I think that it is spreading like period-doubling in bifurcation diagram.

I think that multi-fractals techniques may work if we try to study problem globally.

Possibly, u are referring to the viral replication within an individuals lung (and all the above answers would be relevant) ??

There are many ways to study it.

How is spreading it at different locations by infected persons or associated things?

After infection, how spreads in body depending on environment or non-availability of medical treatment?

After infection, how to control this chaotic coronavirus?

For data reports on coronavirus situations, see it

https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/

No, it has nothing to do with chaos or fractals. Despite the common use of the term "fractal dimension" by scientists who seem to know the formal definition of it, there are no fractals in the observable universe, from the sub-atomic to cosmic, including all biological and physical systems. Once you understand the definition you will see why.

Mathematical chaos, on the other hand, is present in many physical dynamically changing systems. Again, one needs to know the definition of the term to use it. However, in the spread of coronavirus, simple mathematical modeling of epidemics produces the results we are seeing. I devised such a model several weeks ago and predicted on an almost day to day basis, the progression of the epidemic in the US. I have adjusted some parameters in my model to more closely fit the data but in broad terms it was spot on.

We have now started to intervene in drastic ways through a "semi-lockdown" policy and mass testing (rather late in the game unfortunately for those who became seriously ill or died). This changes the parameters in my model but if the president had continued to ignore science or if he forces workers back to work en mass, then in just ONE week from today there will be over 500,000 cases in the US and over 8000 deaths.

BTW, there is no period-doubling because there are no periodic motions in the number of infected, only increases. That it is increasing yet bounded (by total population size) rules out any chaotic behavior.

All the above possibly come into play when looking at the spread within an organism. Also, possibly, the above may come into play when looking at how the above epidemic suddenly "emerged" and looking at the various factors (including socio-economic) that contributed to it. Also, it is worth looking at the PDE's governing the diffusivity of the virus in various surroundings and how it can be slowed (other than social "distancing")

Transport of a cloud of viruses might be by diffusion in an ideal setting but in fact they move more on droplets through small currents of air after being breathed, coughed, or sneezed out of an infected person. An equally important means of transport is through {hand of infected} to surface to hand to face of susceptible person.

Those are details, however, that can be modeled at the population level as diffusion of infected people through a community of susceptible people. A good spatial model would have several islands (population centers) of differing sizes connected by lines of travel: A network of islands. Then one could simulate the spread throughout the world. Transportation along lines connecting islands could be modeled as one-dimensional diffusion, with differing diffusivities according to how busy each route was. E.g. Hong Kong to LA would have a high diffusivity coefficient while LA to Belfast, Ireland would have low diffusivity. That model could probably be derived from a stochastic system accounting for movement of individuals.

In the body of a person, the reproduction and transport is fairly well understood, I believe, and diffusion really doesn't play much of a role except for localized spread from infected tissue.

I think the system of PDE's modeling the world as a large collection of population centers (with diffusion within each of these) connected in a network of one dimensional diffusive channels is definitely worth studying and might have broad interest. Within each island (population center) the population should be age-structured (time scale of tens of years say) and the infected population should be (time from infection)-structured with a time scale of weeks, say. There should also be sub-populations getting quite ill and those having very slight symptoms but still being infectious. Testing would need to be included as a parameter to place people in another sub-population, those under care.

In any case, it is clearly quite complex but computationally just a bit messy with many parameters. It should be coded and run on a parallel computer system I think. But a toy model with about four centers could be run on a laptop, I think.

Peter William Bates "No, it has nothing to do with chaos or fractals."

With this I agree. It can be modeled with low-order differential equations that are integrable.

"Despite the common use of the term "fractal dimension" by scientists who seem to know the formal definition of it, there are no fractals in the observable universe, from the sub-atomic to cosmic, including all biological and physical systems. Once you understand the definition you will see why."

In the way physicist use the term "fractal", there are most definitely fractals in the universe. For a physicist, fractality is restricted to "above a certain length-scale" and "below a second length-scale", so all fractal behaviour occurs in a certain finite range. Below and above, objects become compact, possibly with different integer dimensionalities. Physicists deal with "real-world fractals", including so-called statistical fractals, not with ideal mathematical objects.

