Benutzer:Pjotr Hambach/Artikelentwurf

A general simple approach is presented allowing a numerical estimation of the time evolution of an viral infection process influencing the population of a biological system. Recursive methods allow the determination of important parameters like the growth rate of the infection process.

General Approach Bearbeiten

In the following approach, the infection process is considered as a time dependent evolution of a simple biological system, influenced by a virus infection process. The time evolution for this infection process is considered along an equidistant time grid $t_{n}$ with counting the steps in time evolution. An arbitrary point along the time grid (evolution step) is then given as

(1) $\ \ \ t_{n}=\tau \cdot n,\ \ n=0,\ 1,\ 2,\ ...$ ,

where the constant $\tau$ is the characteristic time interval for an evolution step depending on the infection process. Hence, the evolution process is represented by a discrete sequence of states of the biological system along the time grid:

(2) $\ \ \ State_{1}\longrightarrow State_{2}\longrightarrow \ ...\ State_{n-1}\longrightarrow State_{n}\longrightarrow \ ...$

where $State_{n}$ means the state of the biological system at time $t_{n}$ .

As a biological system, let us consider a population of individuals influenced by a virus infection process. The total number of individuals of the population may be $N_{0}$ . The infection process may be caused by an virus with the natural reproduction factor $R_{0}$ . At time $t_{0}$ , the evolution of the infection process is starting with the total number of actual infected individuals $A_{0}$ (identical with the initial number of spreaders). In the actual approach, a recursive formula for the growth rate $\Delta _{n}=\Delta (t_{n})$ (number of new infected individuals at time $t_{n}$ ) will be presented, so that $S_{n}=S(t_{n})$ , the total number of infected individuals at time step $t_{n}$ , can be determined. At time $t_{0}$ , the growth rate is defined by $\Delta _{0}=A_{0}$ .

For $n=1,2,3\ ...$ the recursive formula for the growth rate $\Delta _{n}$ (number of new infected individuals) at time $t_{n}$ (evolution step $n$ ) is

(3) $\ \ \ \Delta _{n}={N_{0}-S_{n-1} \over N_{0}}\cdot R_{0}\cdot F_{n-1}\cdot A_{n-1}$ ,

where $A_{n}=A(t_{n})$ is the number of actual infected individuals (identical with number of actual spreaders) at time $t_{n}$ :

(4) $\ \ \ A_{n}=\sum _{i=\max(0,n-D)}^{n}\Delta _{i}$

$D=0,\ 1,\ 2,\ ...$ is a positive integer constant due to the properties of the virus: $D+1$ represents the average duration of the infection for an individual in terms of the number of time intervals $\tau$ . Example: If $D=1$ , the average duration of the infection is then $2\cdot \tau$ . The total number of infected individuals $S_{n}$ at time $t_{n},\ n=1,\ 2,\ 3\ ...,$ is regarded to describe the time evolution of the biological system along the time grid (1). $S_{n}$ is given by the propagation formula

(5) $S_{n}=S_{n-1}+\Delta _{n}$ ,

where the initial value at the begin of the infection process is $S_{0}=A_{0}$ and the growth rate $\Delta _{n}$ is given by the propagation formula (3) above. Finally, the time dependend factor $F_{n}=F(t_{n})\leq 1$ is introduced to represent the actions for damping the spread of the infection process at time $t_{n}$ , for example social distancing.

By choosing the total number of infected individuals $S_{n}$ as the state of the biological system at evolution step $n$ , formula (5) can be considered as the equation for the time evolution of the system, acting $\Delta _{n}$ as the time dependent propagator to transform the state from $S_{n-1}$ to $S_{n}$ .

Simple Conclusions Bearbeiten

Independent from concrete calculations for the time evolution based on (3) and (4) using the characteristic system parameters $\tau$ , $N_{0}$ and $A_{0}$ , a general simple analysis concerning the dynamic of the system can be performed as follows.

We are concerned with the question, under what condition the growth rate does decrease without any actions for damping. With other words: When would the natural growth of infection stop? This occurs, when the number of new infected individuals at a certain point in time $t_{m}$ is smaller than the number of spreaders in the previous time period $t_{m-1}$ :

(6) $\ \ \ {{\Delta _{m}} \over {A_{m-1}}}<1$

Applying equation (6) and assuming that no actions for damping are taken ( $F=const=1$ ), we get

(7) $\ \ \ {{\Delta _{m}} \over {A_{m-1}}}={N_{0}-S_{m-1} \over N_{0}}\cdot R_{0}<1$ .

Rearranging the inequality yields the following condition for the total number of infected individuals

(8) $\ \ \ S_{m-1}>N_{0}\cdot {\Bigl (}1-{{1} \over {R_{0}}}{\Bigr )}$ .

For example, if the natural reproduction rate is $R_{0}=3$ , the total number of infected individuals must arrive at ${\tfrac {2}{3}}$ or approx. $66\%$ of the population $N_{0}$ to reverse the growth process.