**3. Lissajous parametric equations: the requirements for describing virus-mutation evolution relative to infection and evolution, thermodynamically**

The Lissajous figures graphically represent the relationship between two quantities that have an oscillatory behavior as a function of a certain variable, usually but

**Figure 3.** *Example of a Lissajous figure for variables that oscillate with different periods.*

not necessarily, time. In general, suppose that the two quantities have amplitudes *S*<sup>0</sup> 1 and *S*<sup>0</sup> <sup>2</sup> and repeat themselves with periods T1 and T2. Their behavior as a function of time can be represented by the two relations:

$$\mathcal{S}\_1(t) = \mathcal{S}\_1^0 \bullet \sin\left(2\pi \frac{t}{T\_1}\right) \tag{1}$$

$$S\_2(t) = S\_2^0 \bullet \sin\left(2\pi \frac{t}{T\_2} + \rho\right) \tag{2}$$

The quantities in brackets are called phases, and the term *ϕ* is the delay with which the variable S2 follows the variable S1. The Lissajous figures are completely determined by the amplitudes *S*<sup>0</sup> <sup>1</sup> and *S*<sup>0</sup> <sup>2</sup> , by the periods T1 and T2 and by the delay *ϕ*; these parameters can be reconstructed from their shape.

In particular, the Lissajous figures have nodes (see A in **Figure 3**) when the periods of oscillation *T*<sup>1</sup> and *T*<sup>2</sup> are not equal. In the case of the interaction between host and virus it is very unlikely that this will happen, since the mutations of one are strictly a consequence of the other and it is improbable that the mutations of the virus (for example, represented by the thermodynamics that describes specific genes involved in mutator mutations), oscillate many times within a single oscillation of mutations of the host. This requires evaluation of evidence from experimental data. For now, we will limit ourselves to the case where *T*<sup>1</sup> ¼ *T*<sup>2</sup> and call *TR* their common period of oscillation, which in this case is the repetition time for repeated infectious outbreaks. In general it will depend on the mutation rate *μ* (representing linear, binary, lytic or other type of RNA mutation), so we can generically indicate the phasic term 2*π=TR* as *δ μ*ð Þ.

We therefore obtain:

$$\mathcal{S}\_1(t) = \mathcal{S}\_1^0 \bullet \sin\left(\delta(\mu)\right) \tag{3}$$

*Trajectories of RNA Virus Mutation Hidden by Evolutionary Alternate Reality Thermodynamic… DOI: http://dx.doi.org/10.5772/intechopen.100481*

$$\mathcal{S}\_2(t) = \mathcal{S}\_2^0 \bullet \sin\left(\delta(\mu) + \rho\right) \tag{4}$$

When the two curves have the same period, the shape of the Lissajous figure can be a straight line (such as direct infection of an immediate host as seen in person-toperson pathogenesis), or an ellipse or a circle, depending on the amplitudes *S*<sup>0</sup> <sup>1</sup> and *S*0 <sup>2</sup> and on the delay *ϕ* over increasing time. This takes on an important meaning: it is the speed with which the viral mutation responds to that of the host. For ecothermodynamic stability to be preserved, the range of mutations and spread in virus outbreaks would be isenthalpic, such that the sum of each mutation (infection) redistributes the energy equivalent to the volume of excess, so that the enthalpy remains unchanged.

Human dysbiosis is known to be created by the presence of anaerobic proteobacteria species [53–55] that are found in the hindgut and gut of livestock, mentioned earlier. Specifically, we noted that they include the same species upon which SARS-CoV-2 virus infect leading to disease in their human hosts: we observe virus-infection represents symbiont-driven breakdown within the human orotracheal and gastrointestinal tracts [56], and not only for COVID19 disease [57–59] repeatedly in locations of high-resource consumption (slaughterhouse districts) that continue to operate during and following a severe drought.

