SISHD Model: Epidemic & Insurance Dynamics

• The SISHD model is a compartmental epidemic framework that extends the SIS paradigm by incorporating hospitalized individuals and tracking disease-induced mortality.
• It mathematically delineates population dynamics through differential equations that quantify transitions among susceptible, infectious, hospitalized, and deceased states.
• The model’s analytical structure underpins actuarial applications by enabling precise health insurance pricing based on epidemic metrics.

The SISHD model is a compartmental epidemic framework designed to extend the classical susceptible-infectious-susceptible (SIS) paradigm by explicitly incorporating hospitalization for treatment and disease-induced mortality. The model aims to capture epidemiological dynamics critical for actuarial applications, such as health insurance pricing, by providing a more realistic representation of disease progression and its economic consequences (Do et al., 5 Jan 2026).

1. Mathematical Formulation of the SISHD Model

The SISHD model stratifies the population at time $t$ into four compartments:

• $S(t)$: Number of susceptibles,
• $I(t)$: Number of infectious (not hospitalized),
• $H(t)$: Number of hospitalized,
• $D(t)$: Cumulative number of disease-induced deaths.

The total living population is $N(t) = S(t) + I(t) + H(t)$. The system dynamics are governed by the following system of ordinary differential equations: $$\begin{cases} \dot S = \Lambda - \beta I S - \beta \epsilon H S + \alpha_I I + \alpha_H H - \mu S, \ \dot I = \beta I S + \beta \epsilon H S - (\alpha_I + \delta + \gamma_I + \mu)I, \ \dot H = \delta I - (\alpha_H + \gamma_H + \mu) H, \ \dot D = \gamma_I I + \gamma_H H. \end{cases}$$ where the parameters (all strictly positive) are:

• $\Lambda$: Recruitment (birth) rate,
• $\mu$: Natural death rate,
• $\beta$: Transmission rate from $I$ to $S$,
• $\epsilon \in [0,1]$: Relative infectiousness of $H$ versus $I$,
• $\alpha_I$: Recovery rate from $I$ to $S$,
• $\delta$: Hospitalization rate ($I \to H$),
• $\gamma_I$: Disease-induced death rate of $I$,
• $\alpha_H$: Discharge (recovery) rate from $H$ to $S$,
• $\gamma_H$: Disease-induced death rate of $H$.

This system generalizes the classical SIS model by accounting for hospitalized individuals (compartment $H$) with distinct transmission characteristics and explicitly tracking cumulative disease-related deaths ($D$), thereby enriching the model’s applicability to scenarios with significant morbidity and mortality.

2. Reproduction Number and Threshold Behavior

The basic reproduction number $\mathcal R_0$ provides a threshold criterion for the persistence or extinction of the infection. Using the next-generation matrix methodology, considering infected compartments $X = (I, H)$, one obtains: $$\mathcal R_0 = \frac{\beta\,(\Lambda/\mu)\,[(\alpha_H + \gamma_H + \mu) + \epsilon \delta]}{(\alpha_I + \delta + \gamma_I + \mu)(\alpha_H + \gamma_H + \mu)}.$$ The term $\mathcal R_0$ encapsulates contributions from both non-hospitalized and hospitalized infectives, modulated by their relative infectiousness and transition rates. $\mathcal R_0 < 1$ corresponds to eventual disease eradication, while $\mathcal R_0 > 1$ implies endemicity. The analytical derivation leverages the structure of the linearized system at the disease-free equilibrium.

3. Equilibria and Stability Analysis

Two principal equilibria are established for the SISHD model:

• Disease-Free Equilibrium (DFE):

$$E_0 = \left( S_0 = \frac{\Lambda}{\mu},\ I_0 = 0,\ H_0 = 0,\ D_0 = 0 \right).$$

This equilibrium is locally and globally asymptotically stable when $\mathcal R_0 < 1$ within the biologically feasible region ($S, I, H \geq 0;\ S+I+H \leq \Lambda/\mu$). For $\mathcal R_0 > 1$, $E_0$ becomes unstable.

• Endemic Equilibrium (EE):

When $\mathcal R_0 > 1$, a unique endemic equilibrium exists,

$E_* = (S_*, I_*, H_*, D_*),$

with components

$$S_* = \frac{(\alpha_I + \delta + \gamma_I + \mu)(\alpha_H + \gamma_H + \mu)}{\beta \left[ (\alpha_H + \gamma_H + \mu) + \epsilon \delta \right]},$$

$$I_* = \frac{\Lambda - \mu S_*}{(\gamma_I + \delta + \mu) - \frac{\alpha_H \delta}{\alpha_H + \gamma_H + \mu}},$$

$$H_* = \frac{\delta I_*}{\alpha_H + \gamma_H + \mu}.$$

The disease-induced deaths term $D_*$ is determined such that $\dot D = 0$. The local asymptotic stability of $E_*$ is established via the Routh–Hurwitz criterion. Numerical evidence supports global attractivity of $E_*$ in the interior whenever $\mathcal R_0 > 1$.

