 0.$$

Hence, for all Re λ ≥ 0,

$$\det (K-\lambda I_4)\ne 0.$$

Appendix A.2. Construction of the Schur complement

Define

$$S(\lambda )=J_{11}-\lambda I_3-L{(K-\lambda I_4)}^{-1}M.$$

Write

$$S(\lambda )=\left(\begin{array}{lll}{S}_{11}& S_{12}& S_{13}\\ S_{21}& S_{22}& S_{23}\\ S_{31}& S_{32}& S_{33}\end{array}\right).$$

Direct symbolic computation yields the following exact entries:

$$S_{11}=-d-\omega -\lambda ,S_{21}=(1-\delta )\omega ,S_{31}=\delta \omega ,$$

$$S_{22}=-d-\xi -{\eta }_1-\lambda ,S_{32}=\xi .$$

For the third column,

$$S_{23}=m{a}^{\ast }-\frac{m{a}^{\ast }\rho }{\rho +\lambda }=\frac{m{a}^{\ast }\lambda }{\rho +\lambda },$$

and

$$S_{33}=m{i}^{\ast }-d-m-{\eta }_2-\lambda -\frac{m{i}^{\ast }\rho }{\rho +\lambda }=-(d+m+{\eta }_2+\lambda )+\frac{i^{\ast }m\lambda }{\rho +\lambda }.$$

The entries S₁₂(λ) and S₁₃(λ) are rational functions of λ. Their denominators are products of D_B(λ), d + σ + λ, and ρ + λ, which are nonzero for Re λ ≥ 0 by Appendix A.1. Moreover, their degrees at the numerator do not exceed those of the denominator. Hence S₁₂(λ) and S₁₃(λ) are well-defined and bounded on the closed right half-plane $\{\lambda \in \mathbb{C}:\text{Re}\lambda \ge 0\}.$ Accordingly, define

$${\mathcal{C}}_2≔{\sup }_{\text{Re}\lambda \ge 0}|S_{12}(\lambda )|,$$

and

$${\mathcal{C}}_3≔{\sup }_{\text{Re}\lambda \ge 0}\left(|S_{13}(\lambda )|+|S_{23}(\lambda )|\right).$$

Appendix A.3. Column diagonal dominance of S(λ)

We now verify that S(λ) is strictly column diagonally dominant for all λ satisfying Re λ ≥ 0.

First column

For every λ with Re λ ≥ 0,

$$|S_{11}|=|-d-\omega -\lambda |=|d+\omega +\lambda |\ge d+\omega .$$

Since

$$|S_{21}|+|S_{31}|=(1-\delta )\omega +\delta \omega =\omega ,$$

it follows that

$$|S_{11}|\ge d+\omega >\omega =|S_{21}|+|S_{31}|.$$

Thus the first column is strictly diagonally dominant.

Second column

For every λ with Re λ ≥ 0,

$$|S_{22}|=|-d-\xi -{\eta }_1-\lambda |=|d+\xi +{\eta }_1+\lambda |\ge d+\xi +{\eta }_1.$$

Moreover, by definition of ${\mathcal{C}}_2$,

$$|S_{12}|\le {\mathcal{C}}_2,|S_{32}|=\xi .$$

Hence,

$$|S_{12}|+|S_{32}|\le {\mathcal{C}}_2+\xi .$$

Therefore, if

$${\mathcal{C}}_2<d+{\eta }_1,$$

then

$$|S_{12}|+|S_{32}|\le {\mathcal{C}}_2+\xi <d+{\eta }_1+\xi \le |S_{22}|.$$

Thus the second column is strictly diagonally dominant.

Third column

By the preceding analysis of S₃₃(λ), we have

$$\mathcal{L}≔{\inf }_{\text{Re}\lambda \ge 0}|S_{33}(\lambda )|>0,$$

and therefore

$$|S_{33}(\lambda )|\ge \mathcal{L},\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$

By definition of ${\mathcal{C}}_3$,

$$|S_{13}|+|S_{23}|\le {\mathcal{C}}_3.$$

Hence, if

$${\mathcal{C}}_3<\mathcal{L},$$

then

$$|S_{13}|+|S_{23}|\le {\mathcal{C}}_3<\mathcal{L}\le |S_{33}|.$$

Thus the third column is strictly diagonally dominant.

Conclusion

Under the conditions

$${\mathcal{C}}_2<d+{\eta }_1,{\mathcal{C}}_3<\mathcal{L},$$

each column of S(λ) is strictly diagonally dominant for every λ satisfying Re λ ≥ 0. Therefore, by the strict column diagonal dominance criterion,

$$\det S(\lambda )\ne 0,\forall \lambda \in \mathbb{C},\text{Re}\lambda \ge 0.$$