generated from jamesmbaazam/QuartoPresentationTemplate
-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy path_06_controlling_epidemics.qmd
223 lines (143 loc) · 7.63 KB
/
_06_controlling_epidemics.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
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
::: callout-note
For a recap on the various control measures, refer to @sec-control-measures.
:::
------------------------------------------------------------------------
## Vaccination
::: {callout-note}
For a background on vaccination, refer to @sec-vaccination.
:::
------------------------------------------------------------------------
::: columns
::: {.column width="40%"}
{width="100%"}
:::
::: {.column width="60%"}
- Vaccination is one of the most effective ways to control infectious diseases.
- Conceptually, vaccination works to reduce the number of susceptible individuals, $S$.
:::
:::
------------------------------------------------------------------------
- There are different types of vaccination strategies, including:
- [pediatric]{style="color:tomato"} vaccination: vaccinating children to prevent the spread of diseases.
- [mass/random]{style="color:tomato"} vaccination: vaccinating a large proportion of the population.
- [targeted]{style="color:tomato"} vaccination: vaccinating specific groups of individuals, example, healthcare workers.
- [pulse]{style="color:tomato"} vaccination: periodically vaccinating a large number of individuals.
------------------------------------------------------------------------
- Let's consider the case of mass/random vaccination.
- Compartmental models can be extended to capture this by adding a new compartment, $V$.
- Let's consider the SEIR model with vaccination.
------------------------------------------------------------------------
### The Susceptible-Exposed-Infected-Recovered-Vaccinated (SEIRV) Model
------------------------------------------------------------------------
::: columns
::: {.column width="40%"}
:::
::: {.column width="60%"}
- The SEIRV model is simply the SEIR model with a vaccinated compartment, $V$.
- The vaccinated compartment represents previously susceptible individuals who have been vaccinated and are immune to the disease.
:::
:::
------------------------------------------------------------------------
- The vaccinated compartment is:
- not infectious and does not move to the exposed or infectious compartments.
- replenished by the rate of vaccination, $\eta$.
------------------------------------------------------------------------
The model diagram and equations are as follows:
::: columns
::: {.column width="40%"}
```{=tex}
\begin{align}
\frac{dS}{dt} & = -\beta S I - \eta S \\
\frac{dE}{dt} & = \beta S I - \sigma E \\
\frac{dI}{dt} & = \sigma E - \gamma I \\
\frac{dR}{dt} & = \gamma I \\
\frac{dV}{dt} & = \eta S
\end{align}
```
:::
::: {.column width="60%"}
:::
:::
where $\eta$ is the rate of vaccination.
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Discussion
- What are some of the assumptions of the SEIRV model?
- What are the implications of these assumptions for the model's predictions?
:::
------------------------------------------------------------------------
### Numerical simulations
#### R Practicals
- We can use the same approach as the SIR and SEIR models to simulate the SEIRV model.
- Modify the `seir.Rmd` script to simulate the SEIRV model.
------------------------------------------------------------------------
## Non-Pharmaceutical Interventions (NPIs)
::: {callout-note}
For a background on non-pharmaceutical interventions, refer to @sec-npi.
:::
------------------------------------------------------------------------
- Conceptually, NPIs usually act to either reduce the transmission rate, $\beta$ or prevent infected individuals from transmitting.
- NPIs like isolation, social distancing and movement restrictions can reduce the transmission rate, $\beta$, by reducing the contact rate between susceptible and infectious individuals.
- Hygiene measures reduce the probability of transmission per contact, thereby reducing the transmission rate, $\beta$, since $\beta = c \times p$.
------------------------------------------------------------------------
- Let's consider two scenarios that will extend the SIR model to include NPIs:
1. Modifying the transmission rate, $\beta$.
2. Preventing infected individuals from transmitting through isolation.
------------------------------------------------------------------------
### Modifying the transmission rate
- NPIs such as social distancing, mask-wearing, and hand hygiene can reduce the transmission rate, $\beta$.
- We can model this by making the transmission rate a function of time, $\beta (t)$.
---
The modified SIR model with a reduced transmission rate is as follows:
```{=tex}
\begin{align*}
\frac{dS}{dt} & = -\beta (t) S I \\
\frac{dI}{dt} & = \beta (t) S I - \gamma I \\
\frac{dR}{dt} & = \gamma I
\end{align*}
```
---
- The simplest form is to reduce $\beta$ by an NPI efficacy, say $\epsilon$.
- Assuming the NPI is implemented between $t_{\text{npi\_start}}$ and $t_{\text{npi\_end}}$, it means that $\beta$ remains the same before that period and is modified to $(1- \epsilon)\beta$ during the period of the NPI, where $0 \leq \epsilon \leq 1$.
- With this knowledge, we can define $\beta (t)$ mathematically as:
```{=tex}
\begin{equation*}
\beta(t) = \begin{cases}
\beta & \text{if } t < t_{\text{npi\_start}} \text{ or } t > t_{\text{npi\_end}} \\
(1 - \epsilon) \beta & \text{if } t_{\text{npi\_start}} \le t \le t_{\text{npi\_end}}
\end{cases}
\end{equation*}
```
------------------------------------------------------------------------
#### R Practicals
- Let's open the script file `sir_npi.R` and follow along.
------------------------------------------------------------------------
### NPIs as compartments
- In the previous example, we retained the SIR model structure and modified the transmission rate.
- We can also model NPIs as compartments in the model. This is useful when we want to treat individuals affected by the NPIs differently.
------------------------------------------------------------------------
- For example, isolation is an NPI that prevents infected individuals from transmitting the disease.
- This means that infected individuals in isolation do not contribute to the transmission of the disease and need to be removed from the infected compartment.
- Moving isolated individuals to a separate compartment allows us to track them separately in the model and apply relevant parameters.
------------------------------------------------------------------------
- We can model this by introducing a new compartment, $Q$, for isolating infected individuals.
- Let's assume, infected individuals move to the isolated compartment at a rate, $\delta$.
- Infected individuals in the isolated compartment do not transmit the disease.
------------------------------------------------------------------------
The modified SIR model with isolated is as follows:
```{=tex}
\begin{align*}
\frac{dS}{dt} & = -\beta S I \\
\frac{dI}{dt} & = \beta S I - \gamma I - \delta I \\
\frac{dR}{dt} & = \gamma I + \tau Q \\
\frac{dQ}{dt} & = \delta I - \tau Q
\end{align*}
```
where $\delta$ is the rate at which infected individuals move to the isolated compartment, and $\tau$ is the rate at which individuals recover from isolation.
------------------------------------------------------------------------
##### R Practicals
- We can use the same approach as the SIR model to simulate the model with isolation.
- Modify the SIR model in `sir.Rmd` to incorporate the isolated compartment, $Q$ and the relevant parameters.