2.1. The linear chain trick: Erlang Sojourn Times as an Exact ODE reformulation

Why Erlang-2 for hospital and ICU stays. The choice of Erlang-2 distributions is motivated by three independent arguments.

Why two ward compartments (, Erlang-2) and not one or three?. The model uses two successive general ward compartments (and similarly for intensive care). We clarify at the outset that H₁, H₂ (and H₃ below) denote compartments of the model, not the modeling hypotheses of Section 2 (which carry the sans-serif labels H1, H2). The question is: why use an Erlang-2 chain (two stages) rather than a single exponential stage (Erlang-1) or three stages (Erlang-3)? The answer is quantitative:

1. 1. Statistical inadequacy of the exponential. A single exponential stage produces a memoryless sojourn time with coefficient of variation CV  =  1. For a mean hospital stay of 11.8 days, this implies of patients discharged within 3 days — clinically unrealistic. The table below compares the three leading ODE-compatible choices:

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https://doi.org/10.1371/journal.pone.0352960.t007

Erlang-2 is therefore the minimal ODE-compatible extension beyond exponential that substantially reduces variance, at the cost of only one additional equation per clinical sub-chain.

1. 2. Empirical fit. COVID-19 hospital and ICU length-of-stay distributions are unimodal with CV  < 1, inconsistent with the exponential but consistent with Erlang-2 [3,37]. Higher-order Erlang chains () are not supported by available length-of-stay data [3,37].
2. 3. ODE tractability and exact equivalence with the convolution framework. Among distributions with CV  < 1, Erlang distributions provide a standard and parsimonious class that admits an exact finite-dimensional ODE reformulation via the linear chain trick (Proposition 2.1). More general distributions (Weibull, log-normal) would require integro-differential equations with memory kernels, as studied by Granger et al. in the retarded-transition-rate framework of [24,25]. Those works obtain and endemic equilibria for arbitrary sojourn distributions, at the cost of working in an infinite-dimensional function space. Our Erlang-2 chain is the exact ODE realization of the special case where the memory kernel is a Gamma- density (see Proposition 2.1 and its proof by variation of constants): the two frameworks are mathematically equivalent for this kernel, but our ODE formulation yields the explicit closed-form results of Sections 3–5 that are not available in full generality from the integro-differential approach. Erlang-2 achieves the best statistical improvement over the exponential at minimal model complexity (one additional ODE per clinical chain).

The linear chain trick: exact equivalence with integro-differential equations. Erlang-distributed sojourn times are, in principle, non-Markovian: they correspond to residence-time distributions with memory, whose natural formulation is a Volterra-type integro-differential equation [24, 25]. The key insight, established in [28,29], is that this non-Markovian system admits an exact finite-dimensional ODE reformulation via the linear chain trick.

Proposition 2.1 (Linear chain trick, [28,29]). Let Z(t) represent the total occupancy of a compartment with Erlang-2 sojourn time distribution, mean , and exogenous inflow u(t). The following two formulations are mathematically equivalent:

• Integro-differential form: (1)
  where is the Erlang-2 probability density function (residence-time kernel).
• ODE chain form: (2)
  Both formulations produce identical output Z(t) for any input u(t) and zero initial conditions. The sojourn time of an individual entering the chain (2) is distributed as , with mean and variance .

Proof 2.2 We use the method of variation of constants (variation of parameters) to solve the ODE chain explicitly, then read off the equivalence with the integro-differential form and the sojourn time distribution.

Step 1: Solution of the first stage by variation of constants. The equation is a first-order linear ODE. The associated homogeneous equation has the fundamental solution . By variation of constants, we set and require:

With zero initial condition X₁(0) = 0 (so c(0)=0):

(3)

Step 2: Solution of the second stage by variation of constants. The equation is driven by X₁(t) computed in (3). Its homogeneous fundamental solution is again . Setting and requiring :

With X₂(0) = 0, substituting (3) into the above:

Exchanging the order of integration (Fubini; ):

Step 3: Total occupancy and the occupancy kernel. Adding both stages:

(4)

The kernel is the survival function of the distribution:

where is the standard Erlang-2 CDF. Hence equation (4) is the occupancy formula : the number of individuals present at time t equals the sum over all past arrivals of the probability of still being in the chain at time t.

Step 4: Equivalence with the integro-differential form. Differentiating (4) with respect to t and using h(0) = 1, :

which is exactly the integro-differential form (1). The equivalence is therefore a direct consequence of the variation-of-constants solution and the identity .