4. Health Insurance Pricing Framework

The SISHD model enables direct actuarial application through explicit modeling of epidemiological states relevant to benefit triggers and liabilities. Under the equivalence principle, the level premium rate $\pi$ (per time-unit, per susceptible) is set so that the expected present value (PV) of premiums equals the expected PV of benefit outgo, over a fixed time-horizon $T$: $$\pi \int_0^T S(t)\,dt = b_I \int_0^T I(t)\,dt + b_H \int_0^T H(t)\,dt + d \int_0^T D'(t)\,dt,$$ where

• $b_I$: per-unit time benefit to infectious (non-hospitalized),
• $b_H$: per-unit time benefit to hospitalized,
• $d$: lump-sum death benefit.

As $D'(t) = \gamma_I I(t) + \gamma_H H(t)$,

$$\pi = \frac{b_I \int_0^T I(t)\,dt + b_H \int_0^T H(t)\,dt + d \int_0^T [\gamma_I I(t) + \gamma_H H(t)]\,dt}{\int_0^T S(t)\,dt}.$$

Reserve adequacy is defined via

$$V(t) = \pi \int_t^T S(\tau)\,d\tau - \left[ b_I \int_t^T I(\tau)\,d\tau + b_H \int_t^T H(\tau)\,d\tau + d \int_t^T (\gamma_I I(\tau) + \gamma_H H(\tau))\,d\tau \right],$$

with the admissible premium satisfying $V(t) \geq 0$ for all $t \in [0, T]$: $$\pi^* = \max_{t \in [0, T]} \frac{b_I \int_t^T I(\tau)\,d\tau + b_H \int_t^T H(\tau)\,d\tau + d \int_t^T [\gamma_I I(\tau) + \gamma_H H(\tau)]\,d\tau}{\int_t^T S(\tau)\,d\tau}.$$ This formulation permits explicit insurance pricing under dynamic morbidity and mortality patterns generated by the SISHD dynamics.

5. Numerical Exploration and Validation

Simulations were conducted using a classical fourth-order Runge–Kutta method for the reduced $(S,I,H)$ system over a one-year horizon with step size $h=10^{-3}$. Two distinct parameter regimes were explored:

• Subcritical Regime ($\mathcal R_0 < 1$): Five parameter sets $\{A_1, \ldots, A_5\}$, exhibiting convergence to the DFE $(\Lambda/\mu, 0, 0)$ from various initial conditions, confirming theoretical predictions of global eradication.
• Supercritical Regime ($\mathcal R_0 > 1$): Five sets $\{B_1, \ldots, B_5\}$, with each trajectory converging to its respective endemic equilibrium $E_*$. Time-series exhibit initial transient growth in $I$ and $H$ before stabilization.

For example, for parameter set $B_2$:

• $\Lambda=20$, $\mu=0.02$, $\beta=0.000253$, $\epsilon=0.25$, $\alpha_I=0.03$, $\gamma_I=0.06$, $\delta=0.03$, $\gamma_H=0.04$, $\alpha_H=0.01$,
• $\mathcal R_0 \approx 2.0$,
• $E_* \approx (500, 94.6, 40.5)$.

The corresponding insurance pricing ($b_I=1$, $b_H=20$, $d=100$) yielded a zero-profit premium $\pi \approx 3.68$ and a minimal solvency premium $\pi^* \approx 5.67$. Only premiums $\pi \geq \pi^*$ maintained non-negative reserves $V(t)$ throughout. These results empirically validate analytical thresholds and demonstrate the utility of SISHD-informed actuarial pricing.

6. Interpretation and Significance

The SISHD framework advances the mathematical epidemiology of insurance-linked risks by explicitly representing hospitalization and mortality within epidemic models. This enables more granular, state-contingent assessment of both disease and financial dynamics versus standard SIS or SIR models. The analytical tractability of equilibrium and threshold analyses ensures rigor in both theoretical and applied settings. Notably, the integration of SISHD outputs into explicit premium and reserve formulas ensures actuarial solvency and fair pricing in rapidly evolving epidemiological contexts, with numerical methods validating theoretical predictions for a broad range of scenarios (Do et al., 5 Jan 2026).

The approach provides an adaptable toolset for stakeholders navigating insurance pricing and reserve management in the presence of endemic or epidemic disease, underpinning both risk assessment and contractual design in health insurance products.