PModel (Aedes Albopictus)

1. Overview

Model name: am_aedes_albopictus

Description: am_aedes_albopictus is as climate-driven model designed to...

Key features

• Uses climate and population datasets as inputs.
• Implements a system of differential equations representing mosquito development, mortality, and hatching rates.
• Outputs gridded estimates of adult mosquito abundance in NetCDF4 format.

Reference Publication:

• DOI: https://doi.org/10.1038/s43247-025-02199-z
• Supplementary material

---

2. Quickstart

How to run the model

1. Locate the file src/heiplanet_models/global_settings.yaml.

2. Modify the path where you have the data, for instance:

ingestion:
    path_root_datasets: "data/in/"

1. Run the Python script src/heiplanet_models/Pmodel/run_model.py

python run_model.py

---

3. Model Architecture

Inputs

Dataset	Description	Format
ERA5Land Temperature	Gridded temperature data	NetCDF4
ERA5Land Precipitation	Gridded precipitation	NetCDF4
Human population	Population distribution	NetCDF4
Initial conditions (opt)	Starting model state	NetCDF4

Outputs

Dataset	Description	Format
Mosquito Abundance	Gridded mosquito abundance for the Aedes Albopictus specie	NetCDF4

Workflow

You can find a dataflow diagram here.

4. Mathematical Model

Description

Six stage differential equation model

• Three aquatic stages:

  • Egg $E$
  • Diapaussing egg $E_{\text{dia}}$
  • Juvenile stage (Larval stage + Pupal stage)

• Three aerial stages:

  • Emerging adult $A_{\text{em}}$
  • Blood fed adults $A_{b}$
  • Ovipositing adults $A_{0}$

System of Equations

Differential Equation
$\dot{E} = \beta(T)(1-\omega(\overline{T}, \overline{S}))M - Q(W,P)\delta_{E}(T)E - m_{E}(T)E$
$\dot{E_{\text{dia}}} = \delta_{E}(T)\omega(\overline{T}, \overline{S})E_{\text{dia}} - \sigma(\overline{T},S)Q(W,P)\delta_{E}(T)E_{\text{dia}} - m_{Ed}(T)E_{\text{dia}}$
$\dot{J} = Q(W,P)\delta_{E}(T)E + \sigma(\overline{T},S)Q(W,P)\delta_{E}(T)E_{\text{dia}} - \delta_{J}(T)J - \left( \frac{J}{K_{J}(W,P)} + m_{J}(T) \right)J$
$\dot{A_{\text{em}}} = \frac{1}{2}\delta_{J}(T)J - \delta_{A_{\text{em}}}(T)A_{\text{em}} - m_{A}(T)A_{\text{em}}$
$\dot{A_{b}} = \delta_{A_{\text{em}}}(T)A_{\text{em}} + \delta_{A_{o}}A_{o} - \left( m_{A}(T) + r + \delta_{A_{b}}(T) \right)A_{b}$
$\dot{A_{o}} = \delta_{A_{b}}(T)A_{b} - \left( m_{A}(T) + r + \delta_{A_{o}}A_{o} \right)$

Variables and Parameters

State Variables

Variable	Description	Units
$E$	Egg population	individuals
$E_{\text{dia}}$	Diapaussing egg population	individuals
$J$	Juvenile population (larvae + pupae)	individuals
$A_{\text{em}}$	Emerging adult population	individuals
$A_{b}$	Blood fed adult population	individuals
$A_{o}$	Ovipositing adult population	individuals

Environmental Inputs

Variable	Description	Units
$T$	Temperature	°C
$S$	Photoperiod (day-light length). Using Forsythe model	hours
$W$	Precipitation/Rainfall	mm
$P$	Human population density	people/km²
$\overline{T}$	Mean temperature over previous 7 days	°C
$\overline{S}$	Mean day-light over previous 7 days	hours

Additional equations

Parameter	Description	Function	Units
$\underline{S}$	unknown probably a typo	unknown	unknown
$\underline{T}$	unknown probably a typo	unknown	unknown

