simplying approach.md

Author: paigemiller

docs/approach.md

@@ -1,64 +1,46 @@
 # Using Next Generation Matrices
 
-The NGM for a 4-Group Infectious Disease Model with compartments $S_g$, $I_g$, $R_g$: Susceptible, Infected, and Recovered compartments in the $G$ groups $g = 1, \dots, G$. Where possible we try to be general, otherwise we will take $G = 4$
+The NGM for a 4-Group Infectious Disease Model with compartments $S_i$, $I_i$, $R_i$: Susceptible, Infected, and Recovered compartments.
 
-Dynamics for $I_g$ in each group given by:
+Dynamics for $I_i$ in each group given by:
 
 $$
-\frac{d I_g}{dt} = \sum_{j} \frac{\beta_{jg} S_j I_j}{N_j} - \gamma_g I_g
+\frac{d I_i}{dt} = \sum_{j} \frac{\beta_{ij} S_i I_j}{N_j} - \gamma_i I_i
 $$
 
 where:
 
-- $\beta_{jg}$: Transmission rate from group $j$ to group $g$,
+- $\beta_{ij}$: Transmission rate from group $j$ to group $i$,
 - $S_j$: Susceptible population in group $j$,
 - $N_j$: Total population in group $j$,
-- $\gamma_g$: Recovery rate in group $g$.
+- $\gamma_i$: Recovery rate in group $i$.
 
 The NGM is calculated at the disease free equilibrium (DFE) where
 
 $$
-I_g = 0, S_g = N_g \  \text{for all\ } g
+I_i = 0, S_i = N_i \  \text{for all\ } i
 $$
 
----
-
 And then the NGM `R` is given by:
 
 $$
-\mathbf{R} = \mathbf{F} \mathbf{V}^{-1}
-$$
-
-where $\mathbf{F}$ is the matrix of new infections and $\mathbf{V}$ is the matrix of transitions between compartments, not representing new infections.
-
-The elements of $\mathbf{F}$ are
-
-$$
-\mathbf{F}_{ij} = \frac{\beta_{ij} S_{j}}{N_{j}}
+\mathbf{R}_{ij} = \frac{\beta_{ij} S_{i}}{\gamma N}
 $$
 
-while $\mathbf{V}$ is a diagonal matrix with $\mathbf{V}_{ii} = \gamma$ where the recovery rate is shared among all groups $g$.
-
-Since $\mathbf{V}$ is diagonal, its inverse is as well, with $(\mathbf{V}^{-1})_{ij} = 1 / \gamma_i$.
-
-If all $\gamma_i = \gamma$, $\mathbf{R}$ is given by
-
-$$
-\mathbf{R}_{ij} = \frac{\beta_{ij} S_{j}}{\gamma N_{j}}
-$$
+If all $\gamma_i = \gamma$ and we assume SIR dynamics.
 
 The basic reproductive number $R_0$ is calculated as the dominant eigenvalue of $R$.
 
-This model incorporates vaccination by recalculating the distribution of susceptible individuals $S_g / N_g$ when $v_g$ vaccinations have been administered to each group $g$ (assuming all or nothing vaccination, with vaccine efficacy given by $ve$):
+This model incorporates vaccination by recalculating the distribution of susceptible individuals in each group $S_{i}^{vax}$ (assuming all or nothing vaccination, with vaccine efficacy given by $ve$):
 
 $$
-\mathbf{S_{g}^{vax}} = S_{g}^{initial} - v_{g} * ve
+\mathbf{S_{i}^{vax}} = S_{i}^{initial} * (1 - v_{i} * ve)
 $$
 
 Then $R_ij$ with vaccination factored in is given by
 
 $$
-\mathbf{R}_{ij}^{vax} = \frac{\beta_{ij} S_{j}^{vax}}{\gamma N_{j}^{vax}}
+\mathbf{R}_{ij}^{vax} = \frac{\beta_{ij} S_{i}^{vax}}{\gamma N}
 $$
 
 

widget/widget.py

@@ -17,10 +17,10 @@ def app():
 
     # Population size
     st.sidebar.subheader("Population Sizes")
-    default_values = [100, 100, 50, 1000]
+    default_values = np.array([0.2, 0.2, 0.005, 0.595]) * 1000000
     N = np.array(
         [
-            st.sidebar.number_input(f"Population ({group})", value=default_values[i], min_value=0)
+        st.sidebar.number_input(f"Population ({group})", value=int(default_values[i]), min_value=0)
             for i, group in enumerate(group_names)
         ]
     )