Step 5: Sojourn time distribution. Set u(t)=0 and consider a single individual entering X₁ at t = 0 (i.e., X₁(0)=1, X₂(0)=0). Its sojourn in stage 1 is (constant departure rate ); similarly, its sojourn in stage 2 is , independent of T₁. The total time has density:

confirming . The survival function is , consistent with Step 3. The moments follow:

Application to the two clinical chains of model (5).

General ward chain . The inflow to the ward sub-system is , corresponding to severely infectious individuals admitted to the general ward. By (3)–(4) applied with :

The total ward occupancy satisfies, for zero initial conditions:

and its dynamics obey the integro-differential equation:

The sojourn time in the ward is with mean days and (consistent with French COVID-19 hospitalization data [3,37]).

ICU chain . The sub-chain receives inflow from patients who deteriorate from the general ward to intensive care. By (3)–(4) with :

and the total ICU occupancy satisfies:

The ICU sojourn time is with mean days and .

Conclusion. In both chains, the ODE system of model (5) is the exact variation-of-constants representation of the corresponding Volterra integro-differential equation with Erlang-2 kernel. The transition (and ) is therefore not a clinical event but a mathematical stage: both stages together constitute a single continuous episode whose total duration is Erlang-2 distributed.

3.1 Well-posedness of the model

To ensure the mathematical validity of the non-age-stratified Kermack-McKendrick-like model, we analyze its well-posedness, establishing the existence, uniqueness, non-negativity, and boundedness of solutions. The model is described by the following system of ordinary differential equations:

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

where S, E, I, , , , I_icu, H₁, H₂, H_icu, ICU₁, ICU₂, and R represent the susceptible, exposed, infectious, severe infectious, mild infectious, hospitalized, ICU-admitted, and removed compartments, respectively. Parameters , , and govern transmission, progression, and branching probabilities.

Let the state vector be . The biologically meaningful domain is:

where N > 0 is the constant total population (H1), and initial conditions .

Theorem 3.1. For any initial condition , the system (6)–(18) admits a unique solution defined for all . Moreover, the solution is non-negative and bounded, with .

Proof 3.2. The system can be written as , where is defined by the right-hand sides of (6)–(18). The function consists of linear and bilinear terms (e.g., , , ). Since is continuously differentiable on , it is locally Lipschitz continuous in any bounded region, including . By the Cauchy-Lipschitz theorem, there exists a unique local solution for any . To ensure global existence, we demonstrate boundedness below.

To show non-negativity, consider the behavior of each compartment on the boundary of . If S(t) = 0, then , so S(t) remains non-negative. If E(t) = 0, then , since . Similarly:

• I(t) = 0: ,
• : ,
• : ,
• : ,
• I_icu(t) = 0: ,
• H₁(t) = 0: ,
• H₂(t) = 0: ,
• H_icu(t) = 0: ,
• ICU₁(t) = 0: ,
• ICU₂(t) = 0: ,
• R(t) = 0: .

Thus, the non-negative orthant is positively invariant.

To prove boundedness, compute the total population dynamics:

Summing the right-hand sides:

All terms cancel, yielding . Thus, N(t) = N (constant, consistent with H1). Since all compartments are non-negative, , ensuring boundedness.

Since remains in the compact set , the solution cannot escape to infinity in finite time, guaranteeing global existence. Hence, is positively invariant, and the solution is unique, non-negative, and bounded for all .

The well-posedness of the model ensures that it is mathematically robust for further analysis, such as stability of equilibria or numerical simulations, and biologically meaningful, as solutions remain in the epidemiologically relevant domain .

3.2 Basic reproduction number

We use the next-generation matrix method [14] on the infected subsystem at the DFE . Only E receives new infections; the other equations represent transitions. The infection matrix F and transition matrix V are:

(19)

V is lower-triangular with positive diagonal, hence non-singular. Its inverse is computed by forward substitution (, solved column by column):

(20)

The next-generation matrix has rank 1. Its (1,1) entry is:

Since has rank 1, , giving:

(21)

Sensitivity The main partial derivatives are

(22)

(23)

These signs confirm that epidemic control requires reducing transmission or shortening the community infectious periods. By contrast, parameters governing downstream clinical management after hospital admission affect healthcare burden but do not enter in this DAG-structured model.

Theorem 3.3 (Global Convergence to the DFE). Consider the system with permanent immunity (). For any initial condition in , the trajectory converges to the disease-free equilibrium point , where is the unique solution of equation (28) and .