"Mathematical chaos, on the other hand, is present in many physical dynamically changing systems."

And a chaotic attractor usually is a fractal. So how can there be no fractals in the observable universe? Does the object that an attractor constitutes in space-time not belong to the observable universe?

"However, in the spread of coronavirus, simple mathematical modeling of epidemics produces the results we are seeing. I devised such a model several weeks ago and predicted on an almost day to day basis, the progression of the epidemic in the US. I have adjusted some parameters in my model to more closely fit the data but in broad terms it was spot on."

How many parameters does your model have? The US curve is still not far from the exponential growth range, so it seems difficult to predict deviations, and, in particular, the level where the cumulative number of infected persons will saturate. I have done some extremely simple-minded modeling myself and I find that for the US and Germany the predicted saturation level still changes with each new data point, so there is no real predictive power yet. Interestingly, for Italy no new fitting of the curve to the data was necessary the last two days. So the model did predict these two data points correctly. If that stays so, then in Italy the number of once-infected persons will saturate slightly below 200000.

Of course, changes in policy by the authorities may affect the expected numbers in both ways.

Professor Kassner, I am first surprised because I did not know this was a public forum. I thought two people had asked me a question and I had answered it to the best of my knowledge. Now the surprise has worn off a little, I'll get to your comments/questions.

I am glad that you agree with me about the absence of chaos and fractals in this case. And you sort of partially agree with my statement "Despite the use of the term ... by scientists who know the definition...". You do this by pointing out, as I implied, that some physicists choose another, looser definition for the term "fractal." But even with your description of this looser "definition" used by physicists, can you state in precise and unambiguous terms the physicists definition of "fractal"? One that is not trivial. One that would exclude an annulus, say, which is one circle inside another having the same centers. It looks self-similar to me in a certain range of length scales.

About my model for the viral epidemic in the US,

I have subdivided the population into 8 disjoint subsets according to susceptible, infected for less than a week (all undiagnosed and infectious), those infected for a time between one and two weeks and undiagnosed, those infected for more than two weeks and undiagnosed, those diagnosed and in their first week of care, those diagnosed and in their second week or more of care, those who have recovered, the dead. There are 20 or 21 parameters. I could include more to account for health care workers getting infected from those in quarantine, for instance, but I didn't. I also did not break down each of these sets by age, which really should be done.

Rather than a continuous dynamical system, I just used a discrete DS which I put into a spreadsheet.

Of course you are correct about saturation and my model includes this by having susceptibles move to infected according to mass-action kinetics (the probability that when two individuals meet, one infects the other is the fraction of infected who are loose in the population multiplied by the fraction of susceptibles in the non-quarentined (or dead) population). There is a prefactor parameter representing the average infectivity of an infected person. So, as the number of suscptibles gets below a certain threshold, the rate of increase of infected people decreases. Doing this in integer arithmetic causes the virus to die out before the entire population becomes infected. I have not used any curve fitting algorithm to tune my parameters as more data arrives but I have made adjustments according to intuition. One of my parameters is testing rate which in turn impacts the three rates in which undiagnosed infected people move into quarantine and care.

I can think of ways to improve the model, as I just pointed out, but I am doing this for my own information and not to publish or make claims to society. Even though this modeling is very important to society, there are specialists who I am sure have made more sophisticated and more accurate models than mine, so I feel I might do more harm than good by publicizing my amateur attempts.

I hope this more than answers your question about my model and doesn't bore you unduly.

Regards,

Peter

PS to anyone else reading,

I do not want to share my model because I do not want to spread my almost-idle thoughts and have people act upon them, having seen a version that has passed through N (even well-scrubbed) hands.

Would it be simpler, to use an Excel Spreadsheet, and put the number infected each day for an area--like a country--then extrapolate out, how many days to double those numbers? Then, compare the extrapolated numbers with the future daily numbers, to see when and of the curve starts to flatten? I am only charting total infected, to make it easy. You can get the daily data from https://www.worldometers.info/coronavirus/

In February I charted China's total cases, and saw when the curve started flattening very rapidly near the end of the month, when they went to total-lockdown, instead of the "Lockdown-Lite" we seem to have here in California and "ZERO Lockdown" in most of the USA still.

