Leptospirosis Simulation

Decoding of Infection And Transmission Dynamics

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

Model Architecture

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Differential Equations

Humans

$$ \begin{aligned} \frac{dS_h}{dt} &= \Lambda_h + \omega_h R_h - \beta_{hb} S_h \frac{L_b}{k+L_b} - \beta_{hh} I_h S_h - \beta_{hr} I_r S_h - \beta_{ha} I_a S_h - \mu_h S_h \\ \frac{dE_h}{dt} &= \beta_{hb} S_h \frac{L_b}{k+L_b} + \beta_{hh} I_h S_h + \beta_{hr} I_r S_h + \beta_{ha} I_a S_h - \sigma_h E_h - \mu_h E_h \\ \frac{dI_h}{dt} &= \sigma_h E_h - r_h I_h - (\mu_h+\alpha)I_h \\ \frac{dR_h}{dt} &= r_h I_h - \omega_h R_h - \mu_h R_h \end{aligned} $$

Animals

$$ \begin{aligned} \frac{dS_a}{dt} &= \Lambda_a + \omega_a R_a - \beta_{ab} S_a \frac{L_b}{k+L_b} - \beta_{aa} I_a S_a - \mu_a S_a \\ \frac{dI_a}{dt} &= \beta_{ab} S_a \frac{L_b}{k+L_b} + \beta_{aa} I_a S_a - r_a I_a - \mu_a I_a \\ \frac{dR_a}{dt} &= r_a I_a - \omega_a R_a - \mu_a R_a \end{aligned} $$

Rodents

$$ \begin{aligned} \frac{dS_r}{dt} &= \Lambda_r - \beta_{rb} S_r \frac{L_b}{k+L_b} - \beta_{rr} I_r S_r - \mu_r S_r \\ \frac{dI_r}{dt} &= \beta_{rb} S_r \frac{L_b}{k+L_b} + \beta_{rr} I_r S_r - \mu_r I_r \end{aligned} $$

Environment

$$ \begin{aligned} \frac{dL_b}{dt} &= \eta_a I_a + \eta_r I_r - (\varepsilon+\tau)L_b \end{aligned} $$

Equation Parameterization

Sl. No	Parameter	Description
1	b₁	Human recruitment rate
2	b₂	Animal/Vector recruitment rate
3	Ω	Human recruitment rate
4	Ωₑ	Rodent recruitment rate
5	β₁	Human-to-human transmission/contact rate
6	β₂	Vector-to-human transmission/contact rate
7	β₃	Human-to-vector transmission/contact rate
8	β₁₁	Environmental-to-human transmission rate
9	β₁₂	Environmental-to-rodent transmission rate
10	β₂₁	Rodent-to-human transmission rate
11	β₂₂	Rodent-to-rodent transmission rate
12	α	Human incubation/progression rate (E → I)
13	τ	Rodent incubation/progression rate (Eₑ → Iₑ)
14	γ	Human recovery rate
15	γₕ	Human recovery transition (Iₕ → Rₕ)
16	λₕ	Recovery rate of infected persons
17	σ	Loss of immunity (waning immunity)
18	μ	Human natural death rate
19	μₕ	Human natural mortality rate
20	ε	Rodent natural death rate
21	γᵥ	Vector natural mortality rate
22	δ	Human disease-induced mortality
23	δₕ	Human disease fatality rate
24	δᵥ	Vector disease-induced mortality
25	δᵣ	Rodent disease-induced mortality
26	μₗ	Environmental Leptospira removal rate
27	C	Environmental Leptospira concentration
28	η₁	Contribution of infected humans/hosts to environmental Leptospira
29	η₂	Contribution of infected rodents to environmental Leptospira

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