Kyasanur Forest Disease(KFD) Decoding Infection and Transmission Dynamics
SI (Monkey), SIR (Human), and SI (Mammalian Hosts and Ticks Vector) Models
Model Architecture
Differential Equations
Ticks (Larva–Nymph–Adult)
$$ \begin{aligned} \frac{d l_S}{dt} &= (f_T+f_h+f_r)\!\left(\frac{E_p p_h}{\tau} f m (1-f)\right)a_S - r_1 p_S m_S (1-\eta) l_S G - \frac{r_2 \beta_T p_I}{N_p} l_S - \mu_T l_S, \\ \frac{d n_S}{dt} &= r_1 p_S m_S (1-\eta) l_S G - G(1-\varepsilon) n_S - \frac{r_2 \beta_T p_I}{N_p} n_S - \mu_T n_S, \\ \frac{d a_S}{dt} &= G(1-\varepsilon) n_S - (f_T+f_h+f_r)\!\left(\frac{E_p p_h}{\tau} f m (1-f)\right)a_S - \mu_T a_S, \\ \frac{d l_I}{dt} &= (f_T+f_h+f_r)\!\left(\frac{E_p p_h}{\tau} f m (1-f)\right)a_I - r_3 p_S m_S l_I G - \delta_T l_I, \\ \frac{d n_I}{dt} &= r_3 p_S m_S l_I G - G n_I - \delta_T n_I, \\ \frac{d a_I}{dt} &= G n_I + \frac{r_2 \beta_T p_I}{N_p} n_S - (f_T+f_h+f_r)\!\left(\frac{E_p p_h}{\tau} f m (1-f)\right)a_I - \delta_T a_I \end{aligned} $$Primates (Monkeys)
$$ \begin{aligned} \frac{d P_S}{dt} &= B_P - \left( \frac{\beta_{lP} l_I \theta}{N_T} + \frac{\beta_{DP}(P_I+P_D)}{N_P} \right) P_S - \mu_P P_S, \\ \frac{d P_I}{dt} &= \left( \frac{\beta_{lP} l_I \theta}{N_T} + \frac{\beta_{DP}(P_I+P_D)}{N_P} \right) P_S - \delta_P P_I - \mu_P P_I, \\ \frac{d P_D}{dt} &= \delta_P P_I - \kappa P_D \end{aligned} $$Mammals
$$ \begin{aligned} \frac{d M_S}{dt} &= B_M - \left( \frac{\beta_{am} a_I \theta}{N_T} + \frac{\beta_{DM}(P_I+P_D)}{N_P} \right) M_S - \mu_M M_S, \\ \frac{d M_I}{dt} &= \left( \frac{\beta_{am} a_I \theta}{N_T} + \frac{\beta_{DM}(P_I+P_D)}{N_P} \right) M_S - \delta_M M_I - \mu_M M_I \end{aligned} $$Humans
$$ \begin{aligned} \frac{d H_S}{dt} &= B_H - \left( \frac{\beta_{nH} n_I \theta}{N_T} + \frac{\beta_{MH} M_I}{N_M} + \frac{\beta_{DH} P_D}{N_P} \right)(1-\alpha) H_S - \alpha H_S - \mu_H H_S, \\ \frac{d H_I}{dt} &= \left( \frac{\beta_{nH} n_I \theta}{N_T} + \frac{\beta_{MH} M_I}{N_M} + \frac{\beta_{DH} P_D}{N_P} \right)(1-\alpha) H_S - \gamma H_I - \delta_H H_I - \mu_H H_I, \\ \frac{d H_R}{dt} &= \gamma H_I + \gamma_H H_V - \mu_H H_R, \\ \frac{d H_V}{dt} &= \alpha H_S - \gamma_H H_V - \mu_H H_V \end{aligned} $$Model Parameters
| SL No | Parameter | Description |
|---|---|---|
| 1 | H_S, M_S, P_S | Susceptible population of humans, mammals, and primates |
| 2 | H_I, M_I, P_I | Infectious population of humans, mammals, and primates |
| 3 | l_S, n_S, a_S | Susceptible population of larva, nymph, and adult ticks |
| 4 | l_I, n_I, a_I | Infected population of larva, nymph, and adult ticks |
| 5 | H_V, P_D, H_R | Vaccinated humans (H_V), carcasses of dead monkeys (P_D), recovered humans (H_R) |
| 6 | r_1, r_2, r_3 | Contact rates between tick stages and monkey/mammal hosts: r1 (S-vector→S-host), r2 (S-vector→I-host), r3 (I-vector→I-host) |
| 7 | alpha, gamma | Human vaccination rate (α) and recovery rate (γ) |
| 8 | mu_T, mu_M, mu_P, mu_H | Natural death rates of tick, mammals, primates, and humans |
| 9 | B_M, B_P, B_H | Birth rates of mammals, primates, and humans |