Rounded out to the nearest thousands of total cases on March 26,

--USA was 85K, then maybe by March 31 = 275K ?

--New York 39K, maybe by March 31 = 80K ?

--Italy 63K, and maybe by March 31 = 150K?

--Spain 58K, and maybe by March 31 = 132K

Seems like the China method of "Total Lockdown", massive testing and isolation, is the only way to go, to help flatten the curve?

Yes it would be easier but far less accurate for predicting months ahead.

One possibility in the context of fractal surfaces is that of anomalous diffusion especially sub-diffusion which would be useful in the current context to slow the diffusion of the virus. Of course, such surfaces would have to be induced at least temporarily. Other interesting possibilities relate to the role played by inter viral communication (similar to inter bacterial communication) / horizontal gene transfer etc and diffusion on arbitrary manifolds (such as mobius strips etc)

Until the curves start getting flattened, the Excel chart and extrapolating out about 3-4 weeks using whatever the doubling rate is, can give you a very conservative predicted trend, unless all control is lost, like in New York State today. NY State has had shifting doubling rates--Doubling every day between March 18-19, then doubling every 2 days between 19th and 23rd. Then slowed down to doubling every 4 days from the 23rd and 26th. And today, is still tracking at a doubling of a little less than 4 days, so that means by April 1st 100,000 cases.

Here, I am writing chaotic situation based on sensitive to initial condition.

We can observe that, after some days, there is sudden dramatic changes in the coronavirus disease if we do not have any control. These sudden changes can be work like fractal branching. It is spreading among people like as branching trees which looks like fractal branching.

I think that if we compute Lyapunov exponents about spread of coronavirus, then it would be positive which also represents chaotic behaviour.

Since we have only very short data, so we can not easily predict about this critical situation regarding spread of coronavirus.

Since coronavirus is exponentially growing in short interval of parameter at different locations after certain time interval, the dynamical system may lead to chaotic situation. This type of family of functions can be construct to obtain mathematical modeling on infection by coronavirus.

I forgot to address Professor Kassner's last comment about the possibility of me giving contradictory statements about the absence of fractal sets in nature and about the presence of chaotic motions in nature. He mentioned the fact that some chaotic attractors have fractal boundaries. I believe here that he is speaking of Julia sets for discrete dynamical systems, such as iterations of maps into the plane. Three comments: 1. In nature, we do not view time as discrete. 2. The boundary of the attractor is only observed by taking time to infinity along trajectories. 3. Matter may be discrete and with particles having sizes bounded below. Therefore, one cannot magnify indefinitely and continue to observe self-similar patters. Matter may instead be a field (continuum) but with finitely many local "peaks" in any bounded volume. Again, self-similarity is impossible at all degrees of magnification.

Now I must confess, as will most physicists, that I really do not know the complete nature of matter and I am eager to be enlightened by new experimental evidence that will change our concept of matter. Perhaps fractals will exist if matter itself has a fractal nature. At the moment with technology as advanced as it is, we can actually see self-similarity break down at high enough resolution.

@Mohammad Sajid

Chaotic dynamical systems have infinitely many unstable periodic orbits. If you model the dynamics of this epidemic from more or less first principles, you will see that there are no periodic orbits (people either recover or die). Therefore it is not chaotic.

@Craig Carlton Dremann

If you model the epidemic using first principles and not just trying to find a reasonable curve that passes through data points, you will see that the doubling time necessarily changes, since the population size is finite and the number of cases is increasing. At some point the number of infected people could become greater than half the population size, in which case there can be no doubling until the population size increases by birth rates being greater than death rates.

Correct, however, you hope is that each country is able to start instituting adequate and effective virus management measures like China did, within a month or so of having their first few doubling times, so you do not get a run-away pandemic?

Craig Carlton Dremann Yes, anyone could see the doubling time from such data and one would hope that any administration would act quickly and responsibly to save lives after seeing the data and having scientific advisors to explain it. Unfortunately, the US and UK do not seem to have sensible leadership at the moment.