Equations

Number	Function	Description	Units
1	$\beta(T)$	Egg per female per day	1/day
2	$\omega(\overline{T}, \overline{S})$	Diapausing egg proportion	NA
3	$Q(W,P)$	Hatching fraction depending in human density and rainfall	NA
4	$\delta_{E}(T)$	Egg development rate	1/day
5	$m_{E}(T)$	Egg mortality rate	1/day
6	$\sigma(\overline{T}, S)$	Spring hatching rate	1/day
7	$m_{Ed}(T)$	Diapausing egg mortality rate	1/day
8	$\delta_{J}(T)$	Juvenile development rate	1/day
9	$K_{J}(W,P)$	Juvenile Carrying Capacity	NA
10	$m_{J}(T)$	Juvenile Mortality rate	1/day
11	$\delta_{A_{em}}(T)$	Emerging adult development rate	1/day
12	$m_{A}(T)$	Adult Mortality rate	1/day
13	$\delta_{A_{o}}$	Subsequent blood meal rate after oviposition	1/day
14	$r$	Mortality rate associated with long distance travel and search behavior	1/day
15	$\delta_{A_{b}}(T)$	Blood fed adult development rate	1/day
16	$f(X)$	Sigmoidal "step-function"	dimensionless
17	$\check{S}_{a}$	Critical day-light length in autumn	hours
18	$\check{T}_{D}$	Critical diapause temperature (°C)	°C
x	$M$	unknown	unknown

1. Egg per female per day

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \beta(T) &= \max(-0.0163 + 1.2897T -15.837T^{2}, 0) \end{align} $$

Constant	Description	Typical Value
$k1$	---	-0.0163
$k2$	---	1.289
$k3$	---	-15.837

---

2. Diapausing egg proportion

Description: put a description here to improve or change the docstrings.

!!! warning The LaTeX equation derived from the Octave code should appear as follows.

$$
\omega(T, \text{lat}, t) =
\begin{cases}
0.5, & \text{if } S(\text{lat}, t) \leq \text{CPP}(\text{lat}) \text{ and } t > 183 \\
0,   & \text{otherwise}
\end{cases}
$$

$$ \begin{align} \omega(\underline{T}, \underline{S}) = 0.5 \times f\left(\underline{S} - \check{S}{a}\right)\, f\left(-\underline{T} - \check{T}{D}\right) \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.5

---

3. Hatching fraction depending in human density and rainfall

Description: put a description here to improve or change the docstrings.

$$ \begin{align} Q(W, P) = 0.8 \left( \frac{2.5\, e^{-0.05\,(W(t)-8)^2}}{e^{-0.05\,(W(t)-8)^2} + 1.5} \right) + 0.2 \left( \frac{0.01}{0.01 + e^{-0.01 P(t)}} \right) \end{align} $$

Constant	Description	Typical Value
$E_{\text{opt}}$ taken from octave code water_hatch.m	---	8
$E_{\text{var}}$ taken from octave code water_hatch.m	---	0.05
$E_{\text{0}}$ taken from octave code water_hatch.m	---	1.5
$E_{\text{rat}}$ taken from octave code water_hatch.m	---	0.02
$E_{\text{dens}}$ taken from octave code water_hatch.m	---	0.01
$E_{\text{fac}}$ taken from octave code water_hatch.m	---	0.01

---

4. Egg development rate

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \delta_{E}(T) = 0.5070\left( - \left( \frac{T-30.85}{12.82} \right)^2 \right) \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.5070
$k2$	---	30.85
$k3$	---	12.82

---

5. Egg mortality rate

Description: put a description here to improve or change the docstrings.

$$ \begin{align} m_{E}(T) = -\ln\left( 0.955\, \exp\left[ -0.5 \left( \frac{T - 18.8}{21.53} \right)^{6} \right] \right) \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.955
$k2$	---	0.5
$k3$	---	18.8
$k4$	---	21.53

---

6. Spring hatching rate

Description: put a description here to improve or change the docstrings.

