# A Brief Theory of Epidemic Kinetics

## Abstract

**:**

_{0}, depending on contamination and recovery factors. Various properties of the attractor are examined, and particularly its relations with R

_{0}. Decreasing this ratio below a critical value leads to a tipping threshold beyond which the epidemic is over. By contrast, significant values of the above ratio may bring the system through a bifurcating hierarchy of stable cycles up to a chaotic behaviour.

## 1. Introduction

_{0}below 1. The difference between the basic and effective reproductive factors will also be discussed in terms of the attractor properties. Finally, the possible occurrence of multi-stable cycles and chaotic behaviour during a sudden change in control parameter values will be investigated.

## 2. Contamination Kinetics

_{C}of them being contagious at a given time t. Neglecting the (nevertheless unfortunate) number of fatalities (about 2% in the case of COVID-19), the remainder (N-Nc) is made of exposed people (i.e., liable to infection), but also of individuals protected by long-term immunity, resulting from previous contamination and recovery (still debated for COVID-19), or from vaccination (if available and efficient).

_{C}/dt)

^{+}is proportional to the product ${N}_{C}\left(1-\xi \right)\left(N-Nc\right)$ of the number of contagious individuals by that of exposed ones in close contact with them at a given time. It is also taken proportional to the duration δt of the contact, to the turnover rate ν of exposed/contagious pairs, and inversely proportional to a protection efficiency factor p (e.g., mask wearing, lockdown measures, and so forth).

## 3. Recovery Kinetics and Global Evolution

_{C}of contagious individuals, and is controlled by a recovery parameter D, which may be understood as the average reciprocal duration (1/τ) of the disease hosted by individuals.

_{C}/N)/dt as a function of Nc/N for three different cases. For high contamination, slow recovery and low immunity (overcritical case), the parabola intersects the horizontal axis at two Nc/N values, Nc/N = 0 and Nc/N = N

_{C}*/N. called “fixed points”. The derivative d(Nc/N)/dt being zero at such points, Nc/N does not vary with time. However, slight variations of Nc/N around N

_{C}*/N or around zero would result in quite different behaviours. Starting from A, a fluctuation of Nc/N to the left would bring the system in a region where d(Nc/N)/dt is positive, which would push Nc/N back to the fixed point “A”. Conversely, a fluctuation towards the right, where the derivative is negative, would bring the system to the left, back to “A” again. “A” is called an attractor. For opposite reasons, any fluctuation of the system starting from O (and necessarily to the right) would drive it to the positive derivative zone, up to “A” again. “O” is a repulsor.

_{C}value at the attractor A is easily found from Equation (4), solving the equation:

_{0}as R

_{0}= CN(1 − ξ)/D, which depends on the long-term immunity factor ξ, on contamination and recovery factors, and also on the total population size N.

_{C}increases exponentially with time, a least as long as [CN(1 − ξ) − D] is positive (i.e., R

_{0}> 1). This is actually observed in early stages of COVID-19 in most countries, as shown in [8], and represented by straight lines if the number of infected people is plotted vs. time in a semi-logarithmic scale.

_{0}, and the proportion of non-infected individuals, including both susceptible and immunized ones, is clearly 1/R

_{0}.

_{0}tends to infinity (infinite contamination factors and (or) no recovery), the steady state proportion Nc*/N of contaminated people tends to 1 (100% infected). On the contrary, the fraction Nc*/N of contaminated people at the attractor A is reduced by a decrease in the contamination factor C and an increase in the recovery rate D and in the long-term immunity ξ, as intuitively expected, and eventually reaches zero when R

_{0}= 1.

_{0}= CN(1 − ξ)/D is larger than 1, there is a balance between newly contaminated and recovering people, which may appear at first sight to be equivalent to a 1 to 1 transmission of the epidemic, suggesting a reproductive rate equal to 1. Even so, the epidemic is not over at this stage. The literature appears as somewhat unclear on this point, probably related to a wide variety of definitions of the reproductive rate [9,10,11]. Actually, the situation at the attractor A does not correspond to R

_{0}= 1, but instead to R* = 1, R* being the “current (or effective) reproductive number”, defined as the ratio of the instantaneous number of new infections by that of fresh recoveries. It is often stated in the literature that in this situation, a single individual contaminates a single other one on average, due to a reduction of the number of exposed people. This is often referred to as the “herd (or global or collective) immunity” threshold [9]. Such a terminology may suggest that contamination of a fraction 1 − 1/R

_{0}of the population (which is actually the case at the attractor A) should provide a total immunity to the whole population.