DOES the USA already know what the implications of the virus going to be, because in May 2018, the John Hopkins Center for Health Security ran a simulation called "Parainfluenza Clade X" to determine what the potential would be of a virus pandemic, and they concluded: "...twenty months 150 million people worldwide--two percent of the global population--have died."

"...The global economy has collapsed under the strain, with the Dow Jones average down 90 percent. U.S. GDP down 50 percent, and unemployment at 20 percent. Washington is barely functioning--the president and vice president are both ill, one one-third of Congress is dead or incapacitated."

People involved in that simulation were Tom Daschle (former leader of US Senate), Dr. Julie Gerberding (former head CDC), Jim Talen (former Missouri senator)--Why are all of these people keeping quiet right now, and not telling us what they saw in the global virus pandemic simulation only two years ago, and helping lead us out of this mess, with some new simulations???!!!

This simulation information is from pages 201-203 of the Bryan Walsh book, "END, A Brief Guide to the End of the World: Asteroids, Supervolcanoes, Rogue Robots, and more", published in 2019. The author also predicts on pages 192-194 the various reasons why, he believes that Trump "lacks the talent and the temperament to lead the United States through an outbreak..." plus, "...there are aspects about Trump that .."make him dangerous in the face of a new disease."

Detail about the John Hopkins "CLADE X" exercise can be read at

http://www.centerforhealthsecurity.org/our-work/events/2018_clade_x_exercise/index.html

We can understand about fractal branching (or multi-fractal branching) here. If

someone infected at one location, then it will infect other places some more

persons and this chain will be started. Continuously this chain leads to fractal branching. Of course, here branching is not very similar like mathematical

construction of it. Because, infected person may infect to some people and

these some people may infect less or more than before infected some people and

continue this chain.

I think that it is a big responsibility on Scientists, Statisticians and

Mathematicians to provide some concrete modelling and solutions regarding

control of coronavirus.

As we know that instant and previous data of coronavirus are available at

https://www.worldometers.info/coronavirus/

People forget that the whole point is to reduce fatalities and increase recovery rates to 100 % (like flu) and not to reduce infection rate . So, u could have a model with 3 compartments : Non-Infected, Infected, Recovered. Again, a lot would depend on whether exposure to the disease once offers immunity against it. Technically, there is something called the logistic equation where for certain ranges of the parameters, there is chaos. Typically, the logistic equation is at a single point over a period of time. In our case, we have a logistic equation with parameters specific to each region, a spatio-temporal model with a set of coupled logistic equations (oscillators in some sense).

Also, spreading through cough / sneezing is line of sight while spread through diffusion is multi-directional and slower. So, if we reduce the symptoms (cough / cold), i suppose we could reduce the risk of infection manifold.

Mathematical models should not just be used as a tool for social control but also to aid in the recovery process and developing treatment strategies. The same models that are used for populations could also be used within human beings (within a population of cells for example)

Also, in attributing the deaths due to Covid-19, care has not been taken to isolate all the factors (such as old age, heart disease, diabetes , poverty , homelessness etc) and in determining how many of those deaths were directly due to the virus. This is scientifically and statistically incorrect. A major factor may be that this epidemic gives an impetus to those who are hopeless over their situation and precipitates matters. And many more deaths could occur indirectly due to the isolation (especially in the case of the vulnerable) than the disease itself

Sundaram Ramchandran Please see my answer to Prof Kassner 3 days ago. The usual SIR model (susceptible, infected, recovered) is too simple to model this epidemic in my opinion, that's why my model has 8 sub-populations. It really should have several more because the seriousness of infection depends upon age and preexisting conditions. In your latest post you also mention psychological conditions, which are important. I did not include a provision for hospitals running out of supplies and personnel, which is what is happening now in many places.

Your comments about the logistic equation, I believe, mix up two different equations. There is the one that maps an interval into itself and as you say, for certain parameter ranges the map is chaotic. The other logistic equation is a differential equation that can be used to model growing but self-limiting populations (like infected people - the more infected, the less susceptible people to infect). The first quadratic gives a map and in the second case it gives the rate of change. Also, it is a theorem that chaos cannot occur in 1-dimensional or 2-dimensional dynamical systems (i.e. autonomous ODEs).

