We consider a cohort of university students of total size N(t), partitioned into five mutually exclusive compartments. The transitions among these compartments are illustrated in Figure 1. The partition is driven by two observations that the existing modeling literature has not combined. First, absenteeism is not adopted instantaneously: students typically pass through a latent phase in which peer rationalization of skipping is internalized before any class is actually missed. Second, absenteeism is not a single state: occasional and habitual absence differs both in how readily they are reversed and in how strongly they influence others. The chain below adapts the latent-class structure of Teklu and Terefe [20], formulated there for animosity toward mathematics rather than for absenteeism, and extends the graded treatment of absence introduced by Adu et al. [6]. Table 1 presents descriptions of the state variables used in the study.

The parameters used in the study are described in Table 2.

Regular attendees acquire absenteeism norms at the rate

Equation 1 is a standard-incidence form in which all three behaviorally affected classes contribute, weighted by relative influence: θ is below unity because an exposed student expresses the norm verbally but does not yet model it behaviorally, and σ is at or above unity because established absence is the most visible and most readily imitated signal. Setting θ = 0 and σ = 1 recovers a single-weight model as a special case.

with non-negative initial data. Throughout, the composite parameters are defined in Equation 3 as follows:

Since N is constant and Π = μN, set s=SN, e=EN, a=AN, h=CN, and r=RN. Dividing Equation 2 by N gives

Let ΠN=μ. Summing the five equations then gives

Thus, the simplex defined in Equation 5

is invariant and globally attracting.

Lemma 1. For non-negative initial data, the solutions of Equation 4 remain non-negative for all t > 0, and Ω is positively invariant.

Proof. On s = 0 we have ds/dt = μ+ωr ≥ 0; on e = 0, de/dt = βs(a+σh) ≥ 0; on a = 0, da/dt = κe+ϕr ≥ 0; on h = 0, dc/dt = δa ≥ 0; and on r = 0, dr/dt = γ₁a+γ₂h ≥ 0. The vector field is therefore inward-pointing on every boundary face of ℝ+5, so no trajectory leaves the positive orthant. Invariance of the simplex follows from the conservation identity in Section 2.3.

Setting the right-hand side of Equation 4 to zero with e = a = h = r = 0 gives the unique absenteeism-free equilibrium E0=(1,0,0,0,0).

We compute R0 by the next-generation matrix method, taking the behaviorally affected compartments in the order x = (e, a, h, r)^⊤. The class r is retained in this subsystem: because re-engaged students relapse directly into occasional absence at rate φ, the re-engaged class is not absorbing and contributes to onward transmission. This is a structural consequence of assumption (v) with no counterpart in either antecedent model. Since new adoption of absenteeism norms occurs only through entry into the exposed compartment, only the first row of the new-adoption matrix (F) contains non-zero entries; all entries in rows two to four are zero. The new-adoption and transition matrices at E0 are presented in Equation 6 as follows:

The quantity Δ is defined in Equation 7 as follows:

Lemma 2. Δ > 0 for all admissible parameter values.

Proof. k₂k₃ − (γ₁k₃ + γ₂δ) = (δ + μ)(γ₂ + μ) − γ₂δ = μ(δ + γ₂ + μ) > 0, so k₂k₃ > γ₁k₃ + γ₂δ. Since k₄ = ω + φ + μ > φ, it follows that k₂k₃k₄ > (γ₁k₃ + γ₂δ)φ.

Lemma 2 guarantees detV=k1Δ≠0. Since F has rank one, the spectral radius of FV-1 is its (1, 1) entry, giving

Expression (Equation 8) admits a direct behavioral reading. The first term is the contribution of the latent phase: a student spends a mean time 1/k₁ exposed and transmits at the reduced weight θ throughout. The second is the contribution of active absence, weighted by the probability κ/k₁ of progressing from exposure to absence and by the influence accumulated across the occasional and habitual phases. Setting the direct relapse rate φ = 0 gives the threshold expression presented in Equation 9:

the sum of three stagewise contributions, which confirms the derivation.