!!! warning The LaTeX equation derived from the Octave code should appear as follows.

$$
\begin{cases}
\text{ratio}_{\text{dia\_hatch}}, & \text{if } \overline{T}_{\text{last 7 days}} \geq \text{CTT} \text{ and } D(\text{lat}, t) \geq \text{CPP} \\
0, & \text{otherwise}
\end{cases}
$$

$$ \begin{align} \sigma(\overline{T}, S) = 0.1 \times f(\check{T} - \underline{T})\, f(-\check{S}_{s} - S) \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.1

---

7. Diapausing egg mortality rate

Description: put a description here to improve or change the docstrings.

!!! warning The constants in the code are different than constants reported on the paper.

$$ \begin{align} m_{Ed}(T) = m_{E}(T) = -\ln\left( 0.955\, \exp\left[ -0.5 \left( \frac{T - 18.8}{21.53} \right)^{6} \right] \right) \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.955
$k1$	---	0.5
$k3$	---	18.8
$k4$	---	21.53

---

8. Juvenile development rate

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \delta_{J}(T) = \frac{1}{0.08T^{2} - 4.89T + 83.85} \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.08
$k2$	---	-4.89
$k3$	---	83.85

---

9. Juvenile carrying capacity

Description: put a description here to improve or change the docstrings.

$$ \begin{align} K_{J}(W,P) = \lambda\,\frac{0.1}{1 - 0.9^{t}}\sum_{x=1}^{t} 0.9^{(t-x)}\left(\alpha_{\text{rain}}W(x) + \alpha_{\text{dens}}P(x)\right) \end{align} $$

Constant	Description	Typical Value
$\lambda$	Scaling factor of carrying capacity	$10^6$
$t$	---	-
$\alpha_{\text{rain}}$	Weight for rainfall contribution	$10^{-3}$
$W(x)$	Rainfall at time step $x$	-
$\alpha_{\text{dens}}$	Weight for population contribution	$10^{-5}$
$P(x)$	Humand density population at time step $x$	-
$\gamma$ taken from octave code capacity.m	---	0.9

---

10. Juvenile mortality rate

Description: put a description here to improve or change the docstrings.

!!! warning

Equation in paper do not show the exponential function $exp()$. However, the exponential function is used in the Octave code.

$$ \begin{align} m_{J}(T) = -\ln\left[\,0.977\, \exp\left(-0.5 \left(\frac{T - 21.8}{16.6}\right)^{6}\right)\,\right] \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.977
$k2$	---	0.5
$k3$	---	21.8
$k4$	---	16.6

---

11. Emerging adult development rate

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \delta_{A_{em}}(T) = \frac{1}{0.069T^2 - 3.574T + 50.1} \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.069
$k2$	---	3.574
$k3$	---	50.1

---

12. Adult mortality rate

Description: put a description here to improve or change the docstrings.

!!! warning

The equation reported on paper do not show the exponential $exp()$ found in the Octave/Matlab code. Additionally the equation in Octave should look like this.

$$
m_{A}(T) =
\begin{cases}
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\, T^{0.1}\,\right], & \text{if } T > 0 \\[1.5ex]
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\,\right], & \text{if } T \leq 0
\end{cases}
$$

!!! warning

$T_{\text{mean}}$ is mentioned here for first time. The possible description about this process is this one:

*"Daily time-step simulations are used to solve the model. The resulting output is first aggregated to monthly time steps and then further aggregated to yearly values. To represent diurnal temperature variation, the simulation for each day is divided into 100 time steps according to the numerical solver used (the deSolve package in R), ranging from 0.14, 0.19, …, up to 24.00 hours. The temperature at each of these time points is used to simulate daily development and mortality rates. For additional details on the simulation procedure and a simple example of the model implementation in Octave (v4.2.1), see Metelmann et al.⁵˒¹⁸. Finally, spatial aggregation at the NUTS-3 level (Europe) was performed in R version 4.1 using the raster package."*

$$
m_{A}(T) =
\begin{cases}
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\, T^{0.1}\,\right], & \text{if } T > 0 \\[1.5ex]
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\,\right], & \text{if } T \leq 0
\end{cases}
$$

$$ \begin{align} m_{A}(T_{\text{mean}}) = -\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T_{\text{mean}} - 20.9}{13.2}\right)^{6}\right)\, (T_{\text{mean}})^{0.1}\,\right] \end{align} $$