We can therefore summarily describe three limiting cases that have in fact been matched with infectious spread in the SARS-CoV-2 outbreaks and which make the Lissajous model ideal:


**Figure 4.** *Example of a Lissajous figure with zero delay, degenerating into a line.*

*S*0 <sup>1</sup> and height *S*<sup>0</sup> <sup>2</sup> within repeated intervals. These can be correlated to mutation frequency per 400th infected patient [60] or relative to the habitat as described earlier [19, 60]. Formally we have *<sup>ϕ</sup>* <sup>¼</sup> *<sup>π</sup>* <sup>2</sup>; from the mathematical point of view it corresponds to the fact that the maximum amplitude of the pathogenic mutations and the **maximum rate of change** of the virus mutations coincide (and vice versa) with the eco-thermodynamics of the habitat as it gravitates towards a new evolved thermodynamic equilibrium. This period theoretically is an estimate of the transition time for mutationrelated events (infecting survivors, reduction of population) on behalf of thermodynamic sustainability of the habitat after abiotic stress. This case therefore appears significant from an evolutionary point of view because it indicates that the maximum adaptive effort of the host/virus occurs in response to the peak of virus/host mutations.

In this case, if the phase shift is *<sup>ϕ</sup>* <sup>¼</sup> *<sup>π</sup>* <sup>2</sup> the second function is transformed into the cosine of the phase and the term *ϕ* disappears (**Figure 5**).

3.Response in antiphase. It corresponds to the case in which *ϕ* ¼ *π* and the Lissajous figure again becomes a line, with a negative angle this time. It corresponds to the case in which the maximum of host-virus infectiousmutations, and their rates of change are of the opposite sign: the mutations of the virus increase while those of the host decrease (and vice versa). In fact, the most important part of a public health anti-outbreak effectiveness strategy of a technology would be based on measured reduction of the delay component for a given pathogen outbreak, thermodynamically. This alone, according to the author, means that that the relationship between delay *ϕ* ¼ *π* and a strategy to prevent outbreaks can be evaluated based on a valid virus-specific spreadmoment relationship to the habitat (**Figure 6**).

**Figure 5.** *Example of a Lissajous figure with* <sup>π</sup> <sup>2</sup>*-delay.*

*Trajectories of RNA Virus Mutation Hidden by Evolutionary Alternate Reality Thermodynamic… DOI: http://dx.doi.org/10.5772/intechopen.100481*

**Figure 6.** *Example of a Lissajous figure with* π *delay, again degenerating into a line.*

The real case will be intermediate between the three listed above and the figure of Lissajous will be an inclined ellipse; the most important parameters, namely the delay *ϕ* and the amplitudes *S*<sup>0</sup> <sup>1</sup> and *S*<sup>0</sup> <sup>2</sup> are derived from the figure without need to know the period *TR*. This is significant because the period can be difficult or impossible to determine if the exact time point in the cycle at which the mutations were detected is unknown. If the period is known from the experimental data, it is obviously of great importance too.

Letting *TR* be the repetition time, then using the Lissajous trajectory equation, we know the duration of the outbreak to be based on the virus-bacteria mutation frequency under stressed-habitat response conditions:

*TR* <sup>¼</sup> *<sup>N</sup> μbacteria* <sup>¼</sup> *<sup>N</sup>*�<sup>1</sup> *μvirus* <sup>¼</sup> *N N*ð Þ �<sup>1</sup> *μR* where the duration can be estimated as <sup>1</sup> *μR* and *N* represents the susceptible host-population size.

And so, the Lissajous curve obtained by plotting the characteristic phase associated for expected infectivity might be drawn for mutation rate per virus species, and the corresponding phase angle *δ* and *S* extracted from the solution describing, for example, *S μhost* ð Þ and *S μvirus* ð Þ based on these equations.

Using the lissajous trajectory model in three dimensions, however, reveals a new opportunity to include evolution as both lateral and vertical. We can also define the amplitude *<sup>S</sup>*<sup>0</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* as the volume of the stress response for an ecothermoydnamic habitat as that which is vulnerable to evolution. If so, we can consider the phase of intersecting signals within the lissajous model as genomic natural mutation frequency, *wx*, affected by the intensity of stress. The intensity of thermodynamic stress frequency includes an interesting ratio, *nx* : *ny* : *nz* which would describe the periodicity of infections in order to achieve the maximum number of points of intersection (infection), for specific evolutionary-spacetime stages (planes). This ratio is a new property that is very powerful in understanding the habitat stress relationship to resulting infections, and not found in any prior reference in the research by the author, to-date. Evolution is described by the position of the set of planes defining the thermodynamic stable balance between the habitat

