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VECTRI Core Equations – Single Location

This document computes key VECTRI quantities for a single grid cell using explicit parameters and formulas, without classes or helper functions. You will learn to implement VECTRI equations from scratch and visualize the physical processes.

📋 Outline

This tutorial covers:

1. Imports

import math
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

2. Parameter Definitions

We define each parameter explicitly and comment its meaning.

2.1 Larval development

# ---------------------------
# Larval development
# ---------------------------

T_L_min = 16.0   # [°C] Minimum water temperature for larval development
K_L     = 90.9   # [degree-days] Degree-days needed to complete larval stage
T_L_max = 37.0   # [°C] Lethal upper water temperature (no larvae survive)

# Larval development equation:
#   R_L = (T_wat - T_L_min) / K_L    if T_L_min < T_wat < T_L_max, else 0
# where R_L is the fraction of the larval life cycle completed per day.

2.2 Larval survival, crowding, and flushing

# ---------------------------
# Larval survival
# ---------------------------

P_L_surv0   = 0.825  # Base daily larval survival (no crowding, no flushing)
M_L_max     = 300.0  # [mg m^-2] Larval biomass capacity (carrying capacity)
tau_flush   = 50.0   # [mm/day] Rainfall scale for flushing
K_flush_inf = 0.4    # Survival of early larvae under very heavy rain (asymptote)

# We will compute:
#   crowd_term = 1 - M_L / (w * M_L_max)        (bounded to [0,1])
#   P_L_surv_crowd = crowd_term * P_L_surv0     (crowding-only survival)
#   K_flush = L_f + (1 - L_f) * [ (1 - K_flush_inf)*exp(-R_d/tau_flush) + K_flush_inf ]
#   P_L_surv = P_L_surv_crowd * K_flush         (total survival)

2.3 Gonotrophic cycle (egg development)

# ---------------------------
# Gonotrophic cycle (egg development)
# ---------------------------

T_gono_min = 7.7    # [°C] Minimum temperature for egg development
K_gono     = 37.1   # [degree-days] Degree-days needed to complete gonotrophic cycle

# Gonotrophic rate:
#   R_gono = (T_eff - T_gono_min) / K_gono   if T_eff > T_gono_min, else 0
# Gonotrophic period (days) = 1 / R_gono

2.4 Sporogonic (parasite) cycle

# ---------------------------
# Sporogonic (parasite) cycle
# ---------------------------

T_sporo_min = 16.0  # [°C] Minimum temperature for parasite development
K_sporo     = 111.0 # [degree-days] Degree-days for sporogonic cycle (EIP)

# Parasite development rate:
#   R_sporo = (T_eff - T_sporo_min) / K_sporo   if T_eff > T_sporo_min, else 0
# Extrinsic incubation period (EIP, days) = 1 / R_sporo

2.5 Adult vector survival (Martens II) and lifespan

# ---------------------------
# Adult vector survival (Martens II)
# ---------------------------

K_mar2_0 = -4.4
K_mar2_1 =  1.31
K_mar2_2 = -0.03

# Daily survival probability:
#   den = K_mar2_0 + K_mar2_1 * T_eff + K_mar2_2 * T_eff^2
#   P_V_surv = exp(-1 / den)  if den > 0, else 0
#
# Approximate expected lifespan (days):
#   lifespan = 1 / (1 - P_V_surv)   if P_V_surv < 1, else infinite

2.6 Indoor temperature and effective mosquito temperature

# ---------------------------
# Indoor temperature & effective temperature
# ---------------------------

T0_indoor = 10.33  # [°C] Intercept for indoor temperature
K_indoor  = 0.58   # [-]  Slope relating outdoor T to indoor T

# Indoor temperature:
#   T_indoor = T0_indoor + K_indoor * T2m
#
# Effective temperature (fraction of time indoors = beta_indoor):
#   T_eff = beta_indoor * T_indoor + (1 - beta_indoor) * T2m