Constant	Description	Typical Value
$k1$	---	0.677
$k2$	---	0.5
$k3$	---	20.9
$k4$	---	13.2
$k5$	---	0.1

---

13. Subsequent blood meal rate after oviposition

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \delta_{A_{o}} = k_{1} \end{align} $$

Constant	Description	Typical Value
$k_{1}$	---	0.1

---

14. Mortality rate associated with long distance travel and search behavior

Description: put a description here to improve or change the docstrings.

$$ \begin{align} r = k_{1} \end{align} $$

Constant	Description	Typical Value
$k_{1}$	---	0.8

---

15. Blood fed adult development rate

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \delta_{Ab} = \frac{T-10}{77}+0.2 \end{align} $$

Constant	Description	Typical Value
$k_{1}$	---	10
$k_{2}$	---	77
$k_{3}$	---	0.2

---

16. Sigmoidal "step function"

Description: put a description here to improve or change the docstrings.

$$ \begin{align} f(X)=\frac{1}{1+e^{20X}} \end{align} $$

Constant	Description	Typical Value
$k_{1}$	---	20

17. Critical day-light length in autumn

Description: put a description here to improve or change the docstrings.

$$ \begin{aligned} \check{S}_{a}=10.058+0.08965 \times Latitude(degrees) \end{aligned} $$

Constant	Description	Typical Value
$k_{1}$	---	10.058
$k_{2}$	---	0.08965

---

18. Critical diapause temperature (°C)

Description: put a description here to improve or change the docstrings.

$$ \begin{align} \check{T}{D} = k{1} \end{align} $$

Constant	Description	Typical Value
$k_{1}$	---	21

---

x. unknown 1

Description: put a description here to improve or change the docstrings.

!!! warning This M appears in the Egg $E$ differential equation; however, it is not listed in Table S11 of Supplementary material.

$$ \begin{align} M = ? \end{align} $$

---

Equations

!!! warning This is just a demo table.

Stage	Symbol	Description	Python Function	Units
Birth	$\omega(\underline{T}, \underline{S})$	Diapausing egg proportion	mosq_dia_lay	N/A
Birth	$\sigma(\underline{T}, S)$	Spring hatching rate	mosq_dia_hatch	$\frac{1}{day}$
Birth	$Q(W,P)$	Hatching fraction (rainfall & population)	water_hatching	N/A
Development	$K_L(W,P)$	Juvenile carrying capacity	carrying_capacity	N/A
Development	$\delta_J(T)$	Juvenile development rate	mosq_dev_j	$\frac{1}{day}$
Development	$\delta_{Aem}(T)$	Emerging adult development rate	mosq_dev_i	$\frac{1}{day}$
Mortality	$m_E(T)$	Egg mortality rate	mosq_mort_e	$\frac{1}{day}$
Mortality	$m_J(T)$	Juvenile mortality rate	mosq_mort_j	$\frac{1}{day}$
Mortality	$m_A(T_{mean})$	Adult mortality rate	mosq_mort_a	$\frac{1}{day}$

---

5. Examples

TODO: Add some plots

---

Authors & Contact

Here is a markdown table template for author information:

Author	GitHub Username	Email	Affiliation
Robert Koch	@rkochdeutschland	robert.koch@koch-institute.de	Robert Koch Institute

---

[LegacyCode]

Workflow description

a. Entry point name: aedes_albopictus_model.m

1. Create variables to assign the prefix of each dataset that will be used matlab tmean_prefix = 'ERA5land_global_t2m_daily_0.5_'; pr_prefix = 'ERA5land_global_tp_daily_0.5_'; dens_prefix = 'pop_dens_';

