Spatiotemporal SEIQR Epidemic Model

• Spatiotemporal SEIQR models are compartmental epidemic systems with S, E, I, Q, and R states that integrate spatial diffusion and diverse quarantine interpretations.
• They employ reaction-diffusion PDEs, network formulations, and nonstandard finite difference schemes to ensure positivity, stability, and biological fidelity.
• Optimal control strategies, such as vaccination, treatment, and social distancing, are integrated to minimize outbreak costs and guide intervention designs.

Searching arXiv for the supplied papers and closely related spatiotemporal SEIQR models. I’m going to look up the cited arXiv papers and nearby work on spatiotemporal SEIQR/SQEIAR epidemic models so the article can be grounded in current preprints. A spatiotemporal SEIQR epidemic model is a compartmental infectious-disease model in which the state variables $S,E,I,Q,R$ depend on both time and space, so that transmission, quarantine, recovery, and population movement are represented jointly rather than only through temporal incidence curves. In the recent arXiv literature, the dominant formulation is a reaction-diffusion system of semilinear parabolic PDEs with homogeneous Neumann boundary conditions, although closely related spatial models replace diffusion by interconnected networks or graph-based spillover operators. The class therefore includes a canonical SEIQR PDE model, numerical structure-preserving discretizations, and adjacent variants in which quarantine is applied to susceptibles, hospitalized cases play the role of an isolated class, or network mobility replaces Laplacian transport (Zinihi et al., 12 Jul 2025, Zinihi et al., 4 Aug 2025).

1. Conceptual scope and model family

The canonical spatiotemporal SEIQR formulation uses five compartments: susceptible $S$, exposed $E$, infected $I$, quarantined $Q$, and recovered $R$, each written as $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$. In this formulation, exposure may arise through contact with exposed or infected individuals, infected individuals enter quarantine, quarantined individuals either recover or return to susceptibility depending on the model, and spatial movement is represented by compartment-specific diffusion coefficients (Zinihi et al., 12 Jul 2025).

Related preprints show that the term is used somewhat more broadly in practice. One paper replaces the strict SEIQR structure by an SQEIAR system in which $Q$ denotes quarantined susceptible individuals and asymptomatic infection $A$ is explicit; another uses an SEIRAH network model where $H$ functions as a quarantine/hospitalization class; and a graph-hybrid SEIR model is presented as directly adaptable to SEIQR by adding a $S$0 compartment and modifying the discrete recursion (Mohammed et al., 2024, Deng et al., 2021, Zheng et al., 2020).

Model source	State space	Spatial mechanism
(Zinihi et al., 12 Jul 2025)	$S$1	reaction-diffusion PDE with controls
(Zinihi et al., 4 Aug 2025)	$S$2	reaction-diffusion PDE with NSFD discretization
(Mohammed et al., 2024)	$S$3	diffusion plus regional quarantine via $S$4
(Deng et al., 2021)	$S$5	interconnected residence-work small-world networks
(Zheng et al., 2020)	$S$6 with $S$7 inflow	graph-based hybrid recursion with edge-RNN

This variety matters because a frequent misconception is that “spatiotemporal SEIQR” denotes a single fixed dynamical law. The recent literature instead uses the label for a family of spatial epidemic systems that share quarantine-aware compartmental structure but differ in the semantics of $S$8, the treatment of asymptomatic infection, and the mechanism of spatial coupling. This suggests that the most stable defining feature is not a unique equation set, but the joint treatment of compartmental disease progression and explicitly spatial transmission structure.

2. Reaction-diffusion SEIQR formulation

In the five-compartment PDE formulation, the uncontrolled dynamics on $S$9 are

$E$0

with $E$1 bounded and smooth, $E$2, homogeneous Neumann boundary conditions, and positive initial data (Zinihi et al., 4 Aug 2025).