2.7 Host–vector system & transmission

# ---------------------------
# Host–vector system
# ---------------------------

tau_zoo = 50.0   # [people/km^2 or people] Scale for zoophily/anthropophily
P_hv    = 0.2    # P(mosquito infected | bite on infectious host)
P_vh    = 0.3    # P(host infected | infectious bite)

# Mean human biting rate:
#   hbr_bar = (1 - exp(-H / tau_zoo)) * V_biting / H
#
# Daily EIR:
#   EIR_d = hbr_bar * CSPR
#
# Vector -> host infection probability per day:
#   P_v_to_h = 1 - exp(-EIR_d * P_vh)

2.8 Simple pond hydrology (temporary ponds)

# ---------------------------
# Simple pond hydrology parameters
# ---------------------------

w_max = 0.04  # Max fractional pond coverage (4% of grid cell)
E     = 5.0   # [mm/day] Evaporation
I     = 245.0 # [mm/day] Infiltration
K_w   = 0.001 # Geometry/scale factor

# Pond fraction evolution (conceptual ODE):
#   dw/dt = K_w * [ rain * (w_max - w) - w * (E + I) ]
#
# We will store:
#   pond_inflow  = rain * (w_max - w_prev)
#   pond_outflow = w_prev * (E + I)
#   pond_balance = pond_inflow - pond_outflow   (mm/day equivalent)

3. Climate Time Series (Single Location)

We create a synthetic 180-day series of daily temperature and rainfall, roughly mimicking a warm tropical site with moderate rainfall variability.

# Time axis: 180 days
dates = pd.date_range("2025-01-01", periods=180, freq="D")
day_of_year = dates.dayofyear.values

# Synthetic air temperature [°C]: seasonal cycle around 24 °C
T2m = 24.0 + 3.0 * np.sin(2 * np.pi * (day_of_year / 365.0))

# Synthetic rainfall [mm/day]: random Gamma distribution (episodic wet days)
rng = np.random.default_rng(42)
rain = rng.gamma(shape=2.0, scale=3.0, size=len(dates))  # mean ~6 mm/day

climate_df = pd.DataFrame({"T2m": T2m, "rain": rain}, index=dates)
climate_df.head()

4. Location-Specific Constants

We hold these fixed over time for this demo. In a full VECTRI run, many of these are dynamic or spatially varying.

# Larval & hydrological state
M_L      = 3.0   # [mg m^-2] Larval biomass (assumed constant for simplicity)
L_f      = 0.25  # [-] Early larval stage (0 = very early, 1 = about to emerge)
delta_Tw = 1.5   # [°C] Water temperature offset relative to air

# Ponds: initial fraction
w0 = 0.0         # start completely dry

# Host & vector
H            = 200.0  # Humans in the cell
H_inf_frac   = 0.10   # Fraction infectious
V_biting     = 50.0   # Biting mosquitoes (number)
CSPR         = 0.10   # Circumsporozoite protein rate (infectious fraction)
beta_indoor  = 0.5    # Fraction of time mosquitoes spend indoors

H_inf = H * H_inf_frac  # number of infectious humans

5. Core Equations Applied Day by Day

We loop through each day and compute all quantities using the equations written explicitly above.

n = len(climate_df)

# Hydrology
w = np.zeros(n)           # pond fraction
w[0] = w0
pond_inflow  = np.zeros(n)  # mm/day equivalent inflow
pond_outflow = np.zeros(n)  # mm/day equivalent outflow
pond_balance = np.zeros(n)  # inflow - outflow

# Temperatures
Twat   = np.zeros(n)  # water temperature
T_ind  = np.zeros(n)  # indoor temperature
T_eff  = np.zeros(n)  # effective mosquito temperature

# Larval development & survival
R_L                = np.zeros(n)
larval_period_days = np.zeros(n)
K_flush            = np.zeros(n)
P_L_surv_crowd     = np.zeros(n)
P_L_surv           = np.zeros(n)