2. Create a for loop to iterate over years. The following operations are done in each loop:

   a. The name for each dataset is completed with the year and extension: matlab tmean = [tmean_prefix, num2str(years), '.nc']; pr = [pr_prefix, num2str(years), '.nc']; dens = [dens_prefix, num2str(years), '_global_0.5.nc'];

   b. Log the year that is being processed: matlab year = years; % change years according to file name disp(['Year in Process: ', num2str(years)]);

   c. Run the model that calculates the differential equations. This model receives the following datasets and variable:

   • dataset tmean: mean temperature dataset
   • dataset pr: precipitation dataset
   • dataset dens: population density dataset
   • variable year: year of the dataset

   matlab %model_run(tmax,tmin,tmean, pr,dens,year); model_run(tmean, pr,dens,year); % for tmean only with NO DTR (Tmax or Tmin is not present) disp(['Year done: ', num2str(years)]); 3. End the program

b. Name: model_run.m

1. Create a variable name for the output file. ```matlab outfile = strcat('Mosquito_abundance_Global_', num2str(year), '.nc') % change the output file name as required

if exist(outfile) == 2 delete(outfile) end 2. Copy the dataset that contains the precipitation (one of the ERA5*.nc) and store this copy using the variable name for the output file.matlab copyfile(pr, outfile); % overwriting the file gives us advantage that we can loose the ;latitude and longitude information of Tmax, % Tmin file and only work with time varible in 3rd dimension and then rewrite this over same nc file 3. Rename the internal variable (`pr`) of this copy with the name `adults`. It means, we have a new .nc file that contains `longitude`, `latitude` and the variable name `adults`. With this we can just overwrite the variable adult with the calculations we intend to do.matlab ncid = netcdf.open(outfile,'NC_WRITE'); netcdf.reDef(ncid) netcdf.renameVar(ncid,3,'adults'); netcdf.endDef(ncid); netcdf.close(ncid); ```

1. Defines a delta time step that will be important for solving the system of differential equations. matlab step_t = 10;

2. Load variables from each dataset involved (tmean, pr, dens)

   a. Load temperature variables and applies some preprocessing steps with load_temp2(tmean, step_t) function matlab [Temp, Tmean] = load_temp2(tmean, step_t); % Without DTR if Tmean is only available, no Tmax or Tmin

   b. Load population density variables and applies some preprocessing steps with load_hdp(dens) matlab DENS = load_hpd(dens);

   c. Load Precipitation variable with load_rainfall(pr) function matlab PR = load_rainfall(pr);

   d. Load Latitude variable from the tmean dataset with the function load_latitude(tmean) matlab LAT = load_latitude(tmean);

3. Calculate Juvenile Carrying Capacity: $K_{L}(W,P)$

   • $W$: Rainfall accumulation, this the information we get from PR

   • $P$: Human density, this is the information we get from DENS

   matlab CC = capacity(PR, DENS);

   - file associated: `capacity.m`
   
   - Equation in Supplement Information: 14
   

4. Calculate Hatching fraction depending in human density and rainfall: $Q(W,P)$ $$ Q(W,P) = (1 - E_{rat}) \left( \frac{(1+E_{0})e^{\left(-E_{var}(W(t)-E_{opt})\right)^2}}{e^{\left( -E_{var} (W(t)-E_{opt})^2 \right)}} \right) + E_{rat} \left( \frac{E_{dens}}{E_{dens} + e^{-E_{fac}P}} \right) $$

   Where:

   • $W$: Precipitation, we get this information from PR

   • $P$: Human density, we get this information from DENS

   • $E_{opt}$ = 8;
   • $E_{var}$ = 0.05;
   • $E_{0}$ = 1.5;
   • $E_{rat}$= 0.2;
   • $E_{dens}$ = 0.01;
   • $E_{fac}$ = 0.01;

   matlab egg_active = water_hatch(PR, DENS); - file associated: water_hatch.m - Equation in Supplementary Information: 13

5. Create a Vector Initial population: $V_{0}$ matlab previous = 'no_previous'; v0 = load_initial(previous, size(Temp));

   • File associated: load_initial.m
   • TODO: Ask about the name and meaning of this variable.