The compartmental interpretation is explicit. $E$3 is the recruitment rate, $E$4 and $E$5 are transmission rates from susceptible-exposed and susceptible-infected contact, $E$6 is the rate at which exposed individuals become infectious, $E$7 is the rate at which infected individuals are quarantined, $E$8 is the recovery rate of quarantined individuals, $E$9 is the rate at which non-infected quarantined individuals return to susceptible, $I$0 is the natural death rate, and $I$1 are diffusion coefficients (Zinihi et al., 12 Jul 2025).

The modeling assumptions are equally important. Spatial movement is isotropic and represented by the Laplacian $I$2; homogeneous Neumann conditions encode a closed domain with no flux across $I$3; parameters are taken to be positive constants in one formulation and nonnegative, spatially constant in another; and positivity of initial data is assumed and then proved to persist (Zinihi et al., 12 Jul 2025, Zinihi et al., 4 Aug 2025). Because one paper keeps parameters spatially constant, it notes specifically that diffusion does not alter the basic reproduction number $I$4 in that setting (Zinihi et al., 4 Aug 2025).

The epidemiological interpretation of $I$5 is model-dependent. In the canonical SEIQR PDE, $I$6 is an isolated compartment fed by $I$7. In the SQEIAR extension, by contrast, $I$8 is a quarantined susceptible compartment populated by a regional control $I$9, while symptomatic and asymptomatic infectious individuals are tracked separately and reinfection $Q$0 is allowed (Mohammed et al., 2024).

3. Controlled dynamics and intervention design

The optimal-control SEIQR model introduces three time- and space-dependent controls: $Q$1 interpreted respectively as vaccination / preventive intervention, treatment of quarantined individuals, and social distancing / public awareness / non-pharmaceutical prevention. The controlled system replaces infection terms by $Q$2 and $Q$3, moves susceptibles to recovery through $Q$4, and accelerates recovery from quarantine through $Q$5 (Zinihi et al., 12 Jul 2025).

The admissible controls satisfy

$Q$6

and the objective functional is

$Q$7

with

$Q$8

and

$Q$9

This is a linear-in-control cost structure rather than a quadratic penalization (Zinihi et al., 12 Jul 2025).

A different control architecture appears in the SQEIAR model. There, the two controls are $R$0 for treatment of infected individuals and $R$1 for regional quarantine of susceptibles, with

$R$2

The control enters through a characteristic function $R$3, where $R$4 and the $R$5 are pairwise disjoint. This makes quarantine region-based and spatially distributed rather than population-wide (Mohammed et al., 2024).

The first-order optimality structures also differ. In the controlled SEIQR PDE, when terminal control costs vanish, the optimal controls are bang-bang. In the SQEIAR model, the optimal controls are characterized by projection formulas,

$R$6

$R$7

The contrast is useful: one formulation favors saturating on-off control, while the other yields projected interior values determined by state-adjoint interactions (Zinihi et al., 12 Jul 2025, Mohammed et al., 2024).

4. Analytical properties and threshold quantities

The recent PDE literature places strong emphasis on semigroup-based well-posedness. For the controlled SEIQR reaction-diffusion system, all solutions are bounded in $R$8, remain positive for positive initial data, and yield a unique bounded positive global solution with componentwise regularity

$R$9

Additional estimates are also provided for $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$0, $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$1, $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$2, and $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$3 norms (Zinihi et al., 12 Jul 2025).

The numerics-oriented SEIQR PDE establishes a comparable result: for positive initial data, the system admits a unique global solution $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$4 that remains positive and bounded on $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$5, with the same class of regularity, and is explicitly described as Hadamard well-posed because it has a unique global bounded nonnegative solution depending continuously on initial data (Zinihi et al., 4 Aug 2025). The proof summary invokes semigroup theory for the Laplacian, a priori estimates for boundedness, comparison principles and Gronwall inequalities for positivity, and strong ellipticity with Lipschitz continuity for existence and uniqueness (Zinihi et al., 4 Aug 2025).

Threshold analysis is present but not universal. For the reaction-diffusion SEIQR model with spatially constant parameters, the disease-free equilibrium is

$S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$6

and the next-generation matrix method yields

$S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$7

The standard interpretation is stated: if $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$8, the disease-free equilibrium is locally asymptotically stable; if $S(t,x),E(t,x),I(t,x),Q(t,x),R(t,x)$9, the disease may persist and an endemic equilibrium may emerge (Zinihi et al., 4 Aug 2025).