# Adult & parasite
R_gono      = np.zeros(n)
gono_period = np.zeros(n)
R_sporo     = np.zeros(n)
EIP_days    = np.zeros(n)
P_V_surv    = np.zeros(n)
lifespan    = np.zeros(n)  # approximate expected lifespan (days)

# Host-vector
hbr_bar = np.zeros(n)
EIR_d   = np.zeros(n)
P_h2v   = np.zeros(n)   # host -> vector infection prob per bite
P_v2h   = np.zeros(n)   # vector -> host infection prob per day

for i, (date, row) in enumerate(climate_df.iterrows()):
    # --- Climate inputs ---
    T2  = row["T2m"]
    Rd  = row["rain"]

    # --- Hydrology: pond fraction and water balance ---
    if i == 0:
        w_prev = w0
    else:
        w_prev = w[i-1]

    # Components of the water balance (mm/day equivalent)
    inflow  = Rd * (w_max - w_prev)
    outflow = w_prev * (E + I)
    pond_inflow[i]  = inflow
    pond_outflow[i] = outflow
    pond_balance[i] = inflow - outflow

    # ODE for pond fraction: dw/dt = K_w * (inflow - outflow)
    dw = K_w * (inflow - outflow)
    w[i] = w_prev + dw
    if w[i] < 0.0:
        w[i] = 0.0
    if w[i] > w_max:
        w[i] = w_max

    # --- Temperatures ---
    # Water temperature: T_wat = T2m + delta_Tw
    Tw = T2 + delta_Tw
    Twat[i] = Tw

    # Indoor temperature: T_ind = T0_indoor + K_indoor * T2m
    Ti = T0_indoor + K_indoor * T2
    T_ind[i] = Ti

    # Effective temperature: T_eff = beta_indoor*T_ind + (1 - beta_indoor)*T2m
    Teff = beta_indoor * Ti + (1.0 - beta_indoor) * T2
    T_eff[i] = Teff

    # --- Larval development ---
    if Tw <= T_L_min or Tw >= T_L_max:
        RL = 0.0
    else:
        RL = (Tw - T_L_min) / K_L
    R_L[i] = RL
    if RL > 0.0:
        larval_period_days[i] = 1.0 / RL
    else:
        larval_period_days[i] = math.inf

    # --- Flushing factor ---
    # K_flush = L_f + (1 - L_f) * [ (1 - K_flush_inf)*exp(-Rd / tau_flush) + K_flush_inf ]
    inner = (1.0 - K_flush_inf) * math.exp(-Rd / tau_flush) + K_flush_inf
    Kf = L_f + (1.0 - L_f) * inner
    K_flush[i] = Kf

    # --- Larval survival components ---
    if w[i] <= 0.0:
        # No ponds: no larvae survive
        P_L_surv_crowd[i] = 0.0
        P_L_surv[i]       = 0.0
    else:
        # crowd_term = 1 - M_L / (w * M_L_max)
        crowd_term = 1.0 - M_L / (w[i] * M_L_max)
        if crowd_term < 0.0:
            crowd_term = 0.0
        if crowd_term > 1.0:
            crowd_term = 1.0

        # Crowding-only survival
        PL_crowd = crowd_term * P_L_surv0
        if PL_crowd < 0.0:
            PL_crowd = 0.0
        if PL_crowd > 1.0:
            PL_crowd = 1.0
        P_L_surv_crowd[i] = PL_crowd

        # Total survival including flushing
        PL = PL_crowd * Kf
        if PL < 0.0:
            PL = 0.0
        if PL > 1.0:
            PL = 1.0
        P_L_surv[i] = PL

    # --- Gonotrophic cycle ---
    if Teff <= T_gono_min:
        Rg = 0.0
    else:
        Rg = (Teff - T_gono_min) / K_gono
    R_gono[i] = Rg
    if Rg > 0.0:
        gono_period[i] = 1.0 / Rg
    else:
        gono_period[i] = math.inf