6. Calculate the parameters for each time step and run ODEs matlab v = call_func(v0, Temp, Tmean, LAT, CC, egg_active, step_t);

   Input variables:

   • initial vector: v0
   • temperature: Temp
   • mean temperature: Tmean
   • latitude: LAT
   • juvenile carrying capacity: CC
   • hatching fraction depending in human density and rainfall: egg_active
   • time step: step_time

   • File associated: call_func.m

   • Once all the variables are calculated, write to the output file. matlab ncwrite(outfile,'adults', permute(v(:,:,5,:),[1,2,4,3]));

c. Name: call_func.m

Description: This file the main file that contains the ODEs to calculate the 6 differential equations proposed in the paper.

Input Variables:

• v
• Temp
• Tmean
• LAT
• CC
• egg_activate
• step_t

Process:

1. Calculate the Diapause Lay

   • File associated: mosquito_dia_lay.m
   • TODO: ask about the equation.

2. Calculate the Diapause Hatch

   • File associated: mosquito_dia_hatch.m
   • TODO: ask about the equation

3. Calculate Diapausing egg mortality rate : $m_{Ed}(T)$ $$ m_{Ed}(T) = -\ln\left( 0.955 \left(-0.5 \left(\frac{T-18.8}{21.53}\right)^6 \right) \right) $$

   • Input Variables:
     • Temp
     • step_t
   • File associated: mosq_surv_ed.m
   • Comments: hardcoded values do not correspond to the values in the paper.

4. Create a n-dimensional matrix v_out full of zeros.

5. Create a for loop that iterates over the time:

   5.1. Creates a temperatures vector

   octave T = Temp(:,:,t);

   5.2. Calculates the Mosquito Birth

   octave birth = mosq_birth(T);

   • Inputs:
     • T

   • File associated: mosq_birth.m

   • Comments:

     • TODO: This equation is not present in the suplement material.

   5.3. Calculates the Mosquito development j

   • Inputs:
     • T
   • File associated: mosq_dev_j.m
   • Comments:
     • TODO: This equation is not present in the suplement material.

   octave dev_j = mosq_dev_j(T);

   5.4. Calculates the Mosquito development i

   • Inputs:
     • T
   • File associated: mosq_dev_i.m
   • Comments:
     • TODO: This equation is not present in the suplement material.

   octave dev_j = mosq_dev_i(T);

   5.4. Other calculations ```octave dev_e = 1./7.1; %original function of the model dia_lay = diapause_lay(:,:,ceil(t/step_t)); dia_hatch = diapause_hatch(:,:,ceil(t/step_t)); ed_surv = ed_survival(:,:,t); water_hatch = egg_activate(:,:,ceil(t/step_t)); mort_e = mosq_mort_e(T); mort_j = mosq_mort_j(T);

   T = Tmean(:,:,ceil(t/step_t)); mort_a = mosq_mort_a(T); ```

   5.5. ODE octave vars = {ceil(t/step_t), step_t, Temp, CC, birth, dia_lay, dia_hatch, mort_e, mort_j, mort_a, ed_surv, dev_j, dev_i, dev_e, water_hatch}; v = RK4(@eqsys, @eqsys_log, v, vars, step_t); % Runge-Kutta 4 method % v = FE(@eqsys, @eqsys_log, v, vars, step_t); % Forward Euler method

   5.6. Additional calculations ```octave if mod(t/step_t,365) == 200 v(:,:,2) = 0; end

   if mod(t,step_t) == 0 if mod(ceil(t/step_t),30) == 0 disp(['MOY: ', num2str(t/step_t/30)]); end for j = 1:5 %v_out(:,:,j,t/step_t) = v(:,:,j); v_out(:,:,j,t/step_t) = max(v(:,:,j), 0); % if any abundance is negative it will make it zero end end

