My Covid-19 ODE model
Used in my project. Modified based on SIR model. Author: Me with Claude(I asked for some suggestions)
Extended SEQIJR+A+V Model
State variables
| Variable | Description |
|---|---|
| $S$ | Susceptible |
| $V$ | Vaccinated |
| $E$ | Exposed (latent, infected but not yet infectious) |
| $Q_S$ | Quarantined, uninfected (close contact) |
| $Q_E$ | Quarantined, infected (latent) |
| $I$ | Symptomatic infectious (community) |
| $A$ | Asymptomatic infectious (community) |
| $J$ | Isolated confirmed |
| $R$ | Recovered |
| $N$ | Total population ($N' < 0$ due to deaths) |
Force of infection
$$\lambda = I + \varepsilon_A A + \varepsilon_J J + \varepsilon_E E + \varepsilon_E \varepsilon_Q Q_E$$
Parameters
| Parameter | Description | Category |
|---|---|---|
| $\beta$ | Base transmission rate per infectious contact per day | Transmission |
| $\varepsilon_E$ | Relative infectiousness of $E$ compared to $I$ | Transmission |
| $\varepsilon_A$ | Relative infectiousness of $A$ compared to $I$ | Transmission |
| $\varepsilon_J$ | Leakage of infectiousness from isolated confirmed $J$ | Transmission |
| $\varepsilon_Q$ | Leakage of infectiousness from quarantine | Transmission |
| $\varepsilon_V$ | Vaccine breakthrough coefficient (relative susceptibility) | Transmission |
| $\kappa_1$ | Inverse of latent period ($1/\text{days}$) | Duration |
| $\kappa_2$ | Inverse of quarantine period ($Q_S$ and $Q_E$ shared) | Duration |
| $\alpha_1$ | Exit rate from $I$ (recovery or death) | Duration |
| $\alpha_A$ | Recovery rate from $A$ | Duration |
| $\alpha_2$ | Exit rate from $J$ (recovery or death) | Duration |
| $\omega$ | Waning immunity rate for recovered | Duration |
| $\omega_V$ | Waning immunity rate for vaccinated | Duration |
| $v$ | Daily vaccination rate | Intervention |
| $\gamma_1$ | Contact tracing quarantine rate ($S$ and $E$ shared) | Intervention |
| $\gamma_2$ | Detection and isolation rate of symptomatic $I$ | Intervention |
| $\delta$ | Mass testing detection rate of asymptomatic $A$ | Intervention |
| $p$ | Probability of developing symptoms after infection | Branching |
| $d$ | Probability of testing positive during quarantine | Branching |
| $f_1$ | Recovery probability for $I$ ($1 - f_1$ = death) | Branching |
| $f_2$ | Recovery probability for $J$ ($1 - f_2$ = death) | Branching |
Equations
$$S' = -\beta(N) S \lambda - v S - \gamma_1 S + \kappa_2 Q_S + \omega R + \omega_V V$$
$$V' = v S - \varepsilon_V \beta(N) V \lambda - \omega_V V$$
$$E' = \beta(N) S \lambda + \varepsilon_V \beta(N) V \lambda - \kappa_1 E - \gamma_1 E$$
$$Q_S' = \gamma_1 S - \kappa_2 Q_S$$
$$Q_E' = \gamma_1 E - \kappa_2 Q_E$$
$$I' = p \kappa_1 E - \gamma_2 I - \alpha_1 I$$
$$A' = (1-p) \kappa_1 E + (1-p)(1-d) \kappa_2 Q_E - \delta A - \alpha_A A$$
$$J' = p \kappa_2 Q_E + (1-p) d \kappa_2 Q_E + \gamma_2 I + \delta A - \alpha_2 J$$
$$R' = f_1 \alpha_1 I + \alpha_A A + f_2 \alpha_2 J - \omega R$$
$$N' = -(1-f_1) \alpha_1 I - (1-f_2) \alpha_2 J$$
Notation remark: beta(N)=beta/N;