Theorem 1. System (Equation 4) has no absenteeism-persistent equilibrium when R0≤1 and exactly one when R0>1, given by

where

Proof. At equilibrium, the second equation of Equation 4 gives e*=λ*s*/k1. The third, fourth, and fifth give h*=δa*/k3, r*=a*(γ1k3+γ2δ)/(k3k4), and k2a*=κe*+φr*. Eliminating r^* gives a*Δ/(k3k4)=κe*, hence Equation 10. Substituting into λ^* = β(θe^*+a^*+σh^*) gives λ*=k1R0e* by Equation 8; combining with e*=λ*s*/k1 yields λ*=λ*s*R0, and since λ*≠0,s*=1/R0. The value of e^* follows from s^*+e^*+a^*+h^*+r^* = 1, and e^*>0 if and only if R0> 1.

Theorem 1 establishes a forward (transcritical) bifurcation at R0=1. Although relapse mechanisms can generate backward bifurcation in some compartmental models [21], the linear structure of the affected subsystem in the present model excludes such sub-threshold persistence. Thus, within the present model, reducing R₀ below unity is sufficient for elimination of the absenteeism-persistent equilibrium and underpins the threshold-based control analysis in Sections 4–6.

Theorem 2. E0 is locally asymptotically stable if R0<1 and unstable if R0> 1.

Proof. Immediate from Theorem 2 of van den Driessche and Watmough [22]: F≥0, V is a non-singular M-matrix by Lemma 2, and the absenteeism-free subsystem ds/dt = μ(1−s) is globally stable at s = 1.

The Jacobian of Equation 4 at a general point is

Because the characteristic polynomial of Equation 11 at E* is quintic with coefficients that do not simplify usefully, we assess local stability of the persistent equilibrium by direct evaluation of the spectrum of Equation 11 rather than by symbolic Routh–Hurwitz conditions. Section 5.4 reports this verification over the sampled parameter region.

The global stability of the absenteeism-free equilibrium is established using the matrix-theoretic Lyapunov construction of Shuai and van den Driessche [23]. The advantage of this approach is that the coefficients of the Lyapunov function are obtained systematically from the next-generation operator rather than selected by inspection.

Let

denote the vector of behaviorally affected compartments, and write

so that the new-adoption matrix can be written in rank-one form as

where e1=(1,0,0,0)⊤. At the absenteeism-free equilibrium, the next-generation matrix is FV⁻¹, and the absenteeism threshold number is

Define

Then

Thus, g^⊤ is a positive left Perron eigenvector of the next-generation matrix FV^{− 1}.

Theorem 3. If R0≤1, then E0 is globally asymptotically stable in Ω.

Proof. Define the Lyapunov function in Equation 12 as follows:

where the coefficients are obtained by solving

Solving the coefficient equations gives the expressions presented in Equation 13:

By Lemma 2, all the coefficients are positive. Hence,

∀ x∈Ω, with L(x) = 0 only when

For the normalized system, the affected subsystem can be written as

Since 0 ≤ s ≤ 1 in the invariant region Ω, and Fx≥0 componentwise,

Differentiating L along the trajectories of the system therefore gives

Because g^⊤ = f^⊤V^{− 1},

Moreover,

Consequently,

Since

it follows that

For R₀ < 1, equality L.=0 requires

and, because β, θ, σ > 0,

For a trajectory to remain in this set, the re-engaged compartment must also satisfy r = 0. Since the normalized variables sum to unity, this subsequently gives s = 1. Therefore, the largest invariant subset of

is the absenteeism-free equilibrium

By LaSalle's [24] invariance principle, E₀ is globally asymptotically stable whenever

That solving V⊤g=f independently returns g1=R0 provides a check on expression (Equation 8).

We introduce three bounded, Lebesgue-measurable controls on [0, t_f ].

The constants ψ₁, ψ₂>0 are the maximum achievable increments in the respective re-engagement rates. The separation of u₂ and u₃ by stage is the purpose of this approach: the controls act on different compartments, so their comparison answers a substantive question about the allocation of institutional resources. The controlled system is

over the admissible set

We minimize

subject to Equation 14, where W₁, W₂ > 0 weight the instructional cost of occasional and habitual absence, with W₂ > W₁ since established absence carries the larger academic penalty, and B₁, B₂, B₃ > 0 are the relative programme costs. The quadratic form reflects increasing marginal cost of intervention intensity.

Theorem 4. There exists an optimal control triple in U with a corresponding state solution minimizing J.