   ```

Table 1. Mapping Equations from Octave/Matlab code to Python code

!!! note

Numbers in brackets refer to the corresponding row numbers in [Table S11 from the Supplementary Material](https://static-content.springer.com/esm/art%3A10.1038%2Fs43247-025-02199-z/MediaObjects/43247_2025_2199_MOESM2_ESM.pdf).

Octave Function	Python Function	Equation Number in Paper
mosq_dev_j.m	mosq_dev_j	[5]
mosq_dev_i.m	mosq_dev_i	[6]
mosq_dev_e.m	mosq_dev_e	[NR 1]
capacity.m	carrying_capacity	[14]
mosq_mort_e.m	mosq_mort_e	[9]
mosq_mort_j.m	mosq_mort_j	[10]
mosq_mort_a.m	mosq_mort_a	[11]
mosq_surv_ed.m	mosq_surv_ed	[NR 2]
mosq_birth.m	mosq_birth	[NR 3]
mosq_dia_hatch.m	mosq_dia_hatch	[3]
mosq_dia_lay.m	mosq_dia_lay	[2]
water_hatch.m	water_hatching	[13]

Table 2. Climate sensitive parameter description and functions (inspired on Table S11 in Supplementary information)

!!! warning

Put special attention to the Not Reported (`[NR <number>]`) equations and the Not Reported units (`[NA]`).

Number	Description	Symbol	Unit
[5]	Juvenile development rate	$\delta_{J}(T)$	$\frac{1}{day}$
[6]	Emerging adult development	$\delta_{Aem}(T)$	$\frac{1}{day}$
[NR 1]	(tentative) Emerging adult development Briere model.		[NR 1]
[14]	Juvenile carrying capacity	$K_{J}(W,P)$	NA
[9]	Egg mortality rate	$m_{E}(T)$	$\frac{1}{day}$
[10]	Juvenile mortality rate	$m_{J}(T)$	$\frac{1}{day}$
[11]	Adult mortality rate	$m_{A}(Tmean)$	$\frac{1}{day}$
[NR 2]	(tentative) Diapausing egg mortality rate		$\frac{1}{day}$
[NR 3]	Not reported	Not reported	[NR 3]
[3]	(tentative) Spring hatching rate	$\sigma(\underline{T},S)$	$\frac{1}{day}$
[2]	(tentative) Diapausing egg proportion	$\omega(\underline{T}, \underline{S})$	NA
[13]	Hatching fraction depending in human density and rainfall	$Q(W,P)$	NA

List of equations to understand mapping between code and paper

• [5] Juvenile development rate: $\delta_{J}(T)$ $$ \delta_{J}(T) = \frac{1}{0.08T^{2} - 4.89T + 83.85} $$

  Parameter	Description
  $T$	Temperature (°C)

• [6] Emerging adult development rate: $$ \delta_{Aem}(T) = \frac{1}{0.069T^{2} - 3.574T + 50.1} $$

  Parameter	Description
  $T$	Temperature (°C)

• [NR 1] (tentative) Emerging adult development Briere model $$ BM = q \cdot T \cdot (T - T_0) \cdot \sqrt{T_m - T} $$

  Parameter	Description
  $q$	Empirical coefficient for development rate
  $T$	Temperature (°C)
  $T_0$	Minimum threshold temperature (°C)
  $T_m$	Maximum threshold temperature (°C)

• [14] Juvenile carrying capacity: $$ K_{J}(W,P) = \lambda\,\frac{0.1}{1 - 0.9^{t}}\sum_{x=1}^{t} 0.9^{(t-x)}\left(\alpha_{\text{rain}}W(x) + \alpha_{\text{dens}}P(x)\right) $$

  Parameter	Description
  $\lambda$	Scaling coefficient
  $t$	---
  $x$	---
  $\alpha_{\text{rain}}$	Weight for rainfall contribution
  $W(x)$	Rainfall at time step $x$
  $\alpha_{\text{dens}}$	Weight for population contribution
  $P(x)$	Population at time step $x$

• [9] Egg mortality rate:

!!! warning

Equation in paper do not show the exponential function $exp()$. However, the exponential function is used in the Octave code.

$$ m_{E}(T) = -\ln\left( 0.955\, \exp\left[ -0.5 \left( \frac{T - 18.8}{21.53} \right)^{6} \right] \right) $$

Parameter	Description
$T$	Temperature (°C)