    # --- Sporogonic cycle (parasite) ---
    if Teff <= T_sporo_min:
        Rs = 0.0
    else:
        Rs = (Teff - T_sporo_min) / K_sporo
    R_sporo[i] = Rs
    if Rs > 0.0:
        EIP_days[i] = 1.0 / Rs
    else:
        EIP_days[i] = math.inf

    # --- Adult vector survival and lifespan ---
    den = K_mar2_0 + K_mar2_1 * Teff + K_mar2_2 * (Teff**2)
    if den <= 0.0:
        PV = 0.0
    else:
        PV = math.exp(-1.0 / den)
    if PV < 0.0:
        PV = 0.0
    if PV > 1.0:
        PV = 1.0
    P_V_surv[i] = PV

    # Approx expected lifespan
    if PV >= 0.999:
        lifespan[i] = math.inf
    elif PV <= 0.0:
        lifespan[i] = 0.0
    else:
        lifespan[i] = 1.0 / (1.0 - PV)

    # --- Host-vector interactions ---
    # Mean human biting rate:
    if H <= 0.0:
        hbr = 0.0
    else:
        phi = 1.0 - math.exp(-H / tau_zoo)
        hbr = phi * V_biting / H
    hbr_bar[i] = hbr

    # Daily EIR: EIR_d = hbr_bar * CSPR
    eir = hbr * CSPR
    EIR_d[i] = eir

    # Host -> vector infection probability per blood meal:
    if H <= 0.0:
        Ph2v = 0.0
    else:
        Ph2v = (H_inf / H) * P_hv
    P_h2v[i] = Ph2v

    # Vector -> host daily infection probability:
    #   P_v2h = 1 - exp(-EIR_d * P_vh)
    Pv2h = 1.0 - math.exp(-eir * P_vh)
    P_v2h[i] = Pv2h

6. Pack Results into a DataFrame

ts_df = pd.DataFrame(
    {
        "T2m": T2m,
        "rain": rain,
        "w": w,
        "pond_inflow": pond_inflow,
        "pond_outflow": pond_outflow,
        "pond_balance": pond_balance,
        "Twat": Twat,
        "T_indoor": T_ind,
        "T_eff": T_eff,
        "R_L": R_L,
        "larval_period_days": larval_period_days,
        "K_flush": K_flush,
        "P_L_surv_crowd": P_L_surv_crowd,
        "P_L_surv": P_L_surv,
        "R_gono": R_gono,
        "gono_period_days": gono_period,
        "R_sporo": R_sporo,
        "EIP_days": EIP_days,
        "P_V_surv": P_V_surv,
        "lifespan_days": lifespan,
        "hbr": hbr_bar,
        "EIR_d": EIR_d,
        "P_h2v": P_h2v,
        "P_v2h": P_v2h,
    },
    index=dates,
)

ts_df.head()

7. Basic Summaries

print("=== Basic summaries over the 180 days ===")
print(f"Mean T2m                : {ts_df['T2m'].mean():.2f} °C")
print(f"Mean Twat               : {ts_df['Twat'].mean():.2f} °C")
print(f"Mean T_indoor           : {ts_df['T_indoor'].mean():.2f} °C")
print(f"Mean larval R_L         : {ts_df['R_L'].mean():.4f} frac/day")
print(f"Mean larval survival    : {ts_df['P_L_surv'].mean():.3f}")
print(f"Mean EIP (finite only)  : {ts_df['EIP_days'].replace(math.inf, float('nan')).mean():.2f} days")
print(f"Mean P_V_surv           : {ts_df['P_V_surv'].mean():.3f}")
print(f"Mean lifespan (finite)  : {ts_df['lifespan_days'].replace(math.inf, float('nan')).mean():.2f} days")
print(f"Mean hbr                : {ts_df['hbr'].mean():.3f} bites/person/day")
print(f"Mean daily EIR          : {ts_df['EIR_d'].mean():.4f} inf. bites/person/day")
print(f"Mean P_v2h (per day)    : {ts_df['P_v2h'].mean():.4f}")

8. Plot Templates and Physical Interpretation

Below are the code snippets used in the notebook to make physically meaningful plots. You can keep them here as templates or run them directly in a notebook.