Proof. We verify the conditions of Fleming and Rishel [25]. The admissible set is non-empty, since u₁ = u₂ = u₃ = 0 is admissible and Equation 14 then reduces to Equation 2, whose solutions exist and are bounded by Lemma 1. U is closed and convex by construction. The right-hand side of Equation 14 is bounded by a linear function of the state and is Lipschitz in the state uniformly in the controls, since all states are bounded by N and controls by unity. The integrand of Equation 15 is convex in the controls, its Hessian being diag (B₁, B₂, B₃) > 0. Finally, the integrand is bounded below by n1(∑i|ui|2)n3/2-n2 with n₃ = 2, n1=12min{c1,c2,c3}, and n₂ = 0, using W₁A+W₂C ≥ 0.

Theorem 5. Let (u1*,u2*,u3*) be optimal with associated states. Then there exist adjoint variables p₁, …, p₅ satisfying

with p_i(t_f) = 0, and the optimal controls are

Proof . The Hamiltonian associated with the optimal control problem is defined in Equation 18 as follows:

System (Equation 16) follows from pi′=-∂H/∂xi, using ∂λ/∂E = βθ/N, ∂λ/∂A = β/N, and ∂λ/∂C = βσ/N. Solving ∂H/∂ui=0 on the interior of U gives c₁u₁ = (p₂−p₁)λS, c₂u₂ = ψ₁A(p₃−p₅), and c₃u₃ = ψ₂C(p₄−p₅); projection onto [0,uimax] gives Equation 17.

The characterizations have a clear interpretation. Control u₂ activates precisely when p₃>p₅, that is, when the shadow cost of an additional occasional absentee exceeds that of an additional re-engaged student, and u₃ when p₄>p₅. The relapse rate φ enters p5′ directly, so a cohort with poor retention carries a higher shadow cost on R, which narrows both gaps and depresses both re-engagement controls. This is the dynamic counterpart of Proposition 1.

The parameter values in Table 3 are assumed and are chosen to be dimensionally coherent for a university cohort observed over 4 academic years, with time measured in months. The parameter values used in the simulations are therefore illustrative and are not estimates derived from empirical attendance data. Consequently, the numerical results should be interpreted as demonstrations of the behavior of the proposed model rather than as predictions for a specific university cohort. The consequence for interpretation is discussed further in Section 7.

Several parameters have direct structural counterparts in earlier behavioral compartmental models. In particular, β, δ, γ₁, and γ₂ represent transmission, progression from occasional to habitual absenteeism, early re-engagement, and remediation processes that are also represented in the absenteeism model of Adu et al. [6]. The latent-stage progression parameter κ is motivated by the exposed-to-active behavioral transition in Teklu and Terefe [20]. The relative influence parameters θ and σ, together with the direct relapse parameter ϕ, are extensions introduced in the present SEACR formulation to distinguish the behavioral influence of exposed and habitual absentees and to represent relapse after re-engagement. The numerical values assigned to these parameters are illustrative because the antecedent studies do not provide empirically calibrated estimates that can be transferred directly to the present five-compartment model.

The optimality system was solved by the forward–backward sweep method [26] with fourth-order Runge–Kutta integration on a grid of 2,400 intervals over t_f = 48 months, controls updated by a weighted convex combination, and convergence declared at a relative tolerance of 10⁻⁴; all runs converged within 31 iterations.

With these values, k₁ = 0.6875, k₂ = 0.250833, k₃ = 0.070833, k₄ = 0.280833, and Δ = 4.11217 × 10⁻³ > 0, consistent with Lemma 2. Expression (Equation 8) gives

Theorem 1 then gives the absenteeism-persistent equilibrium in proportions (s^*, e^*, a^*, h^*, r^*) = (0.4084, 0.0578, 0.1864, 0.2105, 0.1370), equivalently (817, 116, 373, 421, 274) students. Substituting these values into the right-hand side of Equation 4 returns residuals of the order of 10⁻¹⁸, and s*=1/R0 holds to machine precision, confirming the algebra of Theorem 1.

Figure 2 shows the uncontrolled trajectories under two transmission regimes. At β = 0.18, giving R0=2.4487, the regular-attendee class declines from 1,700 to approximately 880 over 4 years while occasional and habitual absence rise toward the predicted equilibrium, confirming Theorem 2. At β = 0.06, giving R0=0.8162, all four affected classes decay toward zero and the cohort returns to full regular attendance, consistent with Theorem 3.

Normalized forward sensitivity indices computed from Equation 8 are given in Table 4. As theory requires, χβR0=1 exactly, since R0 is linear in β; this serves as a check on the numerical differentiation.