Proof 3.4. (Detailed proof in Appendix A.) The proof proceeds in three steps: (i) monotone convergence of S(t) to a limit since ; (ii) application of Barbalat’s lemma to show that as ; (iii) convergence of downstream hospital and ICU chains via linear systems theory with vanishing inputs.

Remark 3.5 (Continuum of equilibria in the classical SIR model). The classical Kermack–McKendrick SIR model admits a continuum of stable disease-free equilibria: every point of the form with is an equilibrium. The final size relation (Theorem 3.3 and equation (28)) identifies precisely which equilibrium is reached from given initial conditions, resolving this non-uniqueness. Adding demographic turnover (births and deaths) breaks this degeneracy and yields a unique DFE or endemic equilibrium; our framework with H1 deliberately excludes this to focus on epidemic timescales.

3.3 First integral and final size

We derive a first integral to determine the final epidemic size. Integrating the equation for from 0 to t yields:

(24)

To express the integral on the right-hand side in terms of state variables, we consider the linear combination of the infected compartments. Integrating the equations for from 0 to , and noting that , we have:

(25)

Similarly, integrating the equations for and :

(26)

(27)

Substituting these expressions into (24) as :

We recognize the definition of in the first term:

Thus, the final size relation is given by:

(28)

The final size relation (28) carries direct public health implications. The quantity is the attack rate of the epidemic. When R₀ > 1, if a fraction v of the population is immunized prior to epidemic onset, the condition yields the herd immunity threshold:

(29)

For R₀ values estimated in the range for COVID-19 [3], this corresponds to immunizing between 29% and 44% of the population. Note that the hospitalization and ICU parameters do not appear in (28): they govern clinical burden without affecting the total number of individuals ultimately infected (see also Section 3.2).

4.1 Global Stability of the Disease-Free Equilibrium

Theorem 4.1 (Global Asymptotic Stability of the DFE under waning immunity). For the system with waning immunity (), the Disease-Free Equilibrium is Globally Asymptotically Stable (GAS) in if .

Proof 4.2. Step 1: Infectious subsystem and comparison system. Let be the vector of active infectious compartments (ordered consistently with system (5)). Their dynamics read:

(32)

where F and V are the next-generation matrices at the DFE (equation (19)):

(33)

Since for all in and , we obtain the comparison inequality:

(34)

Step 2: Spectral properties of M. The matrix is an irreducible Metzler matrix and V is a non-singular M-matrix. By Theorem 2 of [14]:

(35)

The Perron–Frobenius theorem guarantees the existence of a strictly positive left eigenvector with .

Step 3: Lyapunov function. Define , with . Its time derivative along trajectories of (32) satisfies:

(36)

Step 4: LaSalle analysis. The set requires s(M) L = 0 and simultaneously. Since :

• Case : s(M) < 0 forces L = 0, hence .
• Case : either , which gives hence ; or S = N, which forces all other compartments to zero by mass conservation.

In both cases the largest invariant set in is .

Steps 5–6: Downstream convergence. Once , the hospital and ICU chains decay exponentially (their matrices are Hurwitz). Then gives , and by mass conservation .

Theorem 4.3 (Existence and uniqueness of the endemic equilibrium). For and , system (30) admits a unique endemic equilibrium .

Proof 4.4 Setting all derivatives to zero and eliminating as linear multiples of (from equations (39) and following) yields a single affine equation in :

(37)

Since and the right-hand side is strictly positive when , there is a unique solution

(38)

All other components are then determined uniquely as affine functions of with positive coefficients, so the endemic equilibrium is unique.

Uniqueness by monotonicity. The left- and right-hand sides of (37) are both affine (linear) in with the left side having a positive slope and the right side being constant. There is exactly one intersection, confirming uniqueness without any fixed-point argument. This affine structure also excludes backward bifurcation: in models with waning immunity where the transfer graph contains a cycle (), backward bifurcation can in principle arise [38]. However, equation (37) is linear in with a strictly positive coefficient, so there is exactly one intersection and no fold, ruling out multiple endemic equilibria or subcritical persistence.

Remark 4.5 The derivation of the unique in (38) relies on the intermediate equilibrium relations (from the second equation at equilibrium) and, for reference:

(39)

4.2 Local Stability: Schur complement and spectral factorisation

Having established the existence of an endemic equilibrium when (Theorem 4.3), we address its local asymptotic stability. The detailed Schur-complement calculation below is carried out under the calibration constraint , which is the parameterization used in the numerical simulations and in the reference model [3]. The well-posedness, next-generation matrix and final-size results above remain valid for the full three-rate formulation. Under this calibration constraint, set and (equilibrium condition (39)).

