Time Evolution of Infection Processes: A Simple Recursive Method

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

General Approach

In the following approach, the infection process is considered as a time dependent evolution of a biological population underlying a virus infection process. Here the reproduction factor $R_{0}$ is used: It is defined as the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection . An individual, after getting infected, infects exactly $R_{0}$ new individuals only after exactly a time $\tau$ (the serial interval or infection transmission period) has passed.

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

where the constant $\tau$ is the characteristic time period for an evolution step (mean infectious period). 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 $t_{n}$.

The total number of individuals of the population may be $N_{0}$. At time $t_{0}$, the evolution of the infection process is starting with the initial number of infected individuals $S_{0}$ (identical with the initial number of spreaders). In the actual approach, a recursive formula for the growth rate $\Delta _{n}=\Delta (t_{n})$ will be presented. It is defined as the number of new infected individuals counted within the time period $[t_{n-1},t_{n}]$, where $t_{n}-t_{n-1}=\tau$. The goal is to determine $S_{n}=S(t_{n})$, the total number of infected individuals at time step $t_{n}$. At time $t_{0}$, the growth rate is defined by $\Delta _{0}=S_{0}$.

Assuming, that an individual, after getting infected, infects new individuals only after exactly a time $\tau$, the recurrence relation for the growth rate $\Delta _{n}$ (number of new infected individuals) at time $t_{n}$ (evolution step $n$) is

(3) $\ \ \ \Delta _{n}={N_{0}-S_{n-1} \over N_{0}}\cdot R_{0}\cdot F_{n-1}\cdot \Delta _{n-1},\ \ \ n=1,2,3,...$ .

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 then given by the recurrence relation

(4)      $S_{n}=S_{n-1}+\Delta _{n},\ \ \ n=1,2,3,...$ ,

with the initial value $S_{0}$ and the growth rate $\Delta _{n}$ from relation (3). 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 (4) 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

Independent from concrete calculations for the time evolution based on (3) and (4) using the characteristic system parameters $\tau$, $N_{0}$ and $\Delta _{0}$, 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)$\ \ \ {{\Delta _{m}} \over {\Delta _{m-1}}}<1$

Applying equation (3) and assuming that no actions for damping are taken ($F=const=1$), we get

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

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.

References

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