Let me add that the purpose of mathematical modeling is not to control society (probably you did not suggest this) but to inform society so that sensible decisions can be made. Of course, there may be charlatans who make flawed models in order to change policy in a way that benefits them but then again, fraudsters will always find novel ways to deceive.

I enjoy reading your ideas. The more people who think about this epidemic, the better we will be, provided those ideas are shared and discussed.

The discretization of a simple population model (first order differential equation) gives us the famous logistic equation (real quadratic polynomial). But this simple logistic equation leads to chaotic behaviour. It has been studied widely by scientists and mathematicians. Moreover, lots of applications are available in almost all kinds of sciences, engineering and technologies.

It is not necessary to make model based on only differential equations.

In past, many viruses came into existence and several types of models have constructed by many researchers. Some of them chaotic and some them non-chaotic. Some of them exhibits fractal nature and some of them not.

There are many kinds of possibilities to study coronavirus.

Sometimes, looks like

Simplicity is complexity.

Complexity is simplicity.

I also wonder if the inflammation of the lung and increase in the (alveoli ?) are not adaptations to dealing with oxygen poor environments such as high altitude mountainous areas etc

You are correct in pointing out that the logistic ODE becomes a logistic map when using the Euler scheme. However, if it is to simulate the ODE, then the time step needs to be small, in which case there cannot be chaotic or periodic orbits. There definitely are some interesting dynamics when one increases the step size. There is a sequence of period doubling bifurcations followed by a chaotic region when the step size is sufficiently large.

You make a very good point that sometimes a system looks simple when it is not and sometimes it looks complex when it is actually rather simple. This is why mathematics is so powerful!

That is a very good question about inflammation being in response to oxygen-poor atmosphere. I bet there is some literature on the topic since it is very important.

The other problem is the inference of the parameters. We have to enough data or one has to infer from simulations or from controlled experiments (on cultured cells etc)

My Excel charts are not looking good, with all hell breaking loose in the next 30 days, just because the Lockdown was NOT put in place immediately when the first cases appeared in each country. And then, when the spread started, did not go from Lockdown to complete quarantine like China did for 2 months in Wuhan.

Everyone MUST immediately go into quarantine in each country, in order not to suffer to much, the curves are not changing fast enough.

The logistic map

https://www.youtube.com/watch?v=ovJcsL7vyrk

This map will change how you see the world.

If xn denotes number infected , xn denotes uninfected and r denotes infectivity , shouldn't one add xn to the equation, i.e x(n+1) = x(n) + r*x(n)*(1-x(n)) where the second term on the rhs denotes interaction between infected and uninfected in a cell. If we throw in spatial terms and death and recovery , then we have at each position x(i)(n+1) = x(i)(n) + r*x(n)*(1-x(n)) - d*x(i)(n) - R*xi(n) - Sum across the neighborhood cells [D(i)(j) (x(i)(n) - x(j)(n)] where D(i)(j) is diffusivity constants , R is recovery rate , d is death rate

About the curve--California is tracking to double its cases every three days, but in order for the curve to bend, the days between doubling must be 30 more more, which is going to need a total China-style quarantine lockdown.

Of course, the uninfected say y(i)(n) = 1 - x(i)(n)

Can we conclude something about coronavirus based on the following:

Define what is chaos?

Chaos is the phenomenon of occurrence of bounded nonperiodic evolution

in completely deterministic nonlinear dynamical systems with high sensitive

dependence on initial conditions.

This is called deterministic chaos since the governing equations are deterministic.

Chaos - Butterfly Effect - Weather Predictions

To describe the dramatic extreme sensitive dependence of chaotic solution

Lorenz coined the term butterfly effect. In his own words, it reads as follows

"As small a perturbation as a butterfly fluttering its wings somewhere

in the Amazons can in a few days time grow into a tornado in Texas" .

That is even a minute perturbation can cause realizable effects in a finite time under chaotic evolution.

Need more discussion on it.

GOOD NEWS from California and the USA on April 2----The days to doubling of total cases is lengthening. In California our total case numbers were doubling like clockwork, every 3-4 days between March 8 and March 29th.

We put on "Lockdown Lite" on the 16th.