Step 1: The Jacobian . Differentiating system (30) at gives the matrix (variables ordered as in (5)):

(40)

where , , , , .

Step 2: Similarity transformation . The permutation P moves R from position 13 to position 6 (simultaneously permuting rows and columns, so each entry retains its value and ). With partition and :

(41)

where is lower-triangular with diagonal , hence Hurwitz: . The coupling has nonzero entries only in the column (, ). The epidemiological block is:

(42)

Note ( and appear in the row), so is not block lower-triangular. The naive factorisation would be incorrect; the correct approach uses the Schur complement.

Step 3: Schur complement factorisation.

Lemma 4.6 (Schur complement factorisation). For all , the characteristic polynomial of satisfies:

(43)

where is the Schur complement of in .

Proof 4.7 Apply the block determinant identity to the block matrix with lower-right block (invertible for ):

This is the standard block determinant identity.

Explicit form of . Since and with nonzero -column , one computes:

where , are standard basis vectors and is the clinical cascade transfer function:

(44)

Thus is a rank-1 perturbation of :

(45)

differing from only in entry (6,4). By the matrix determinant lemma:

(46)

Since all poles of lie in and f(0)>0, is analytic on with as .

Theorem 4.8 (Local asymptotic stability of the endemic equilibrium). Assume , , and the calibration constraint . Then is locally asymptotically stable.

Proof 4.9 Since , by (43) it suffices to show for all with . By (46), either or , where .

Case 1: . A Routh–Hurwitz analysis of shows: and . All six Routh–Hurwitz coefficients () and minors () are strictly positive for R₀ > 1 and (follows from positivity of all rate constants and the equilibrium identity ). Hence , contradicting .

Case 2: with . Since is Hurwitz (Case 1), is invertible for and is bounded. Moreover on (all poles of f have ), and a direct computation using the entries of shows |f(0) g(0)| < 1 for the biologically relevant parameter range, so the equation has no solution with .

Both cases yield a contradiction, so for , and by continuity of the characteristic polynomial (extending from to all via (43)), .

Remark 3. The factorisation holds if and only if , i.e., if and only if (no discharge from H₂ or into R). In the present model , so the spectra of A and are not simply disjoint, but the stability conclusion remains valid by the Schur complement argument above.

4.3 Damped oscillatory dynamics: a spiral sink

Numerical simulations (Fig 4) reveal that convergence to occurs through damped oscillations, identifying as a spiral sink: admits complex conjugate eigenvalues with strictly negative real parts. The mechanism is the negative feedback loop: the waning immunity term replenishes susceptibles with a time lag, triggering secondary waves of decreasing amplitude [38,39].

To characterize this analytically, we project onto the reduced system with and , setting and under the calibration constraint:

(47)

One verifies: and , confirming local stability. The discriminant

(48)

determines the oscillatory nature: when (i.e., , which holds when , as typical for COVID-19 where ), the eigenvalues are complex conjugate and is a spiral sink.

Theorem 4.11 (Spiral sink at the endemic equilibrium). Assume , , and (equation (48)). Then is a locally asymptotically stable spiral sink: all nearby trajectories converge to through damped oscillations. No Hopf bifurcation occurs in this regime, and sustained oscillations are excluded.

For the COVID-19 parameters of Scenario 3 (, , , days⁻¹), one computes , , giving . The eigenvalues are:

corresponding to an oscillation period days and a damping time days, fully consistent with Fig 4.

The global stability question — whether all trajectories in converge to — remains analytically open for the full 13-dimensional system. A rigorous proof, possibly via a Lyapunov–Volterra function adapted to Erlang-chained compartments [40], constitutes a primary direction for future work.

5.1 Basic Reproduction Number: Spectral formula

The matrices (new infections) and (transitions) for the infected subsystem at the DFE are the blocks:

(50)

is block lower-triangular with invertible diagonal blocks; is computed by the same forward substitution as the scalar case (20), with replacing scalars. The next-generation operator has rank n (only its (1,1) block is nonzero after simplification):

so all nonzero eigenvalues come from the (1,1) block, giving:

(51)

This factorises the clinical transmissibility scalar from the social spectral radius . Under the calibration constraint , this reduces to , equivalently with , recovering the formula of [3]. By the Perron–Frobenius theorem, if is irreducible then is associated with a strictly positive eigenvector, ensuring a well-defined threshold.

Analogues of the homogeneous results.

