Leptospirosis Simulation — Kerala Districts

Multi-host SEIRS + Environmental Reservoir · RK4 · Analytical R₀

Quick Guide

Enter district population under Humans / Animals / Rodents — susceptible compartments auto-calculate. R₀ recomputes live as you edit any parameter. The stacked bar chart shows the epidemic composition month-by-month. Click any parameter label to read its meaning.

Parameters

Click parameter name to see meaning

Population — Compartmental Model Input

👥 Human Population (N_h)

Total humans at risk. Susceptible S_h auto-computes as N_h − E_h − I_h − R_h.

N_h (total) S_h (auto) E_h (exposed) I_h (infectious) R_h (recovered)

🐄 Animal Population (N_a)

Cattle & livestock (20th Livestock Census). S_a auto = N_a − I_a − R_a.

N_a (total) S_a (auto) I_a (infectious) R_a (recovered)

🐀 Rodent Population (N_r)

Estimated from field surveys (Kartik et al. 1998). S_r auto = N_r − I_r.

N_r (total) S_r (auto) I_r (infectious) Prevalence % (auto)

💧 Environmental Load (L_b)

Leptospira concentration in water (copies/mL). Range 100–1000.

Simulation Control

Simulation days (T)

RK4 integration horizon, dt=1 day.

100

Recruitment (Auto-computed)

Λ_h — Human Recruitment Rate

1.49

Λ_h = (crude birth rate ÷ 365,000) × N_h. Crude birth rate = 11.06/1000 (Kerala 2023).

Auto-compute from N_h

Λ_a — Animal Recruitment Rate

1.20

Λ_a = μ_a × N_a (steady-state assumption).

Auto-compute from N_a

Λ_r — Rodent Recruitment Rate

2.00

Λ_r = μ_r × N_r (steady-state assumption).

Auto-compute from N_r

Transmission — Human (β)

β_hb — Human infection rate from environment

3.15e-4

1/(copies·day). Range 0.000252 – 0.00378 (Engida 2025).

β_hh — Human-to-human transmission rate

9.8e-8

Zhang et al. 2024. Very low direct contact route.

β_hr — Rodent-to-human transmission rate

1.1e-7

Transmission rate from rodents to humans.

β_ha — Animal-to-human transmission rate

2.4e-7

Transmission rate from animals to humans.

Transmission — Animal (β)

β_ab — Animal infection rate from environment

5.0e-4

Transmission rate from environment to animals.

β_aa — Animal-to-animal transmission rate

4.0e-6

Direct animal-to-animal transmission rate.

Transmission — Rodent (β)

β_rb — Rodent infection rate from environment

1.8e-3

Transmission rate from environment to rodents.

β_rr — Rodent-to-rodent transmission rate

7.0e-8

Direct rodent-to-rodent transmission rate.

Progression & Recovery

σ_h — Human progression rate (incubation)

0.0909

σ_h = 1/incubation days. Range 0.091–0.131 (7–11 days).

r_h (γ_h) — Human recovery rate

0.100

Recovery rate for humans (1/duration).

r_a (γ_a) — Animal recovery rate

0.060

Recovery rate for animals.

ω_h — Human waning immunity rate

1.85e-3

Rate at which recovered humans become susceptible again.

ω_a — Animal waning immunity rate

3.7e-3

Rate at which recovered animals become susceptible again.

Mortality

μ_h — Human natural mortality rate

3.68e-5

Natural mortality rate for humans.

μ_a — Animal natural mortality rate

1.83e-4

Natural mortality rate for animals.

μ_r — Rodent natural mortality rate

1.83e-3

Natural mortality rate for rodents.

α — Disease-induced human mortality rate

5.3e-3

Disease-induced mortality rate for humans.

Environmental

k — Half-saturation constant

10000

Half-saturation constant for environmental transmission.

ε — Leptospira decay rate in environment

0.1125

Natural decay rate of Leptospira in the environment.

τ — Removal rate due to sanitation

0.330

Removal rate due to sanitation/interventions.

Shedding to Environment (η)

η_a — Animal shedding rate to env.

0.100

Shedding rate of Leptospira by infected animals.

η_r — Rodent shedding rate to env.

0.500

Shedding rate of Leptospira by infected rodents.

⚙️ Optimal Control Measures (u₁–u₄)

Enable Control Measures

Toggle ON to run with-vs-without comparison using Pontryagin's Maximum Principle controls.

u₁ — Personal protection / reduced exposure

0.30

Reduces environmental exposure β_hb and β_ab by factor (1−u₁). Range [0,1]. Higher = stronger protection (PPE, boots, awareness).

0 (none)1 (full)

u₂ — Treatment of infected humans

0.25

Accelerates recovery of I_h → R_h (antibiotics, hospitalization). Range [0,1].

0 (none)1 (full)

u₃ — Treatment/vaccination of animals

0.20

Removes infected animals proportional to human cases (u₃·I_h). Range [0,1].

0 (none)1 (full)

u₄ — Rodent control (culling/trapping)

0.15

Removes rodents (S_r and I_r) proportional to human cases (u₄·I_h). Range [0,1].