8.1 Daily Temperatures (Air, Indoor, Effective, Water)

This figure shows how different temperatures relevant to VECTRI relate:

  • T2m: outdoor air temperature.
  • T_indoor: parameterized indoor temperature.
  • T_eff: effective temperature seen by mosquitoes (here 50% time indoors).
  • Twat: water temperature in breeding sites (slightly warmer than air).

These temperatures drive larval development, adult survival, and parasite development.

plt.figure(figsize=(11, 4))
plt.plot(ts_df.index, ts_df["T2m"], label="T2m (air)")
plt.plot(ts_df.index, ts_df["T_indoor"], label="T_indoor")
plt.plot(ts_df.index, ts_df["T_eff"], label="T_eff")
plt.plot(ts_df.index, ts_df["Twat"], label="Twat (water)", linestyle="--")
plt.ylabel("Temperature (°C)")
plt.xlabel("Date")
plt.title("Daily Temperature: Air, Indoor, Effective, and Water")
plt.legend()
plt.tight_layout()
plt.show()

8.2 Rainfall, Water Balance, and Pond Fraction (Breeding Habitat)

Here we show:

  • Rainfall (bars): incoming water from precipitation.
  • Pond inflow / outflow (lines): conceptual water balance terms.
  • Pond fraction w (line): fraction of the cell covered by temporary ponds.

Physically:

  • On days with heavy rain, inflow dominates → pond fraction increases toward w_max (here 4%).
  • In dry spells, outflow (evaporation + infiltration) dominates → ponds shrink and may disappear, reducing breeding habitat and carrying capacity.
fig, ax1 = plt.subplots(figsize=(11, 4))

ax1.bar(ts_df.index, ts_df["rain"], alpha=0.4, label="Rain (mm/day)")
ax1.set_ylabel("Rain (mm/day)")
ax1.set_xlabel("Date")

ax2 = ax1.twinx()
ax2.plot(ts_df.index, ts_df["pond_inflow"], label="Pond inflow", linestyle="-")
ax2.plot(ts_df.index, ts_df["pond_outflow"], label="Pond outflow", linestyle="--")
ax2.plot(ts_df.index, ts_df["w"], label="Pond fraction w", linewidth=2)
ax2.set_ylabel("Inflow/Outflow (mm/day) & w (fraction)")

lines1, labels1 = ax1.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax2.legend(lines1 + lines2, labels1 + labels2, loc="upper right")

plt.title("Rainfall, Water Balance, and Pond Fraction (Breeding Habitat)")
plt.tight_layout()
plt.show()

8.3 Larval Survival Components: Crowding, Flushing, and Total Survival

This figure shows:

  • P_L_surv_crowd: survival due to crowding and base survival only.
  • K_flush: flushing factor (1 = no extra mortality; <1 = some larvae washed out).
  • P_L_surv: total larval daily survival (crowding × flushing).

Physical interpretation:

  • When ponds are small or larval biomass is high, crowding reduces P_L_surv_crowd.
  • On heavy-rainfall days, K_flush drops, especially affecting early instars.
  • The combined effect P_L_surv determines how many larvae survive to the next day.
plt.figure(figsize=(11, 4))
plt.plot(ts_df.index, ts_df["P_L_surv_crowd"], label="P_L_surv_crowd (crowding only)")
plt.plot(ts_df.index, ts_df["K_flush"], label="K_flush (flushing factor)")
plt.plot(ts_df.index, ts_df["P_L_surv"], label="P_L_surv (total)")
plt.ylabel("Probability / Factor")
plt.xlabel("Date")
plt.ylim(0, 1.05)
plt.title("Larval Survival: Crowding, Flushing, and Total Survival")
plt.legend()
plt.tight_layout()
plt.show()

8.4 Development Times: Larval Period, Gonotrophic Period, and EIP

Here we plot:

  • Larval period (if finite): approximate days to complete larval development.
  • Gonotrophic period: days between blood meals and egg laying.
  • EIP (sporogonic period): days for the parasite to develop inside the mosquito.

