# Transportation and Reserve of Emergency Medical Supplies during Public Health Events
## 3 Materials and Methods
### Forecast of Emergency Medical Supply Demand and Model for Mobilization and Stockpiling
#### 3.1.1 Prediction Model of Infected People Based on SEIRD
And make the following assumptions:  
* The susceptible person does not have the possibility of infection when contacting the Exposed, that is, the Exposed will not infect the susceptible person;
* The susceptible person has the possibility of infection after contact with the susceptible person;
* Recovered patients have lifelong immunity. As long as they are cured, they will not be infected again;
* The death population is not infectious;
* The total population remains unchanged during public health issues;
* Only communication between people is considered and environmental communication is not considered;
* At the beginning of public health issues, the number of recovered persons and the number of dead persons were 0. Among them, the susceptible person (\(S\)) will be infected by the infectious person (\(I\)) to become the exposed person (\(E\)). The exposed person (\(E\)) will become the infectious person (\(I\)) after a period of time, and the infectious person (\(I\)) will become the recovered person (\(R\)) or the dead person (\(D\)) after treatment. The specific relationship is shown in Figure 1.  
According to the relationship between the SEIRD model and its variables, the following differential equations are constructed:  
\[\left\{ \begin{array}{rl}\frac{dS}{dt}=-\frac{\beta\times I(t)\times S(t)}{N}\\ \frac{dE}{dt}=-\zeta\times E(t)+\frac{\beta\times I(t)\times S(t)}{N}\\ \frac{dI}{dt}=\zeta\times E(t)-\gamma\times I(t-\tau_{r})-\mu\times I(t-\tau_{ d})\\ \frac{dR}{dt}=\gamma\times I(t-\tau_{r})\\ \frac{dI}{dt}=\mu\times I(t-\tau_{d})\end{array} \right. \tag{1}\]  
Among them:  
* \(S\) refers to susceptible people, \(E\) refers to exposed people, \(I\) refers to infected people, \(R\) refers to recovered people, \(D\) refers to dead people, and \(N\) refers to the total number;
* \(t\) represents the current time;
* \(\beta\) indicates the probability that susceptible people will be converted into exposed people after contacting infected people per unit of time;
* \(1/\zeta\) indicates the average latent period, and people with \(\zeta\times E(t)\) per unit time change from exposed people to infected people;
* \(\gamma\) Indicates the cure rate and the \(\gamma\times I(t-\tau_{r})\) people change from infected to recovered people in unit time \(t\);
* \(\mu\) Indicates the mortality rate and the \(\mu\times I(t-\tau_{d})\) people change from infected to dead people in unit time \(t\). Time delay \(\tau_{r}\) and \(\tau_{d}\) are used for rehabilitation (cure) and death, respectively. Usually, because the epidemic time is very short, it is assumed that the net impact of birth and natural death is zero. Without considering various factors such as birth, natural death, immigration, and emigration, it is assumed that the total number of people \(N\) remains unchanged, and the number of people of various types at \(T\) time is recorded as \(S(T)\), \(E(T)\), \(I(T)\), \(R(T)\), and \(D(T)\), respectively, then \(N=S(T)+E(T)+I(T)+R(T)+D(T)\). To solve the differential equations, transform them into iterative models:  
\[dS=-\frac{\beta\times I(t)\times S(t)}{N}dt \tag{2}\]  
Move items to obtain:  
Integrate \(t\) from \(n-1\) to \(n\) at both ends of the equation:  
\[\int_{n-1}^{n}dS=\int_{n-1}^{n}-\frac{\beta\times I(t)\times S(t)}{N}dt \tag{3}\]  
From the left matrix formula: \(\int_{a}^{b}g(x)\approx(b-a)g(a)\)  
\[S_{n}=S_{n-1}-\frac{\beta\times I_{n-1}\times S_{n-1}}{N} \tag{4}\]  
Figure 1: Diagram of the SEIRD model. In the same way, the residual equation can be obtained:  
\[E_{n}=E_{n-1}+\frac{\beta\times I_{n-1}\times E_{n-1}}{N}-\zeta\times E_{n-1} \tag{5}\]  
\[I_{n}=I_{n-1}+\zeta\times E_{n-1}-\gamma\times I_{n-1}-\mu\times I_{n-1} \tag{6}\]  
\[R_{n}=R_{n-1}+\gamma\times I_{n-1} \tag{7}\]  
\[D_{n}=D_{n-1}+\mu\times I_{n-1} \tag{8}\]