0 (none)1 (full)

📊 Cost Weights (w₁–w₄) & Infection Weights (A₁–A₃)

w₁ (protection cost) w₂ (treatment cost) w₃ (animal vax cost) w₄ (rodent ctrl cost) A₁ (human inf. weight) A₂ (animal inf. weight) A₃ (rodent inf. weight)

R₀ summary + derived indicators + Equilibrium

Peak day: —

R₀ (results) —

Basic reproduction number (β, γ, μ...)

Herd immunity threshold

—

HIT = 1 − 1/R₀

Preparedness Advisory

—

S_h E_h I_h R_h I_a I_r L_b (×0.01) Log Y Day 0

Results — Compartment Sizes (Final Day / Endemic)

Susceptible Humans (S_h*)

—

Exposed Humans (E_h*)

—

Infectious Humans (I_h*)

—

Recovered Humans (R_h*)

—

Infectious Animals (I_a*)

—

Infectious Rodents (I_r*)

—

Environmental L_b*

—

Cumulative Deaths (α · I_h)

—

CFR: —

Cumulative I_h (attack rate)

—

— % of N_h

Peak infected (I_h)

—

Time to peak

—

Epidemic Start/End

—

Endemic starts

—

I_h curve (Infectious Humans)

Markers: Peak (red), Endemic start (green), Epidemic start (blue)

What-If: vary initial Infectious Humans (I_h) and compare curves

Theoretical Endemic Equilibrium Point (E*)

S_h*

—

E_h*

—

I_h*

—

R_h*

—

L_b*

—

S_a*

—

I_a*

—

R_a*

—

S_r*

—

I_r*

—

📐 Differential Equations (click to view)

HUMANS S'_h = Λ_h + ω_h R_h − β_hb S_h L_b/(k+L_b) − β_hh I_h S_h − β_hr I_r S_h − β_ha I_a S_h − μ_h S_h E'_h = β_hb S_h L_b/(k+L_b) + β_hh I_h S_h + β_hr I_r S_h + β_ha I_a S_h − σ_h E_h − μ_h E_h I'_h = σ_h E_h − γ_h I_h − (μ_h + α) I_h R'_h = γ_h I_h − ω_h R_h − μ_h R_h ANIMALS S'_a = Λ_a + ω_a R_a − β_ab S_a L_b/(k+L_b) − β_aa I_a S_a − μ_a S_a I'_a = β_ab S_a L_b/(k+L_b) + β_aa I_a S_a − γ_a I_a − μ_a I_a R'_a = γ_a I_a − ω_a R_a − μ_a R_a RODENTS S'_r = Λ_r − β_rb S_r L_b/(k+L_b) − β_rr I_r S_r − μ_r S_r I'_r = β_rb S_r L_b/(k+L_b) + β_rr I_r S_r − μ_r I_r ENVIRONMENT L'_b = η_a I_a + η_r I_r − (ε + τ) L_b ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WITH OPTIMAL CONTROL (u₁, u₂, u₃, u₄) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ HUMANS S'_h = Λ_h + ω_h R_h − (1−u₁)β_hb S_h L_b/(k+L_b) − β_hh I_h S_h − β_hr I_r S_h − β_ha I_a S_h − μ_h S_h E'_h = (1−u₁)β_hb S_h L_b/(k+L_b) + β_hh I_h S_h + β_hr I_r S_h + β_ha I_a S_h − σ_h E_h − μ_h E_h I'_h = σ_h E_h − r_h I_h − (μ_h + α) I_h − u₂ I_h R'_h = r_h I_h + u₂ I_h − ω_h R_h − μ_h R_h ANIMALS S'_a = Λ_a + ω_a R_a − (1−u₁)β_ab S_a L_b/(k+L_b) − β_aa I_a S_a − μ_a S_a I'_a = β_ab S_a L_b/(k+L_b) + β_aa I_a S_a − r_a I_a − μ_a I_a − u₃ I_h R'_a = r_a I_a + u₃ I_h − ω_a R_a − μ_a R_a RODENTS S'_r = Λ_r − β_rb S_r L_b/(k+L_b) − β_rr I_r S_r − μ_r S_r − u₄ I_h I'_r = β_rb S_r L_b/(k+L_b) + β_rr I_r S_r − μ_r I_r − u₄ I_h ENVIRONMENT L'_b = η_a I_a + η_r I_r − (ε + τ) L_b OBJECTIVE FUNCTIONAL J(u₁,u₂,u₃,u₄) = ∫₀ᵀ [A₁I_h + A₂I_a + A₃I_r·F + ½ Σ wⱼuⱼ²] dt

📊 Year-wise Simulation Validation — Thiruvananthapuram

How Hindcast Validation Works (RK4 Method):
For each historical year: (1) The district's Animal Population (N_a) is used. (2) A year-specific transmission rate (β_ab) is estimated from observed incidence. (3) The full ODE system is solved using RK4 for 365 days. (4) Predicted = Average Monthly I_a.

📂 District Historical Data

Built-in District Data

Select a district to validate.

Upload Custom District CSV

Required columns: Year, Observed_Attacks.

Display Name:

Year From

Year To

Run validation to see hindcast validation against historical outbreaks.

Quick Guide

R₀ summary + derived indicators + Equilibrium

Preparedness Advisory

⚙️ Optimal Control — With vs Without Comparison

📊 Year-wise Simulation Validation — Thiruvananthapuram