Physically:

  • At warmer temperatures, all three periods shorten, increasing vector density and transmission potential.
  • In cooler conditions, the EIP can exceed typical mosquito lifespans, greatly reducing transmission.
plt.figure(figsize=(11, 4))
plt.plot(ts_df.index, ts_df["larval_period_days"], label="Larval period (days)", alpha=0.7)
plt.plot(ts_df.index, ts_df["gono_period_days"], label="Gonotrophic period (days)")
plt.plot(ts_df.index, ts_df["EIP_days"], label="EIP (parasite, days)")
plt.ylabel("Days")
plt.xlabel("Date")
plt.ylim(0, 40)
plt.title("Larval, Gonotrophic, and Sporogonic (EIP) Periods")
plt.legend()
plt.tight_layout()
plt.show()

8.5 Adult Vector Survival and Approximate Lifespan

We plot:

  • P_V_surv: daily survival probability of adult mosquitoes.
  • lifespan_days: approximate expected number of days an adult survives (\(1 / (1 - P_V_surv)\)).

Physical meaning:

  • When P_V_surv is high (e.g. >0.9), typical lifespans are >10 days, allowing mosquitoes to complete multiple gonotrophic cycles and EIP.
  • If P_V_surv drops (e.g. due to unfavorable temperature or interventions), lifespans shorten and many mosquitoes die before becoming infectious.
plt.figure(figsize=(11, 4))
plt.plot(ts_df.index, ts_df["P_V_surv"], label="P_V_surv (daily survival)")
plt.ylabel("Daily survival probability")
plt.xlabel("Date")
plt.ylim(0, 1.05)
plt.title("Adult Vector Daily Survival Probability")
plt.legend()
plt.tight_layout()
plt.show()

plt.figure(figsize=(11, 4))
lifesp = ts_df["lifespan_days"].replace(math.inf, np.nan)
plt.plot(ts_df.index, lifesp)
plt.ylabel("Expected lifespan (days)")
plt.xlabel("Date")
plt.title("Approximate Adult Mosquito Lifespan")
plt.tight_layout()
plt.show()

8.6 Transmission Metrics: Human Biting Rate, Daily EIR, and Vector → Host Infection Probability

This final figure focuses on transmission intensity:

  • hbr: mean human biting rate (bites per person per day).
  • EIR_d: daily entomological inoculation rate (infectious bites per person per day).
  • P_v2h: daily probability that a susceptible human is infected by mosquito bites.

Physical interpretation:

  • As vector density or infectious fraction (CSPR) increases, EIR_d and P_v2h rise.
  • Even modest daily infection probabilities (a few percent) can lead to very high annual infection pressure.
fig, ax1 = plt.subplots(figsize=(11, 4))

ax1.plot(ts_df.index, ts_df["hbr"], label="hbr (bites/person/day)")
ax1.set_ylabel("Bites / Infectious bites per person per day")
ax1.set_xlabel("Date")

ax2 = ax1.twinx()
ax2.plot(ts_df.index, ts_df["EIR_d"], label="EIR_d (infectious bites/person/day)", color="tab:orange")
ax2.plot(ts_df.index, ts_df["P_v2h"], label="P_v2h (vector→host infection prob)", color="tab:green")
ax2.set_ylabel("EIR / Infection probability")

lines1, labels1 = ax1.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax2.legend(lines1 + lines2, labels1 + labels2, loc="upper right")

plt.title("Human Biting Rate, Daily EIR, and Vector → Host Infection Probability")
plt.tight_layout()
plt.show()

9. Exercises

Practice implementing VECTRI equations with these exercises.