Then from March 30 to today, the numbers are now doubling every 12 days. Once every country ON THE PLANET goes on a severe-enough lockdown for at least a month, then the total number of cases and days to double, start stretching out after 2 weeks of Lockdown.

Yesterday, President Trump was still refusing to do any lockdowns in any of the Republican states that support him, which means that a lot of his supporters will be too sick or dead, to be able to vote in November?

"Chaos is the phenomenon of occurrence of bounded nonperiodic evolution" might be a property of a chaotic orbit but such trajectories but bounded and non-periodic orbits need not be chaotic. E.g. x_n =1/n

A key feature of a chaotic system is topological transitivity. Google it.

Positive Lyapunov exponents. Provide us chaos.

Positive Lyapunov exponents in COMPACT phase space produce chaotic orbits but not necessarily otherwise. E.g. x'=2x has Lyapunov exponent 2 but has no chaotic orbits. This of course can be done in any number of dimensions. With a positive LE in a compact space, then trajectories with arbitrarily close initial conditions will separate exponentially fast for a while but because of compactness, they cannot do so indefinitely. They in fact will become arbitrarily close to each other again, infinitely often.

I know that you know this Mohammad, I am just clarifying it for other readers.

Book Prevention and Control of COVID-19

COVID-19 related publications:

https://publons.com/publon/covid-19/?order_by=date

Vol 24, No. 03, March 2020 — For e-subscribers (PDF)

FEATURES

The Modern-Day Nostradamus: George Yuan Xianzhi

https://www.asiabiotech.com/24/2403/24030032x.html?fbclid=IwAR1-XHP6SQfXSqh54Ua_WOeHpD2xcXbZ88SCtmdKeg13oa2A1p-EClVReRw

How is possibility to add nonlinear term for making chaotic model in the following paper:

https://www.frontiersin.org/articles/10.3389/fphy.2020.00127/full?utm_source=ad&utm_medium=fb&utm_campaign=ba_sci_fphy

What were mathematical models used to study PANDEMIC in 1918?

That can help us to study recent PANDEMIC 2019.

Relating to coupled logistic maps (for example in a 2d region), possibly they could be related to reaction diffusion equations where the reaction terms possibly involve the logistic map and could possibly , in certain regimes , demonstrate the phenomenon of travelling waves (Fisher Waves)

Yes, Sundaram. There is a large body of literature on traveling waves for the logistic equation with diffusion as well as for the diffusive bistable equation, where the reaction term is ku(B-u^2) or more generally ku(a+u)(b-u) where k, a, and b are positive constants. There are also several results for traveling waves where the diffusion is non-local,

e.g. Lu(x) := \int J(x-y)u(y) dy - u(x)\int J(x-y)dy, where the integrals are over the spatial domain.

If J is nonnegative with positive integral then L shares some properties with the Laplacian, such as a maximum principle, so one gets comparison results for the parabolic-like equation.

There are also results for cases where the kernel is not so simple and in cases where the coefficients in the reaction term may depend on space and time. Then there are no traveling waves in the strict sense but one may obtain "spreading speed" results. It is a very interesting subject

Somehow, it seems wrong to enforce complete lockdown on a whole metropolis for an extended period , especially in the context of travel within the city , opening of public places (of course with social distancing norms , face masks etc) etc though of course it does make sense to curtail movement in and out of the city. One has to look at dense clusters of population within the various municipal wards of the city that have a higher than larger probability of contracting the disease due to various socio-economic factors (possibly with the help of GIS) . And again, the key is increasing testing so that those who test negative can freely mix with those who test negative . They should encourage those who want to get tested voluntarily (without any obvious symptoms). In this context, instead of or in addition to the antibody based test (which depends on the immune response to the virus and could be delayed), they could test the presence of the peptides / amino-acid sequences / proteins that are characteristic of the virus

I agree with you. However, in the US we do not have the capability of testing everyone who wants to be tested. We do not even have the capability of testing all those who should be tested. It is unfortunate that we do not have a president who pays attention to science or warnings from national security experts.

Not only US but most of larger countries, there is not "capability of testing everyone who wants to be tested." Almost all countries are using lockdown as substitute but it may extend as much as few months only.

