# Modeling and Analyzing Transmission of Infectious Diseases Using Generalized Stochastic Petri Nets
## 3 Modeling the Transmission of Infectious Diseases Using GSPN
### The GSPN-Based General Transmission Model of Infectious Diseases
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Transition** & **Meaning of Transition** & **Category** \\ \hline T1 & Virus to intermediate hosts & Timed \\ T2 & Intermediate hosts to human transmission & Timed \\ T3 & Whether the infected people have symptoms & Timed \\ T4 & Conduct relevant medical examinations & Immediate \\ T5 & Epidemic information collection and reporting & Timed \\ T6 & Human-to-human transmission in the hospitals & Timed \\ T7 & Human-to-human transmission in the infected areas & Timed \\ T8 & Government emergency response & Timed \\ T9 & Intercity transmission & Timed \\ T10 & Human-to-human transmission in the & Timed \\  & non-infected areas & Timed \\ T11 & Spread of the virus & Immediate \\ T12 & Successful vaccine development and inoculation & Timed \\ T13 & Auxiliary transition & Immediate \\ T14 & Auxiliary transition & Immediate \\ T15 & Research specific drugs and formulate drug & Timed \\  & countermeasures & Timed \\ T16 & Treatment of infected people & Immediate \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Meanings of transitions of the GSPN model in Figure 3.  
spread of the virus and has taken corresponding emergency response measures. After T8 is fired, there are two places P11 and P15. P11 stands for the infected people leaving from the infected areas after prevention and control. P15 is an auxiliary place, which means that the GSPN model goes to the development and use of special drugs phase. T9 represents that the infected people move freely between the cities. The process, from T9 through P13, T10, P14, T11 return to P3, illustrates the infected people moving from infected areas to non-infected areas. The virus begins to spread from city to city. This process will be a rapid growth in the number of infected people. (3) The development and use of specific drugs phase. This phase describes the process of research and treatment of specific drugs. T15 is a timed and key transition that represents the research of specific drugs and the formulation of drug countermeasures. Because the development of the specific drugs is a relatively long time, T15 is delayed to fire which may lead to the tokens accumulation. Therefore, P15 and P17 are the auxiliary places. T13 and T14 are the auxiliary transitions. They are used to prevent the tokens redundancy. P16 represents to start effective treatments and drugs research. P18 represents to complete effective treatments and drugs research. The inhibitor arc from P18 to T3 describes that the specific drugs will be able to suppress the incidence of infected people. At the same time, T16 is an immediate transition which means the infected people can be cured. The state of model returns to the beginning of P1. The constructed GSPN model gives the boundary conditions of the system. It can identify the number of possible states of the model. As the token moves, the marking changes. Each marking represents a state of the model. The total possible markings represents the total possible states which can be shown as a \(M_{i}\times P_{s}\) matrix. For the GSPN model of the transmission of infectious diseases in Figure 3, there are \(i=40\) states and \(s=19\) places, and the state reachable graph can be shown as a \(40\times 19\) large matrix. To simplify the matrix representation, in Table 3, we only list the places that have one token in each state \(M_{i}\). In Table 3, the first column is the types of markings, the second column is the name of the states, and the third column is the serial number of the places having one token. For example, on the second row, (P2, P19) means that the places of P2 and P19 have one token in \(M_{1}\) state and the other places have zero token in \(M_{1}\) state.