Global sensitivity analysis was performed by Latin hypercube sampling of 2,000 parameter sets over ±50% of the baseline values, with R0 as the response function; partial rank correlation coefficients (PRCCs) are shown in Figure 3 and Table 5.

The dashed horizontal lines at PRCC ±0.5 are included only as visual reference points for identifying parameters with comparatively strong positive or negative monotonic relationships with R₀; they are not statistical significance thresholds.

The results first show that the early re-engagement rate γ_1 is the most influential protective parameter, with |−0.8944| exceeding the value |−0.8253| of the remediation rate γ_2, and the same ordering appears in the local indices (−0.5523 against −0.4232). A given proportional improvement in the rate at which occasional absentees are recovered reduces the threshold more than the same proportional improvement in the rate at which habitual absentees are remediated. This is a structural result about where in the progression intervention has the most leverage, and it does not depend on any cost assumption. Second, the relapse rate (ϕ) has a substantial positive PRCC of +0.4940, confirming Proposition 1 numerically across the sampled region and establishing relapse as an important driver of absenteeism persistence. The escalation rate (δ) also has a positive association with the absenteeism threshold, indicating that faster progression from occasional to habitual absence contributes to absenteeism persistence.

The positive association between relapse and the threshold indicates that the benefits of re-engagement are less durable when students return rapidly to occasional absenteeism. Third, κ is not significantly associated with R0 (PRCC = −0.0023, p = 0.918), matching its near-zero local index. This is initially surprising but follows from Equation 8 as lengthening the latent phase delays progression to absence without preventing it, and the two effects—longer time transmitting at weight θ and later arrival in the higher-weight classes—very nearly cancel. Within the assumed parameter configuration, slowing progression from exposure to absence has relatively limited influence on the threshold unless progression is also interrupted through re-engagement or related support.

Figure 4 presents contour plots of R0 over (γ₁, γ₂) and over (β, φ), with the critical contour R0=1 marked. Figure 4a shows that the critical contour is reached at more modest values of γ₁ than of γ₂; Figure 4b shows that the transmission rate required for persistence falls as relapse rises.

At the baseline parameter values, the Jacobian (Equation 11) evaluated at E* has the spectrum

all of which have negative real parts, so E* is locally asymptotically stable. The complex pair indicates a damped oscillatory approach, consistent with the shoulder visible in the occasional-absentee trajectory of Figure 2a.

To establish that this is not an artifact of the particular parameter choice, we drew 5,000 parameter sets from the same Latin hypercube region used in Section 5.3, retained the 4,935 for which R0>1, and computed the spectrum of Equation 11 at the corresponding persistent equilibrium in each case. In every one of the 4,935 cases, the maximum real part was strictly negative, the largest value observed across the whole sample being −2.396 × 10⁻⁴. Across the sampled parameter region, all 4,935 parameter sets for which the absenteeism-persistent equilibrium exists produced Jacobian eigenvalues with strictly negative maximum real parts. Thus, the persistent equilibrium is locally asymptotically stable for every parameter configuration examined. This finding provides strong numerical evidence that local stability is robust within the investigated parameter region. It should not, however, be interpreted as a proof of global asymptotic stability. Numerical verification, regardless of the number of parameter sets examined, cannot exclude qualitatively different behavior outside the sampled region or establish convergence of every admissible trajectory to the persistent equilibrium.

Seven strategies were simulated: each control singly, each pair, and all three. Figure 5 shows the occasional and habitual absentee trajectories under each, against the uncontrolled baseline. Figure 6 shows control profiles for Strategies 2, 3, and 7.

The single-control profiles remain close to the upper bound of 0.9 for about 45 months before declining near the end of the intervention period. This pattern reflects the way the assumed absenteeism-burden weights, intervention costs, and terminal conditions interact in the model. It should not be interpreted as a recommendation for universities to maintain maximum intervention intensity continuously for almost 4 academic years. Rather, under the present model assumptions, sustained intervention continues to produce greater benefits than costs for most of the simulation period.

In practice, maintaining such a high level of intervention for this length of time may be difficult. Universities operate under staffing, counseling, monitoring, budgetary, and other implementation constraints that are not explicitly represented in the model. Future extensions could therefore incorporate cumulative budget limits, capacity constraints, time-varying intervention costs, or alternative cost structures to produce control profiles that more closely reflect real institutional conditions. The control trajectories reported here should thus be understood as mathematical optima under the assumed model structure and cost weights, rather than as exact operational schedules.

