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**Kermack-McKendrick epidemic model revisited.**
*(English)*
Zbl 1137.37338

Summary: An epidemy of a highly infectious disease with a fast recovery (or fatality) in a homogeneous population is considered, the influenza being an example of such epidemics. This classical Kermack-McKendrick model assumes a fixed sized population of \(n\) individuals, the population being divided into three subpopulations which change their respective sizes in the running time of the epidemic: Susceptibles (the individuals exposed to the infection), infectives (the infected individuais that are able to spread the disease) and removals (the individuals restored to health not able further to spread the infection or get themselves to be infected again) numbering by \(x(t)\), \(y(t)\) and \(z(t)\) the individuais that are susceptible, infected and removed at some time \(t\geq 0\), respectively. Hence, \(x(t) + y(t) + z(t) = n\), \(x(t)\) and \(z(t)\) being generally a nonincreasing and nondecreasing function, respectively such that \(z(0) = 0\).

The model assumes the dynamics given by the following three dimensional differential equation

\[ \begin{alignedat}{2} & \dot x(t)=-\beta x(t)y(t),&\quad & x(0)=x_0>0,\\ & \dot y(t)= \beta x(t)y(t) - \gamma y(t),&\quad & y(0)=y_0=n-x_0>0,\tag{M1}\\ &\dot z(t)=\gamma y(t), &\quad & z(0)=0,\end{alignedat} \]

where the intensity \(\beta > 0\) is higher for more infectious diseases and the parameter \(\gamma^{-1} > 0\) is proportional to the average duration of the “being infected” status, i. e. to the average time for which an individual is infected.

Our aim is to propose and justify a diffusion version of (M1) model that allows both more general intensities \(\beta\) and to model the spread of epidemic in a population that changes its size \(N_t\) due to a diffusion type emigration and immigration processes defined for example by the Engelbert-Schmidt stochastic differential equation

\[ dN_t = N_t\sigma(N_t) dW_t,\quad N_0=n_0:=x_0+y_0. \]

The model assumes the dynamics given by the following three dimensional differential equation

\[ \begin{alignedat}{2} & \dot x(t)=-\beta x(t)y(t),&\quad & x(0)=x_0>0,\\ & \dot y(t)= \beta x(t)y(t) - \gamma y(t),&\quad & y(0)=y_0=n-x_0>0,\tag{M1}\\ &\dot z(t)=\gamma y(t), &\quad & z(0)=0,\end{alignedat} \]

where the intensity \(\beta > 0\) is higher for more infectious diseases and the parameter \(\gamma^{-1} > 0\) is proportional to the average duration of the “being infected” status, i. e. to the average time for which an individual is infected.

Our aim is to propose and justify a diffusion version of (M1) model that allows both more general intensities \(\beta\) and to model the spread of epidemic in a population that changes its size \(N_t\) due to a diffusion type emigration and immigration processes defined for example by the Engelbert-Schmidt stochastic differential equation

\[ dN_t = N_t\sigma(N_t) dW_t,\quad N_0=n_0:=x_0+y_0. \]

### MSC:

37N25 | Dynamical systems in biology |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

92D25 | Population dynamics (general) |

37H10 | Generation, random and stochastic difference and differential equations |

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\textit{J. Štěpán} and \textit{D. Hlubinka}, Kybernetika 43, No. 4, 395--414 (2007; Zbl 1137.37338)

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