_{0}larger than one, and observe the epidemic evolution from its very beginning (N

_{C}= 0). Equation (4) shows that the epidemic growth rate is a combination of linear and quadratic terms in N

_{C}. Contamination and recovery kinetics behave differently. The initial growth rate (Equation (7)) is linear in N

_{C}, giving the exponential growth discussed above, but due to such competing kinetics, the “growth rate” readily slows down, goes through a maximum, and eventually converges to zero at the attractor. During this whole process, the contamination rate remains larger than the recovery one, but their difference vanishes as the attractor is approached. The attractor is actually a current, stable but dynamical steady state at which the number of new infections and the number of recoveries in a population balance each other, a fraction of the recovered people becoming immune [12]. Recovering people are continuously replaced by newly contaminated ones, a fraction of them being always vulnerable to death. The epidemic is definitely not over and would not die out spontaneously in this stable state, in contrast with other interpretations of the effective reproductive number [13,14]. The classical representation of the effective reproductive rate in which a single individual directly contaminates a single exposed one surrounded by several immunized other ones seems inadequate. At this stage, the apparent 1 to 1 transmission is actually a balance between contamination and recovery kinetics. This is a key point.

_{0}. However, it is worth noting that, during this process, R* always remains equal to 1 whatever the position of A. This operation would merely decrease the flux of sick people travelling through the “attractor box”, and as a consequence reduce the current occupancy rate of hospital beds (which is already not so bad for lack of anything better).

_{0}= 0. In this case, dN

_{C}/dt becomes negative everywhere except at O, and O is now an attractor, corresponding to a number of contaminated people equal to zero from the very first attempt of the epidemic development.

_{0}= 1 is equivalent to the critical transition in a nuclear reaction. In this critical state, only one of the two neutrons produced by every fission of a Uranium nucleus is able (in average) to trigger the fission of another Uranium nucleus, and so forth. For R

_{0}< 1 (undercritical situation), the system remains under control, whereas for R

_{0}> 1 (overcritical situation), it diverges and the nuclear explosion takes place.

_{0}> 1 (large contamination, low recovery), the number of infected people starts increasing exponentially with time (top blue line in Figure 1), but its growth (“explosion”) is gradually hindered by the increasing recovery rate, and also by the decreasing available amount of exposed individuals, at least if a significant proportion of the population is infected (which does not seem to be the case at this stage for the COVID-19). In a same way, R

_{0}< 1 corresponds to the undercritical case: every chain reaction gradually slows down and eventually dies out. The epidemic expansion is over as long as R

_{0}is permanently kept below 1. If not, a new epidemic would be liable to start again somewhere and develop, unless long-term immunity has fully developed, or efficient vaccines have been produced meanwhile.

_{0}goes down, followed by a threshold at R

_{0}= 1 after which the epidemic is over, is a typical non-linear effect.

## 4. Convergence towards the Attractor, a Possible Route to Chaos?

_{0}, the evolution of x with time can be obtained by the recursive relation:

_{n}

_{+1}= x

_{n}, or equivalently:

_{n}values are represented on the horizontal axis, and x

_{n}

_{+1}ones on the vertical axis. Starting the iteration from any x

_{0}value on the horizontal axis (x

_{0}= 0.6 for instance in Figure 2), we draw a vertical line. It intersects the red parabola giving x

_{1}= 0.24. A horizontal line drawn from this intersection to the blue diagonal with unit slope transfers x

_{1}back to the horizontal axis, from which a second iteration is performed, and so forth. The successive x

_{n}values obtained by such iterations are represented by the thin staircase line, that eventually converges to the origin O which is (in this example), the unique fixed point. This is the critical case, corresponding to R

_{0}= 1.

_{0}value, the iteration converges to the fixed point at x* = α/β = 0.9, which is an attractor, whereas O is now clearly a repulsor. In addition, the trajectory to x* is monotonic as in Figure 2.