By contrast, the SQEIAR control paper does not derive an explicit basic reproduction number $Q$0 or a threshold quantity, and instead focuses on existence and boundedness of solutions, optimal control existence, first-order optimality conditions, and numerical impact of controls (Mohammed et al., 2024). A common misconception is therefore that every spatiotemporal quarantine model is accompanied by a threshold computation; the recent literature shows both threshold-centered and control-centered analytical traditions.

5. Numerical schemes and computational behavior

Two computational strands are prominent: direct finite-difference optimal control solvers and structure-preserving NSFD schemes. In the optimal-control SEIQR PDE, spatial discretization is by finite difference method, time stepping is implicit, and optimal controls are solved by forward-backward sweep: a forward solve for the state equations, a backward solve for the adjoint equations, and pointwise control updates (Zinihi et al., 12 Jul 2025).

The numerical study in that paper is conducted on

$Q$1

with $Q$2 cells and $Q$3 days. Eight scenarios are compared: no control; vaccination only $Q$4; treatment only $Q$5; social distancing only $Q$6; vaccination plus treatment; vaccination plus social distancing; treatment plus social distancing; and all three controls. The reported objective values are $Q$7, $Q$8, $Q$9, $A$0, $A$1, $A$2, $A$3, and $A$4, respectively, giving the ordering

$A$5

The combined use of $A$6 is therefore reported as producing the best epidemiological outcome and the lowest cost (Zinihi et al., 12 Jul 2025).

The NSFD paper addresses a different computational problem: standard finite differences can violate positivity. It adopts Mickens-style denominator functions with

$A$7

uses a generalized parabolic mesh ratio $A$8, and analyzes a positivity-preserving but initially inconsistent nonstandard Laplacian. A perturbation $A$9 is then introduced to restore consistency while retaining the positivity-preserving structure. The resulting explicit sequential NSFD update is computed in the order $H$0 (Zinihi et al., 4 Aug 2025).

The paper shows that the naive skew Laplacian discretization is not consistent in the standard sense because a leading truncation term does not vanish as $H$1. It also states a theorem giving first-order accuracy in time and second-order accuracy in space for a denominator-function correction under stated time-step restrictions, but immediately notes that this formal fix collapses to a standard explicit finite-difference-type method and therefore loses unconditional positivity and stability. The modified scheme with $H$2 is then used in the numerical section and is emphasized as positivity preserving, bounded, numerically stable, and more biologically faithful than the standard finite difference method (Zinihi et al., 4 Aug 2025).

The reported experiments reinforce the analytical claims. In 1D, SFD captures the general epidemic evolution but produces negative values in the exposed class $H$3 at later times, whereas NSFD preserves positivity for all compartments and yields smooth, biologically plausible dynamics. In 2D, SFD again yields negative exposed densities near $H$4 to $H$5, while NSFD keeps all variables nonnegative and the spatial propagation smooth and realistic (Zinihi et al., 4 Aug 2025).

A further numerical illustration is provided by the SQEIAR COVID-19 example, where simulations are performed with an explicit finite difference method, the adjoint system is solved numerically, and the controls are updated iteratively. Compared with the uncontrolled case, the paper reports that susceptible individuals decline much more slowly under control, exposed and asymptomatic populations are significantly reduced, infected individuals drop to near zero within about 20 days under control, more than 4300 individuals are quarantined in the controlled case, and the strategy saves more than 80 lives in the simulated population of at most 8000 (Mohammed et al., 2024).

6. Network and graph formulations adjacent to SEIQR

Not all spatiotemporal quarantine-aware epidemic models use diffusion PDEs. One adjacent line models metropolitan spread on interconnected networks. The SEIRAH framework on Tokyo metropolitan areas uses residence-work interconnected networks with several residential networks and one central work network, daily commuting, two daily periods—DayHours(1) and DayHours(2)—and Newman–Watts small-world networks parameterized by $H$6. In that model, $H$7 are infectious classes, $H$8 is the observable effect of propagation, and a time-varying social infectivity $H$9 is inferred from observed daily hospitalized cases $S$00 by a modified binary search minimizing

$S$01

If $S$02, then $S$03 (Deng et al., 2021).