Under Strategy 7, the control profiles begin to taper from about month 10, and each intervention is applied at a lower intensity than when used alone. This is because the three controls act at different stages of the absenteeism process and partly complement or substitute for one another. Notably, u₂ remains the most intensively applied control during the tapering phase, which is consistent with the sensitivity analysis. Over the 48-month simulation period, the uncontrolled cohort accumulates 25,095.5 student-months of absence. Under the assumed parameter values, Strategy 7 averts 24,278.0 student-months, representing 96.7% of the uncontrolled burden and the largest reduction among the seven strategies considered. By comparison, Strategy 3, which relies on remediation alone, averts 15,610.8 student-months of absence and is the least effective of the single-control strategies because it intervenes only after absenteeism has become established.

The average cost-effectiveness ratio (ACER) measures the average intervention cost required to avert one unit of absenteeism and is calculated as the total intervention cost divided by the total absenteeism averted. The incremental cost-effectiveness ratio (ICER) compares two strategies and represents the additional cost required to obtain one additional unit of absenteeism averted when moving from a less effective strategy to a more effective strategy. Accordingly, the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER) are computed using Equation 19 as follows:

Thus, ACER represents the average model-based intervention cost per student-month of absenteeism averted, whereas ICER represents the additional cost per additional student-month of absenteeism averted when one strategy is compared with another.

With total cost CT=12∫0tf(B1u12+B2u22+B3u32)dt, absenteeism averted is measured as ∫0tf[(A+C)no control-(A+C)control]dt, in student-months of absence rather than in headcount. This distinction is stated explicitly because the antecedent literature is not consistent on the measure used.

Strategy 2 has the lowest ACER at 0.027021 and Strategy 3 the highest at 0.095938, a factor of 3.6. The relationship between total intervention cost and absenteeism averted across the seven intervention strategies is illustrated in Figure 7. Strategy 7 costs less than Strategies 4, 5, and 6 while averting more, because the three controls substitute for one another and each is used less intensively, as Figure 6 shows; combined strategies are not simply the sum of their parts. Additionally, under these weights, remediation alone is both the most expensive and the least effective single strategy. Its low effectiveness is a property of the model structure and does not depend on the costing; its high cost does, as Section 6.2 shows.

Recursive ICER elimination proceeds as follows. Strategy 1 is eliminated first, having the highest incremental ratio at 0.7299; then Strategy 5 at 0.2882, Strategy 4 at 0.1553, Strategy 6 at 0.1210, and Strategy 3 at 0.0959. The final comparison is between Strategies 2 and 7, with ICER = 0.0892 for Strategy 7 against an ACER of 0.0270 for Strategy 2. Strategy 2, representing early intervention among occasional absentees, survives the recursive elimination procedure.

The conclusion of Section 6.1 rests on the assumed cost weights (B₁, B₂, B₃) = (50, 30, 80), and its sensitivity to alternative assumptions should be assessed. We therefore recomputed the full seven-strategy comparison across a grid of cost weights. Table 6 reports the ACER-minimizing strategy at selected points of the grid B₂, B₃∈[15, 110] with B₁ fixed at 50.

The ranking is not robust. Hence, early intervention is preferred when it is inexpensive relative to remediation. The combined strategy becomes preferred once the two are comparably priced. Prevention alone is preferred when it is inexpensive relative to both; most strikingly, when remediation is made inexpensive (B₃ = 15), the ACER criterion selects Strategy 3 (remediation alone) despite its averting the least absence of any strategy in Table 7. That last case exposes a limitation of ACER itself as a selection criterion (a strategy can minimize cost per unit averted while leaving most of the burden in place). Since none of B₁, B₂, or B₃ is estimated from data here or in the antecedent literature, the ordering of strategies by ACER should be regarded as conditional on an unverified assumption. This matters beyond the present paper. Adu et al. [6] conclude that class attendance monitoring alone is the most cost-effective strategy for reducing absenteeism, on the basis of a single set of assumed weights and without a robustness check. Our Strategy 2 is the closest analog of their recommended strategy, and we reach the same conclusion under the baseline cost weights. However, the preferred strategy changes under alternative assumed cost configurations. We therefore regard the more general qualitative result as the defensible one: early intervention tends to outperform late remediation under the baseline behavioral parameterization, a finding that is also supported by the PRCC ordering independently of the assumed cost weights. The finer ranking among the remaining strategies should be interpreted as conditional on the assumed programme-cost structure.

