generated from jamesmbaazam/QuartoPresentationTemplate
-
Notifications
You must be signed in to change notification settings - Fork 0
/
_07_heterogeneities.qmd
55 lines (32 loc) · 2.19 KB
/
_07_heterogeneities.qmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
- The models we have discussed so far assume that all individuals in the population are identical.
- However, in reality, individuals differ in their susceptibility to infection and their ability to transmit the disease.
- It is essential to capture this heterogeneity in the models in order for the models to be more realistic and useful for decision-making.
------------------------------------------------------------------------
- This can be captured by incorporating heterogeneity into the models.
- Heterogeneity is often captured by stratifying the population into different groups.
------------------------------------------------------------------------
## Age structure
- For many infectious diseases, the risk of infection and the severity of the disease vary by age.
- Hence, it is essential to capture age structure in the models.
- To do this, we divide the population into different age groups and model the disease dynamics within each age group.
- Let's extend the SIR model to include age structure.
------------------------------------------------------------------------
- We will divide the population into $n$ age groups.
- Because the homogeneous model has 3 compartments, the age structured one will have $3n$ compartments: $S_1, I_1, R_1, S_2, I_2, R_2, ..., S_n, I_n, R_n$.
------------------------------------------------------------------------
- The (compact) model equations are as follows:
\begin{align*}
\dfrac{dS_i}{dt} &= -\sum_{j=1}^{n} \beta_{ji} S_i I_j \\
\dfrac{dI_i}{dt} &= \sum_{j=1}^{n} \beta_{ji} S_i I_j - \gamma I_i \\
\dfrac{dR_i}{dt} &= \gamma I_i
\end{align*}
------------------------------------------------------------------------
- The model can be used to study the impact of age structure on the dynamics of the epidemic.
- For example, we can study the impact of vaccinating different age groups on the dynamics of the epidemic.
------------------------------------------------------------------------
## Other Heterogeneities
- Other forms of heterogeneity that can be incorporated into the models include:
- Spatial heterogeneity
- Temporal heterogeneity
- Contact heterogeneity
- Heterogeneity in host behavior