#### **Figure 7.**

*Schematic demonstration of evolution planes modeled with comparative mutation rates and expected infectiousness (changes in phase). Adapted with Geogebra software [61] and R. Chijner [62].*

*Trajectories of RNA Virus Mutation Hidden by Evolutionary Alternate Reality Thermodynamic… DOI: http://dx.doi.org/10.5772/intechopen.100481*

environment and its incumbent microecology. If so, the lissajous model for the system also could include the phase delay in at least two of those dimensions, such as *ayx* and *ayz*, which represents the expected rate of infectious particle multiplication and the rate of infectious spread, as a function of the varying *Vvulnerable* respectively, throughout the course of evolution from one Alternate Reality to the next stage of evolutionary ecothermodynamic stability.

At any given stage of evolution, the species and the conditions of that habitat may be considered a particular Alternate Reality for which all biodiverse species are sustained and survive. When the conditions requiring new evolution are visible in new stresses, then a new alternate reality evolves and it may be one of many (**Figure 7**).

For exactness, one would expect to find the suitable species mutation and infectiousness rate, for the virus species and host volume. Like livestock at slaughter, this could include the infection of their gut microbiome bacterial organisms dispersed in the habitat shared with human gut deleteriously. Considering that none of these species migrate or evacuate as they would for natural herd immunity, there is no alternative except to endure virus attack in significant pandemics. This means, for a volume of evolution-vulnerable hosts in a given habitat defined by

$$V\_{vulnerable} = A\_{\text{x}} A\_{\text{y}} A\_{\text{z}} \tag{5}$$

We represent

$$\mathcal{S}\_{\mathbf{x}}(\mathbf{x}) = A\_{\mathbf{x}} \cos \left( w\_{\mathbf{x}} \mathbf{x} \right) \tag{6}$$

$$S\_{\mathcal{Y}}(\mathbf{x}) = A\_{\mathcal{Y}} \cos \left( w\_{\mathcal{Y}} \mathbf{x} + a\_{\mathcal{Y}^\mathbf{x}} \right) \tag{7}$$

$$S\_x(\mathbf{x}) = A\_x \cos \left( w\_x \mathbf{x} + a\_{yz} \right) \tag{8}$$

The dimensions for lateral evolution represented by *xz*� and *yz*� planes while vertical evolution in the *xy*-plane. Thus, for each evolution transition, based on stress response that includes arrival of foreign species (vertically) and/or the transition of species within the same habitat (laterally) we observe a snapshot of the same virus spread trajectory involved in the same mutation rate but with different points of intersection and different numbers of intersection. Each represents a characteristic frequency of a specific virus genome: the mutations are consistently carried by the thermodynamic moment of the habitat response to a stress, as the habitat system transitions from one alternate reality to the next. Where researchers traditionally describe mutations as mistakes [63], the lissajous thermodynamic model can disprove the assumption and demonstrate precisions in these genomic departures thorough transitions in alternate reality formation. Otherwise, in twodimension models, the enthalpic energy is represented by amplitudes *S*<sup>0</sup> <sup>1</sup> and *S*<sup>0</sup> 2 , generated by the periods *T*<sup>1</sup> and *T*<sup>2</sup> representing the process of transition from one evolutionary isenthalpic state to another future evolutionary state where energy of the habitat system is fully conserved; and that the delay *ϕ* facilitates the spread width and height for efficient distribution of the mutations necessary. Mutationspread can be described in space time as edges of the trajectory whilst centres of the Lissajous can be correlated to dense locations of chronic mutation-infections. The pathogen's natural moment is self-sustained during the period of an outbreak, until there is no location in the habitat that still needs to evolve towards a stable state. When eco-thermodynamic equilibrium is reached, it returns to zero and represents the point at which mutations cease, theoretically, and the arrival when a new reality following the outbreak is complete. At this point, the evolution process is satisfied by nutrient distribution for the sustainable growth of each species.