• [10] Juvenile mortality rate:

!!! warning

Equation in paper do not show the exponential function $exp()$. However, the exponential function is used in the Octave code.

$$ m_{J}(T) = -\ln\left[\,0.977\, \exp\left(-0.5 \left(\frac{T - 21.8}{16.6}\right)^{6}\right)\,\right] $$

Parameter	Description
$T$	Temperature (°C)

• [11] Adult mortality rate:

!!! warning

The equation reported on paper do not show the exponential $exp()$ found in the Octave/Matlab code. Additionally the equation in Octave should look like this.

$$
m_{A}(T) =
\begin{cases}
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\, T^{0.1}\,\right], & \text{if } T > 0 \\[1.5ex]
-\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T - 20.9}{13.2}\right)^{6}\right)\,\right], & \text{if } T \leq 0
\end{cases}
$$

$$ m_{A}(T_{\text{mean}}) = -\ln\left[\,0.677\, \exp\left(-0.5 \left(\frac{T_{\text{mean}} - 20.9}{13.2}\right)^{6}\right)\, (T_{\text{mean}})^{0.1}\,\right] $$

Parameter	Description
$T_{\text{mean}}$	Temperature (°C)

• [NR 2] (tentative) Diapausing egg mortality rate: !!! warning The constants in the code are different than constants reported on the paper.

$$ m_{Ed}(T) = m_{E}(T) = -\ln\left( 0.955\, \exp\left[ -0.5 \left( \frac{T - 18.8}{21.53} \right)^{6} \right] \right) $$

Parameter	Description
$T$	Temperature (°C)

• [NR 3] !!! warning Missing name

  The following formula has been deducted from octave code

  $$ b(T) = \begin{cases} 33.2\, \exp\left(-0.5 \left(\frac{T - 70.3}{14.1}\right)^2\right)\, (38.8 - T)^{1.5}, & \text{if } T < 38.8 \[1.5ex] 0, & \text{if } T \geq 38.8 \end{cases} $$

• [3] Spring hatching rate

!!! warning The equation in Octave should look like this.

$$
\begin{cases}
\text{ratio}_{\text{dia\_hatch}}, & \text{if } \overline{T}_{\text{last 7 days}} \geq \text{CTT} \text{ and } D(\text{lat}, t) \geq \text{CPP} \\
0, & \text{otherwise}
\end{cases}
$$

$$ \sigma(\underline{T}, S) = 0.1 \times f(\check{T} - \underline{T})\, f(-\check{S}_{s} - S) $$

Parameter	Description
$\check{T}$	missing description
$\underline{T}$	missing description
$\check{S}_{s}$	Critical day-light length in spring (hours)
$S$	Day-light (hours)

• [2] (tentative) Diapausing egg proportion !!! warning The equation in Octave should look like this.

  $$ \omega(T, \text{lat}, t) = \begin{cases} 0.5, & \text{if } S(\text{lat}, t) \leq \text{CPP}(\text{lat}) \text{ and } t > 183 \ 0, & \text{otherwise} \end{cases} $$

$$ \omega(\underline{T}, \underline{S}) = 0.5 \times f\left(\underline{S} - \check{S}{a}\right)\, f\left(-\underline{T} - \check{T}{D}\right) $$

Parameter	Description
$\underline{T}$	missing description
$\underline{S}$	missing description
$\check{S}_{a}$	Critical day-light length in autumn, $\check{S}_{a}=10.058+0.08965 \times Latitude(degrees)$ (hours)
$\check{T}_{D}$	Critical diapause temperature (°C)

• [13] Hatching fraction depending in human density and rainfall:

$$ Q(W, P) = 0.8 \left( \frac{2.5\, e^{-0.05\,(W(t)-8)^2}}{e^{-0.05\,(W(t)-8)^2} + 1.5} \right) + 0.2 \left( \frac{0.01}{0.01 + e^{-0.01 P(t)}} \right) $$

Constants	Description
$W$	Precipitation or Rainfall (mm)
$P$	Human population density ($\frac{\text{people}}{\text{km}^2}$)