Its Tokyo experiment reports that recent $S$04, about 30% of early-outbreak levels; exposed peaks precede hospitalized peaks by about two weeks; asymptomatic infections decline after hospitalized cases fall; and symptomatic infections decline faster than asymptomatic ones. The policy interpretation given is that reducing mobility matters, monitoring asymptomatic spread is critical, early detection and testing are essential, and premature relaxation of restrictions can trigger a second wave (Deng et al., 2021). Although not a standard SEIQR model, it is explicitly described as SEIQR-family in spirit because $S$05 plays a quarantine/confirmed-hospitalization role.

A second adjacent line is graph-hybrid forecasting. IeRNN augments SEIR with latent inflow terms $S$06 and $S$07 from geographically adjacent states, derives a discrete “I-equation” from SEIR, and learns the spatial inflow term through an edge-RNN built from stacked LSTM cells. Training uses mean squared error, with 133 training days and 35 testing days, and the model is evaluated in 1-day and 7-day ahead forecasting (Zheng et al., 2020).

IeRNN is not an SEIQR model as written, but the paper states that the same framework can be extended by adding a $S$08 compartment and modifying the recursion accordingly. The reported experiments show lower MSE than LSTM and ARIMA on U.S. state-level COVID-19 data, while the architecture remains compact relative to full spatiotemporal sequence models. A plausible implication is that graph-based spillover learning and PDE-based diffusion should be viewed as alternative spatial closures for quarantine-aware compartmental dynamics rather than competing definitions of the topic (Zheng et al., 2020).

7. Interpretation, applications, and recurring misconceptions

The principal use of spatiotemporal SEIQR modeling is the analysis of local outbreaks, spatial heterogeneity, diffusion-driven spread, and geographically targeted interventions. In the reaction-diffusion setting, the dependence of both states and controls on $S$09 allows intervention fields to vary over the domain rather than being imposed uniformly. In the SQEIAR setting, the use of $S$10 makes regional quarantine explicit; in the network setting, commuting links and work-network connectivity play the corresponding role (Zinihi et al., 12 Jul 2025, Mohammed et al., 2024, Deng et al., 2021).

Several misunderstandings recur in reading this literature. First, $S$11 does not carry a universal epidemiological meaning. It may denote quarantined infected individuals in SEIQR PDEs, quarantined susceptible individuals in SQEIAR, or be replaced by a hospitalization class $S$12 that acts as an isolated class in a network model (Zinihi et al., 12 Jul 2025, Mohammed et al., 2024, Deng et al., 2021). Second, a lower recovered total under control is not necessarily adverse; in the SQEIAR simulations, recovered totals are lower under control because fewer people become infected (Mohammed et al., 2024). Third, hospital-confirmed or quarantined counts are not synonymous with transmission intensity; in the Tokyo network study, exposed peaks precede hospitalized peaks by about two weeks, and asymptomatic infections may remain after hospitalized cases fall (Deng et al., 2021).

A final misconception is that spatiotemporal epidemic realism is secured solely by adding diffusion. The recent literature instead separates at least three technical objectives: biological fidelity of the continuous model, analytical control of positivity and boundedness, and numerical preservation of those properties under discretization. The NSFD paper makes this especially explicit by showing that a method can capture spatial spread yet still generate nonphysical negative populations unless its discrete structure is designed to preserve the continuous model’s qualitative properties (Zinihi et al., 4 Aug 2025).

Taken together, these works define the spatiotemporal SEIQR epidemic model as a technically heterogeneous but conceptually coherent class: compartmental disease dynamics with an explicit quarantine-aware state structure, coupled to spatial transport through diffusion, commuting networks, or graph spillover, and often embedded in optimal-control or structure-preserving numerical frameworks.