1. DFE stability: The DFE is GAS in if , by the same linear Lyapunov argument as Theorem 4.1, with the left Perron eigenvector of .
2. Final size: System (56) has a unique solution (Theorem 5.1), exactly analogous to the scalar equation (28).
3. Network herd immunity: Epidemic burnout requires , the network analogue of .

5.2 Final size

Following the non-age-stratified approach, for each group i:

(52)

Integrate:

(53)

Sum the first five equations for group i:

(54)

Integrate and combine to obtain the first integral:

(55)

As , the final size system is:

(56)

where .

Theorem 5.1. Assume is irreducible, , and . Then:

1. F(S(0)) = S(0) if and only if .
2. If , there exists a unique fixed point in [0, S(0)].
3. Trajectories converge to a DFE with .

Proof 5.2 Define the map :

(57)

1. 1. Part (i): F(S(0)) = S(0) implies:

(58)

Thus:

(59)

Since is irreducible and non-negative, and the terms are non-negative, .

1. 2. Part (ii): F is monotone increasing: if , then . Since , . Define sequences and . By monotonicity:

(60)

The limits and exist, with . Assume . Then:

(61)

where . Since DF is monotone, for , so:

(62)

By Perron-Frobenius, there exists such that . Thus:

(63)

Since , . But:

(64)

so:

(65)

contradicting . Thus, , and the fixed point is unique.

1. 3. Part (iii): Define:

(66)

where satisfies . Compute:

(67)

Since and :

(68)

If , , but , a contradiction. Thus, .

Remark 5.3 (Two-group illustrative example). To illustrate how age structure modifies epidemic outcomes, consider n = 2 groups: younger individuals (i = 1, fraction p₁ = 0.80 of the population) and older individuals (i = 2, fraction p₂ = 0.20), with an assortative contact matrix calibrated to French COVID-19 contact data [5]:

where by the reciprocity condition, and q = 0.015, yielding . The final size system (56) yields two distinct values , with group-specific attack rates and . Both groups sustain a high epidemic burden, consistent with . The slightly higher attack rate in the old group reflects the large cross-group inflow from the more numerous young group, despite lower intra-group contacts (). The homogeneous model with effective yields an intermediate attack rate that underestimates the burden on one group and overestimates it on the other. This heterogeneity in epidemic burden across age groups has direct implications for targeted vaccination strategies: prioritizing the high-contact younger group reduces R₀ more efficiently than uniform coverage, consistent with findings in [10,11]. This two-group example is illustrated numerically in Fig. 5.

6.1 Scenario 1: Disease Extinction (R₀ < 1)

We first examine the case where control measures reduce the effective transmission rate sufficiently to bring R₀ below unity. The parameter values are detailed in Table 3. We assume permanent immunity () to verify convergence toward the Disease-Free Equilibrium, as predicted by Theorem 3.3.

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Table 3. Biological parameters for Scenario 1 (Disease Extinction, R₀ < 1).

https://doi.org/10.1371/journal.pone.0352960.t003

The simulation depicted in Fig 2 confirms convergence to the DFE when . The total fraction of infectious individuals (sum ) decays exponentially to zero without any oscillatory behavior, the susceptible population stabilizes at a constant positive level, and the recovered fraction reaches a plateau. This is consistent with Theorem 3.3.

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Fig 2. Scenario 1: Disease extinction (, permanent immunity).

All infectious compartments decay exponentially to zero, confirming global asymptotic stability of the DFE (Theorem 3.3).

https://doi.org/10.1371/journal.pone.0352960.g002

6.2. Scenario 2: Epidemic Wave with Permanent Immunity (R₀ > 1)

In this second scenario, we simulate a high-transmission setting yielding R₀ > 1 with permanent immunity (). Parameter values are listed in Table 4. This scenario illustrates the characteristic epidemic wave profile and the healthcare burden predicted by Theorem 3.3.

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Table 4. Parameters for Scenario 2 (Epidemic wave, permanent immunity, R₀ > 1).

https://doi.org/10.1371/journal.pone.0352960.t004

Fig 3 shows the characteristic epidemic wave: the infectious population grows exponentially, reaches a significant peak around t = 25 days, then declines once the susceptible pool drops below the herd immunity threshold . The simulation also highlights the healthcare burden: the hospitalized population (scaled for visibility) exhibits a critical lag behind the infection peak. Since immunity is permanent, the system converges to a DFE — not an endemic equilibrium — consistent with the transient nature of the outbreak described by Theorem 3.3.

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Fig 3. Scenario 2: Epidemic wave (, permanent immunity).

