PPR Scenario

Decoding of Infection And Transmission Dynamics (PPR)

This section contains: Model architecture, Differential equations, and Equation parameterization.

Model Architecture

Architecture image not found at team\ppr(architecture).jpg.
Put the PPR architecture image at that path (recommended: team/ppr(architecture).jpg).

Differential Equations

X: Small Ruminants (Primary host)

$$ \begin{aligned} \frac{dS_X}{dt} &= b_X - \left(\beta_X \frac{I_X}{N} S_X \cdot f_{env}\right) - \omega S_X + \alpha_1 B_X - \mu_X S_X + \epsilon V_X,\\ \frac{dE_X}{dt} &= \left(\beta_X \frac{I_X}{N} S_X \cdot f_{env}\right) - \sigma E_X + \alpha_2 B_X - \mu_X E_X,\\ \frac{dI_X}{dt} &= \sigma E_X - \left(\theta_1 \gamma_1 I_X + \theta_2 \gamma_2 I_X\right) + \alpha_3 B_X - (\mu_X + \delta_X) I_X,\\ \frac{dR_X}{dt} &= \theta_1 \gamma_1 I_X + \theta_2 \gamma_2 I_X - \mu_X R_X + \lambda V_X,\\ \frac{dV_X}{dt} &= \omega S_X - \mu_X V_X - \lambda V_X - \epsilon V_X,\\ \frac{dB_X}{dt} &= \kappa_X - (\alpha_1 + \alpha_2 + \alpha_3 + \mu_X) B_X. \end{aligned} $$

Note: $R_{X/M}$ in your architecture corresponds to the recovered/immune compartment (here shown as $R_X$). Keep the same symbol in your simulator for consistency.

Y: Wild Small Ruminants (Secondary host)

$$ \begin{aligned} \frac{dS_Y}{dt} &= b_Y - \left(\beta_{XY}\frac{I_X}{N} S_Y \cdot f_{env}\right) - \mu_Y S_Y,\\ \frac{dI_Y}{dt} &= \left(\beta_{XY}\frac{I_X}{N} S_Y \cdot f_{env}\right) - \mu_Y I_Y. \end{aligned} $$

Z: Atypical Hosts

$$ \begin{aligned} \frac{dS_Z}{dt} &= b_Z - \left(\beta_{XZ}\frac{I_X}{N} S_Z \cdot f_{env}\right) - \mu_Z S_Z,\\ \frac{dI_Z}{dt} &= \left(\beta_{XZ}\frac{I_X}{N} S_Z \cdot f_{env}\right) - \mu_Z I_Z. \end{aligned} $$

Equation Parameterization

S/N	Parameter	Description
1	b_X	Birth rate of primary host (X: small ruminants)
2	b_Y	Birth rate of secondary host (Y: wild small ruminants)
3	b_Z	Birth rate of atypical hosts (Z)
4	kappa_X	Constant flow rate of small ruminants into restocked population (B_X)
5	beta_X	Effective contact rate / PPR transmission rate (within X)
6	beta_{XY}	Transmission rate from infected primary host (X) to secondary host (Y)
7	beta_{XZ}	Transmission rate from infected primary host (X) to atypical host (Z)
8	S_X	Density of susceptible primary host
9	E_X	Density of exposed primary host
10	I_X	Density of infected primary host
11	R_{X/M}	Density of recovered/immune primary host
12	B_X	Restocked / quarantined primary host class
13	S_Y	Density of susceptible secondary host (wild small ruminants)
14	I_Y	Density of infected secondary host (wild small ruminants)
15	S_Z	Density of susceptible atypical host
16	I_Z	Density of infected atypical host
17	mu_X	Natural mortality rate of primary host
18	mu_Y	Natural mortality rate of secondary host
19	mu_Z	Natural mortality rate of atypical hosts
20	omega	Rate at which susceptible primary hosts are vaccinated (S_X \rightarrow V_X)
21	1-epsilon	Vaccine efficacy (fraction effectively protected)
22	delta_X	Disease-induced mortality rate (primary host)
23	lambda	Rate of recovery from vaccination / immune acquisition from V_X
24	sigma	Progression rate: exposed becomes infectious (E_X \rightarrow I_X)
25	gamma_1	Recovery rate from non-isolated population
26	gamma_2	Recovery rate from isolated population
27	theta_2	Rate (fraction) of isolation
28	theta_1	Rate (fraction) of non-isolation
29	omega_r	Fraction of recruited/restocked small ruminants that are vaccinated (if used in your simulator)
30	alpha_1	Flow rate from B_X to S_X
31	alpha_2	Flow rate from B_X to E_X
32	alpha_3	Flow rate from B_X to I_X
33	N	Minimum number of animals required to sustain spread / scaling population size
34	f_{env}	Environment/fomite-mediated transmission scaling (as used in your equations)
35	epsilon	Vaccine waning/return to susceptibility term used in equations (keep consistent with your model notation)

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