It would be interesting to check how tinkering with the inter-region diffusion parameters could affect the number of infected people especially in densely populated areas with a high proportion of infected people, especially if they are surrounded by areas with a lower density of population. Counter-intuitively, could it lead to reduction in the percentage of infected people and more importantly, slowing the spread of the disease ?? Of course diffusion of the virus itself has to be lowered

I agree that this would be a very interesting project. I hope that you or somebody else has time to investigate. I'd like to see the results and if your conjecture holds true.

Jules, please see my comment of March 26. According to the mathematical sense of the word, there are no fractals appearing in nature, only patterns that up to a certain finite scaling appear fractal-like to the eye. Mathematics is only a model of reality but it does give great insight, including insight into these patterns found in nature.

There were some discussions earlier relating to the existence of fractals and chaotic processes in nature. But then one could ask whether one could model nature as a series of spheres / hemispheres of different radii (again relating to fractals !!) or in terms of polygons, polyhedra or in terms of straight lines or as a combination of all of these (and possibly others). I suppose the latter would be the most accurate and comprehensive answer since the alternative would be to look at is a random noisy jumble which may not be a very constructive approach

I think your prediction is correct. We can relate the spreading of corona virus with fractal branching and chaotic burst.

The epidemic is mainly deterministic, following well-known paths of contagion. Almost the only parts that seem chaotic but are not, are the mixing in the airflow and the mixing of people. OK, there will be something at the atomic scale described by quantum mechanics, but that is about all the stochastic aspect fundamentally involved. That is not to say that stochastic equations cannot be used to SIMULATE the epidemic. They can and should be, since we could not possibly model with great precision the movement of aerosols, droplets, or even people. This however is unrelated to mathematical chaos. Also, as I have pointed out before, there are no fractals observable in spatially discrete systems over finite time intervals.

Chaos is about how deterministic laws lead to surprisingly rich (emergent) behavior. I am wary of using the word unpredictable since the ideas of control etc imply that there is an understanding at some level. I suppose the key is local predictability combined with global (long-term) unpredictability possibly invoking the ideas of differential geometry etc

A dynamical system has unpredictability to a certain extent if it has a positive Lyapunov exponent, since that gives "sensitive dependence on initial conditions". However, if the system does not have a nonempty compact invariant set, it may not be chaotic. Generally, for chaos one needs topological transitivity and a dense set of periodic orbits. Topological transitivity is a fine mixing condition. Something like the existence of orbits that visit every nonempty open set infinitely many times. No need for differential geometry.

I think we can relate the spreading of Corona virus to the definition of self-affinity.

The notion of probability distribution extends from random variables to random functions. Setting B(0)=0, the rescaled function h^(- 1/2)B(ht) has a probability distribution independent of t. This property of scaling is an example of self-affinity.

This spreading is like a probability distribution that apply the transformation in order to prevent the spread of the virus.

Transformation: B(t) - - -> h^(- 1/2)B(ht)

Yes, this is like the heat kernel describing diffusion, a linear process but why should the spread of the virus be like diffusion and a linear process? In fact, it is like that initially on a macro scale (viewing the population as a continuum) but the spread becomes self-limiting due to there being a finite-sized population. This makes it more like a reaction-diffusion equation with logistic nonlinearity, and that would be just the first approximation to the spread. Then would would need to include, feedback from policy decisions such as mask-wearing and social distancing and we would need to make this spatially dependent as not all societies have the same policy. We should add effects of travelers going from one spatial patch to another, with some routes blocked and others not and these being time-dependent. Then we would need to include age-structure to represent contact rates and susceptibility of different age groups.

As you can see, the system would include the self-affinity idea you mention but needs to add so much more complexity to capture the spread, even macroscopically.

I think we could use chaotic attractors in order to get in touch with the virus because it has chaotic behavior such as Lorenz and Rossler dynamical system, I think these system have a common behavior like the virus but we need to look in details for further processing.

Article An analysis of COVID-19 spread based on fractal interpolatio...

https://www.researchgate.net/post/Is_exposing_of_Coronavirus_in_India_like_fractal_branching_and_happened_chaotic_burst