The infectious curve peaks around day 25, then decays to zero. The hospitalized compartment (scaled ) shows a characteristic lag behind the infection peak. The system converges to a DFE with a large recovered fraction, consistent with Theorem 3.3.

https://doi.org/10.1371/journal.pone.0352960.g003

6.3. Scenario 3: Endemic Equilibrium with Temporary Immunity (R₀ > 1)

Finally, we investigate the long-term dynamics when acquired immunity is not permanent, reflecting the reality of respiratory viruses such as seasonal influenza or waning COVID-19 immunity. We maintain but introduce immunity loss at rate days⁻¹ (average immune period months). The simulation is extended to t = 1000 days to visualize convergence to the endemic equilibrium established in Theorem 4.3. Parameters are listed in Table 5.

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Table 5. Parameters for Scenario 3 (Endemic equilibrium, waning immunity, R₀ > 1).

https://doi.org/10.1371/journal.pone.0352960.t005

Fig 4 illustrates the impact of waning immunity. Unlike the permanent immunity scenario, we observe damped oscillations following the initial outbreak peak: the rapid depletion of susceptibles halts the first wave, but the gradual loss of immunity () slowly replenishes the susceptible pool, triggering smaller secondary waves of decreasing amplitude. This oscillatory transient is characteristic of a spiral sink topology at the endemic equilibrium — the Jacobian at admits complex conjugate eigenvalues with negative real parts (Theorem 4.11). As days, the system converges to the endemic state: the infectious and recovered populations stabilize at constant positive values, and the hospitalized compartment (scaled ) stabilizes at a non-zero level reflecting a persistent long-term burden on healthcare facilities. This illustrates both the existence result of Theorem 4.3 and the spiral sink characterization of Section 4.3.

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Fig 4. Scenario 3: Endemic equilibrium (, waning immunity days⁻¹, months).

Damped oscillations converge to a stable endemic state with oscillation period days and damping time days. The horizontal dash-dot line marks the analytic endemic level . The hospitalized compartment (scaled ) stabilizes at a non-zero level, reflecting persistent healthcare pressure. This illustrates Theorem 4.3 and Theorem 4.11.

https://doi.org/10.1371/journal.pone.0352960.g004

6.4. Scenario 4: Two-group age-stratified model

We now illustrate the theoretical results of Section 5 with a concrete two-group simulation. The population is divided into a young group (i = 1, aged <60, fraction p₁ = 80%) and an old group (i = 2, aged , fraction p₂ = 20%), with a contact matrix calibrated to French contact data [5]. The reciprocity condition is enforced, yielding . Parameter values are listed in Table 6. Initial conditions reflect an epidemic already underway: S₁(0)=76%, E₁(0)=3% (young group); S₂(0)=19%, E₂(0)=0.5% (old group).

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Table 6. Parameters for Scenario 4 (two-group age-stratified model).

https://doi.org/10.1371/journal.pone.0352960.t006

Fig 5 illustrates the key structural predictions of Theorem 5.1 and Remark 5.3. Several observations are noteworthy.

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Fig 5. Scenario 4: Two-group age-stratified model ().

Solid lines: susceptible fractions (left axis). Dashed lines: infectious prevalences (right axis). Blue: young group (p₁ = 80%); red: old group (p₂ = 20%). Horizontal dashed lines mark the final susceptible fractions and .

https://doi.org/10.1371/journal.pone.0352960.g005

First, the two groups exhibit markedly different epidemic trajectories despite sharing the same : the young group (p₁ = 80%) drives the epidemic through its higher contact rate c₁₁ = 8, reaching a larger infectious peak earlier, while the old group experiences a delayed and smaller infectious peak due to lower intra-group contacts (c₂₂ = 4) but substantial cross-group exposure.

Second, the final attack rates are group-specific: for the young group and for the old group. These high and similar attack rates reflect the assortative structure of the contact matrix (diagonal dominance) and confirms the theoretical result that depends on the attack rates of all groups via the coupled transcendental system (56).

Third, the susceptible fractions converge monotonically to their respective limits and (horizontal dashed lines), consistent with the permanent immunity assumption and the global convergence result of Theorem 5.1. The effective reproduction number satisfies at the end of the epidemic in both groups, confirming network-wide herd immunity.

These results underscore a key public health implication: protecting the high-risk old group cannot be achieved solely by targeting that group. Controlling transmission in the high-contact young group — which acts as the primary driver of the summation term in (56) — is essential for reducing the overall epidemic burden across the network [10,11].