Exercise 1: Temperature Sensitivity Analysis

Objective: Explore how temperature affects larval development.

Task: 1. Calculate the larval development rate (R_L) for water temperatures ranging from 10°C to 40°C (use 1°C intervals) 2. Plot R_L vs temperature 3. Calculate the corresponding larval period (1/R_L) for each temperature 4. Identify the optimal temperature range for fastest development

Starter Code:

# Temperature range
temps = np.arange(10, 41, 1)
R_L_values = []

for T in temps:
    if T <= T_L_min or T >= T_L_max:
        R_L_values.append(0.0)
    else:
        R_L_values.append((T - T_L_min) / K_L)

# Plot your results here

Questions: - At what temperature is larval development fastest? - What happens at 16°C and 37°C? Why? - How many days does it take for larvae to develop at 25°C? At 30°C?

Exercise 2: Rainfall and Pond Dynamics

Objective: Understand how rainfall patterns affect breeding habitat availability.

Task: 1. Create three rainfall scenarios: - Scenario A: Constant daily rainfall (5 mm/day for 30 days) - Scenario B: Heavy episodic rain (20 mm every 5 days, 0 mm otherwise) - Scenario C: Dry spell (0 mm/day for 30 days) 2. For each scenario, calculate pond fraction (w) over time starting from w0 = 0.02 3. Plot all three scenarios on the same graph 4. Compare maximum pond coverage reached in each scenario

Starter Code:

# Parameters
n_days = 30
w_prev = 0.02

# Scenario A: Constant rain
rain_A = np.full(n_days, 5.0)

# Scenario B: Episodic rain
rain_B = np.zeros(n_days)
rain_B[::5] = 20.0  # Every 5th day

# Scenario C: Dry
rain_C = np.zeros(n_days)

# Calculate pond fraction for each scenario
# Your code here

Questions: - Which scenario produces the highest pond coverage? - How long does it take for ponds to disappear in Scenario C? - What is the equilibrium pond fraction in Scenario A?

Exercise 3: Adult Survival and Lifespan Across Temperatures

Objective: Explore the temperature-dependent survival of adult mosquitoes.

Task: 1. Calculate daily survival probability (P_V_surv) for effective temperatures from 10°C to 35°C 2. Calculate expected lifespan for each temperature 3. Plot both P_V_surv and lifespan vs temperature 4. Identify the optimal temperature for mosquito survival

Starter Code:

Teff_range = np.arange(10, 36, 1)
P_V_surv_vals = []
lifespan_vals = []

for Teff in Teff_range:
    den = K_mar2_0 + K_mar2_1 * Teff + K_mar2_2 * (Teff**2)
    if den <= 0.0:
        PV = 0.0
    else:
        PV = math.exp(-1.0 / den)

    # Calculate lifespan
    # Your code here

Questions: - At what temperature do mosquitoes live longest? - Compare lifespan at 20°C vs 30°C - Why does the relationship show a peak (bell-shaped curve)?

Exercise 4: EIP vs Mosquito Lifespan

Objective: Determine when parasites can complete development before mosquitoes die.

Task: 1. For temperatures 18°C to 32°C (2°C intervals): - Calculate EIP (sporogonic period) - Calculate mosquito lifespan 2. Create a table showing both values at each temperature 3. Identify the "transmission threshold" - lowest temperature where EIP < lifespan

Starter Code:

temps = np.arange(18, 33, 2)
results = []

for T in temps:
    # Calculate EIP
    if T <= T_sporo_min:
        EIP = np.inf
    else:
        R_sporo = (T - T_sporo_min) / K_sporo
        EIP = 1.0 / R_sporo

    # Calculate lifespan
    den = K_mar2_0 + K_mar2_1 * T + K_mar2_2 * (T**2)
    if den <= 0.0:
        lifespan = 0.0
    else:
        PV = math.exp(-1.0 / den)
        lifespan = 1.0 / (1.0 - PV) if PV < 0.999 else np.inf

    results.append({"Temp": T, "EIP": EIP, "Lifespan": lifespan})

df = pd.DataFrame(results)
print(df)

Questions: - At what temperature does EIP first become less than lifespan? - Why is this important for malaria transmission? - What happens at temperatures below 16°C?