To complement the cost-weight analysis, we examined whether the optimal-control profiles and cost-effectiveness ranking were sensitive to uncertainty in key behavioral parameters. The transmission rate β, early re-engagement rate γ₁, remediation rate γ₂, relapse rate φ, and escalation rate δ were each varied independently by ±20% around their baseline values, while all other parameters and cost weights were held fixed. For each scenario, the optimality system was re-solved for all seven strategies and the corresponding absenteeism averted, intervention cost, ACER, and control trajectories were recalculated. The results of the parameter perturbation analysis are summarized in Table 8. Across all 10 perturbation scenarios, Strategy 2 retained the lowest ACER, while Strategy 7 retained the second-lowest ACER. Thus, the leading cost-effectiveness ordering was stable under the moderate one-at-a-time parameter variations considered here. The optimal-control intensities, however, changed quantitatively across scenarios. For Strategy 7, the time-averaged value of u₁ ranged from 0.302 to 0.416, u₂ from 0.509 to 0.556, and u₃ from 0.333 to 0.367 over the 48-month intervention horizon. Higher transmission increased the required prevention intensity, lower baseline early re-engagement increased the optimal intensity of u₂, and higher relapse increased the overall intensity of the combined intervention. These results suggest that, within the ±20% perturbation range examined, uncertainty in behavioral parameters affects the magnitude of the optimal intervention effort more strongly than the leading cost-effectiveness ranking.

The most serious limitation is that no empirical attendance data were used: all parameters in Table 3 are assumed, and the numerical results are therefore illustrative of the model's behavior rather than estimates for any real cohorts. A further theoretical limitation is that global asymptotic stability of the absenteeism-persistent equilibrium has not been established analytically. Although local stability was verified numerically for all 4,935 sampled parameter configurations satisfying the persistence condition, this constitutes numerical evidence rather than a global stability theorem. Consequently, the analysis does not establish that every admissible solution converges to the persistent equilibrium whenever the absenteeism threshold exceeds unity. Establishing such a result through an appropriate Lyapunov, graph-theoretic, or geometric method remains an important theoretical extension of the present model. The assumption of a constant cohort size is reasonable over 1 or 2 academic years but would need relaxing over longer horizons, as would the assumption of homogeneous mixing, which ignores programme, year-group, and residential structure. Transmission is taken to be time-invariant, whereas university attendance is strongly periodic across a semester; a time-dependent β(t) with a semester period would be a natural and, to our knowledge, unexplored extension. Finally, the cost weights are assumed rather than costed, which is why Section 6.2 is included and why we decline to rest any conclusion on the finer ACER ranking. The model also does not capture the full range of benefits that may arise from the interventions considered, since its evaluation is restricted mainly to absenteeism-related transitions and accumulated absence. Broader outcomes, including academic performance, engagement, retention, motivation, and wellbeing, remain outside the present framework.

Fitting β, κ, δ, and γ₁ to attendance registers from a Ghanaian university, with confidence intervals propagated through to the cost-effectiveness ratios, would convert the present illustrative results into estimates. Such empirical calibration would parallel recent developments in dynamical modeling in which model parameters and trajectories are evaluated against observed population data instead of being based exclusively on assumed parameter values [27, 28]. Beyond this, a saturated incidence of the form β(θE+A+σC)/(1+α(A+C)) would relax the assumption that influence scales linearly with prevalence and could admit backward bifurcation, which the present formulation excludes; and a fourth control acting directly on φ would allow retention to be costed against the other three. The present model therefore provides theoretical support for the hypothesis that detecting and responding to absenteeism while it remains occasional may offer greater leverage than intervention initiated only after absenteeism becomes habitual. It also indicates that the effectiveness of re-engagement depends partly on limiting subsequent relapse. Because the model has not yet been calibrated to empirical attendance data, these findings should be interpreted as theoretically derived and numerically illustrated propositions that require validation using longitudinal university attendance records. An important theoretical direction is to establish conditions for global asymptotic stability of the absenteeism-persistent equilibrium when the threshold exceeds unity. The numerical evidence reported in Section 5.4 suggests a stable persistent state throughout the parameter region examined, but an analytical result is still required. Future work may investigate whether the graph-theoretic Lyapunov framework of Shuai and van den Driessche [23], the geometric approach associated with Li–Muldowney-type methods, or another suitable Lyapunov construction can establish global convergence for the present five-dimensional system.