_{0}= 0.3 for instance (black lines), the system does not converge to x*, but instead to a so-called “limit cycle”, for which x oscillates between two different values on both sides of the attractor (bi-stable state). Starting from x

_{0}= 2.8 (green lines), the system also converges to the same limit cycle, though after a larger number of iterations. This kind of result is geometrically obvious as soon as the slope of the parabola at x* becomes less than (−1). In Figure 4, the slope at x* is indeed (−1.1 < −1). It can be shown [9,10,11] that decreasing further f′(x*) (as α goes up) may drive the system through a bifurcating hierarchy of multi-stable cycles that eventually leads to a tipping point and a chaotic behaviour.

_{0}to a value well above 1, is liable to trigger strong instabilities that might be difficult to control, since needs for emergency hospital beds may suffer large periodic or, even worse, chaotic variations. In this case, a temporary subdivision of infected areas into smaller isolated ones may bring the system back to a more stable and manageable state. In any case, the post-crisis release of protection policies should be conducted in a gradual and controlled manner. Other possible consequences of such a route to chaos are worth being studied.

## 5. Conclusions

_{C}/dt of contagious people at a given time results from a competition between two terms: (i) a contamination factor proportional to both the number of contagious individuals and the number of exposed ones, and (ii) a decay factor related to recovery kinetics of contaminated individuals.

_{0}value larger than 1. As a consequence, an “imposed” decrease of R

_{0}through lockdown or vaccination measures for instance would not change the R* value at the attractor, which will stay equal to 1 regardless of the attractor position, but it would reduce the flux of sick people travelling through the “attractor box”, or in other words the current number of sick people, that would help manage the occupancy of hospital beds. The epidemic would really be under control when the reduction of R

_{0}would result in a situation where A and O would merge, i.e., for R

_{0}= R* = 1.

_{0}may lead to a tipping threshold beyond which the epidemic is actually over. This would remain true as long as R

_{0}is kept below 1, i.e. as long as protection measures are enforced, and (or) as long as the long-term immunity of the population (possibly helped by vaccination) has increased enough to be able to take over the lack of protection.

_{0}value may shift the attractor apart from the actual position of the system. We show that, if the system is driven towards the new attractor position by a too large restoring force, for instance after a too rapid release of confinement or other protection measures, it would be brought through a bifurcating hierarchy of stable cycles to a chaotic behaviour, whose management would resultingly be problematic.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Theoretical epidemic growth kinetics. The variation rate d(Nc/N)/dt (Equation (4) normalized by N) of the proportion Nc/N of contagious people vs. Nc/N. In the overcritical case (red curve), a contagious individual contaminates more than another one on average. The red parabola intersects the Nc/N axis at a repulsor O and an attractor A. At the attractor, the steady state value of Nc/N is Nc*/N = 1 − 1/R

_{0}. Decreasing R

_{0}gradually reduces the Nc*/N value down to zero (critical state, green curve), at which A and O merge for R

_{0}= 1. Reducing R

_{0}below 1 brings the system to an undercritical state (blue curve) for which a single contagious individual contaminates less than an exposed one on average; in such conditions, the epidemic eventually dies out.

**Figure 2.**Logistic map for α= 0 and β = 1. Critical case. Starting from any x

_{0}value (0.6 in the figure) the system eventually converges to the origin O, which is the only fixed point (and attractor).

**Figure 3.**Logistic map for α = 0.9, β = 1.O is now a repulsor, and the system converges monotonically towards the attractor at x* = 0.9.

**Figure 4.**Logistic map for α = 2.1, β = 1. The slope of the parabola at x = x* (which is still an attractor) is now less than −1, leading to a bi-stable limit cycle, whatever the starting point (black or green staircase lines).

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Louchet, F.
A Brief Theory of Epidemic Kinetics. *Biology* **2020**, *9*, 134.
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Louchet F.
A Brief Theory of Epidemic Kinetics. *Biology*. 2020; 9(6):134.
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Louchet, François.
2020. "A Brief Theory of Epidemic Kinetics" *Biology* 9, no. 6: 134.
https://doi.org/10.3390/biology9060134