Exercise 5: Sensitivity to Human Population Density

Objective: Explore how human population affects biting rates and transmission.

Task: 1. For human populations H = [10, 50, 100, 200, 500, 1000]: - Keep V_biting = 50, CSPR = 0.10 - Calculate mean human biting rate (hbr) - Calculate daily EIR - Calculate P_v2h 2. Plot hbr vs H 3. Explain the curve shape (hint: zoophily parameter tau_zoo)

Starter Code:

H_values = [10, 50, 100, 200, 500, 1000]
hbr_values = []

for H in H_values:
    phi = 1.0 - math.exp(-H / tau_zoo)
    hbr = phi * V_biting / H
    hbr_values.append(hbr)

# Plot and analyze

Questions: - Does biting rate per person increase or decrease with population? - Why does the curve flatten at high H values? - What is the biological meaning of tau_zoo?

Exercise 6: Combined Effects - Temperature and Rainfall

Objective: Understand how temperature and rainfall interact to affect transmission potential.

Task: 1. Create a 2×2 scenario matrix: - Hot (30°C) + Wet (10 mm/day) - Hot (30°C) + Dry (2 mm/day) - Cool (20°C) + Wet (10 mm/day) - Cool (20°C) + Dry (2 mm/day) 2. For each scenario, calculate over 60 days: - Mean pond fraction - Mean larval development rate - Mean EIP - Mean vector lifespan 3. Rank scenarios by "transmission favorability"

Questions: - Which scenario is most favorable for malaria transmission? - Can you have high transmission with low rainfall? Why or why not? - How does temperature compensate for or exacerbate low rainfall?

Challenge Exercise: Full Seasonal Cycle

Objective: Simulate a complete seasonal cycle and identify peak transmission periods.

Task: 1. Create a full year (365 days) of synthetic climate data with: - Seasonal temperature cycle (20°C to 30°C) - Seasonal rainfall (dry season: 1 mm/day, wet season: 15 mm/day) 2. Run the full VECTRI calculation 3. Calculate monthly mean EIR 4. Identify which months have highest transmission potential 5. Create a comprehensive visualization showing all key variables

Bonus: Add a lag analysis - do peak transmission months align with peak rainfall months?

10. Test Your Knowledge

Ready to test your understanding of VECTRI core equations and processes? Take the interactive quiz to assess your knowledge of model components, parameters, and physical interpretations.

Take the VECTRI Equations Quiz →

📚 Summary

In this tutorial, you've learned:

✅ VECTRI Core Parameters - All key parameters and their physical meanings
✅ Temperature Dependencies - How temperature affects development, survival, and transmission
✅ Hydrological Processes - Pond dynamics and breeding habitat availability
✅ Larval Dynamics - Development, survival, crowding, and flushing
✅ Adult Mosquito Processes - Gonotrophic cycle, survival, and lifespan
✅ Parasite Development - Sporogonic cycle and EIP
✅ Transmission Metrics - Biting rates, EIR, and infection probabilities
✅ Physical Interpretation - Understanding model outputs in real-world context

🔗 Additional Resources

  • VECTRI Model Components Guide
  • The VECTRI online documentation at VECTRI
  • The PDF manual VECTRI_manual_v1.6.pdf VECTRI PDF manual: Introductory sections describing motivation and history.
  • Tompkins & Ermert (2013) - A regional-scale, high resolution dynamical malaria model
  • Bayoh & Lindsay (2003) - Effect of temperature on the development of the aquatic stages of Anopheles gambiae
  • Martens et al. (1995) - Potential impact of global climate change on malaria risk

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