Dynamics of HIV Infection: An entropic-energetic view

Abstract

We propose a time-parametrized analogy between the thermodynamic behavior of a 3-level energy system and the progression of the HIV infection described by the cell population evolution generated by an appropriated cellular automata model. The development of internal energy and its fluctuations, and of the entropy of the 3-level energy system allows the identification of an effective temperature that uniquely characterizes the three main stages of the dynamic process of HIV infection (primary infection, clinical latency, and development of AIDS). Furthermore, this thermodynamical equivalence allows obtaining the instants associated with particular time intervals of the evolution of internal energy and entropy, which are not quantitatively accessible by the usual dynamic models based on differential equations or cellular automata. The maximum entropy point, which marks the threshold between the states of positive and negative temperatures, also corresponds to the onset of the immune system’s exhaustion and the concomitant and inexorable progression to AIDS. Such point lies in the time interval where the energy inversion mechanisms in the populations of non-infected and infected cells occur. This time lag is also characterized by considerable fluctuations of the internal energy when different (patient) samples are compared.

Full text

arXiv:2003.02363v1 [q-bio.PE] 4 Mar 2020
Dynamics of HIV Infection: an entropic-energetic view
Ram´on E. R. Gonz´aleza,∗
, P. H. Figueirˆedoa, S. Coutinhob
aLaborat´orio de Sistemas Complexos e Universalidades. Departamento de F´ısica, Universidade Federal Rural de
Pernambuco, 52171-900, Recife, Pernambuco, Brazil.
bLaborat´orio de F´ısica Te ´orica e Computacional, Departamento de F´ısica, Universidade Federal de Pernambuco,
50670-901, Recife, Pernambuco, Brazil.
Abstract
We propose a time-parameterized analogy between the thermodynamic behavior of a 3-level en-
ergy system and the progression of the HIV infection described by the cell population evolution
generated by an appropriated cellular automata model. The development of internal energy and
its fluctuations, and of the entropy of the 3-level energy system allows the identification of an
effective temperature that uniquely characterizes the three main stages of the dynamic process of
HIV infection (primary infection, clinical latency, and development of AIDS). Furthermore, this
thermodynamical equivalence allows obtaining the instants associated with particular time inter-
vals of the evolution of internal energy and entropy, which are not quantitatively accessible by
the usual dynamic models based on differential equations or cellular automata. The maximum
entropy point, which marks the threshold between the states of positive and negative temper-
atures, also corresponds to the onset of the immune system’s exhaustion and the concomitant
and inexorable progression to AIDS. Such point lies in the time interval where the energy in-
version mechanisms in the populations of non-infected and infected cells occur. This time lag
is also characterized by considerable fluctuations of the internal energy when different (patient)
samples are compared.
Keywords: HIV infection, Dynamics of evolution, Entropic-energetic description, Cellular
Automata
1. Introduction.
The dynamics of HIV infection has been intensively investigated through mathematical mod-
els based on differential equations [1, 2] or by computational simulation of models based on
cellular automata (CA) [3, 4, 5, 6, 7, 8]. The objectives of these studies are quantifying and an-
alyzing the underlying dynamic processes, testing hypotheses and assisting in the establishment
of treatment strategies and protocols. HIV infection leads to a progressive reduction in the num-
ber CD4+T cells. HIV viral load and CD4 cell counts are the main parameters used to describe
the infection process and to define treatment protocols and therapy strategies. The mathematical
models generally describe the evolution of viral burden and the CD4+T cells population in the
healthy, infected and dead cells states.
∗Corresponding author
Email address: ramayo_g@yahoo.com.br (Ram´on E. R. Gonz´alez)
Preprint submitted to Physica A March 6, 2020

In this work, we present a “thermodynamical” vision for the progression of the macroscopic
states of the course of the HIV infection after the primary infection by means of a CA model
designed to describe the dynamics of HIV infection [7]. We assume that at each time step of the
automata evolution (which corresponds to one week) the macroscopic states are viewed as ther-
modynamical equilibrium states with well-defined energy and entropy according to the average
populations of the CD4+T cells in the healthy, infected and dead cells states of the CA model.
The evolution of healthy (non-infected) and infected CD4+T cell concentrations are used to
characterize the three distinct stages of the dynamics of HIV infection, the primary infection (PI),
the clinical latency (CL), and the AIDS development, which occur in two different time scales.
The initial PI phase is quite similar to that in others in more common viral infections: the viral
load reaches its maximum five to six weeks after contamination and endures of the order of 10 to
12 weeks after which the viral load decreases to almost undetectable levels. The clinical latency
is characterized initially by a very high concentration of healthy cells but gradually decreasing in
the time scale of the order of 2 to 10 years or more, leading to impairment of essential functions of
the immune system [9]. During CL phase, the virus remains latent in the body in compartments
where they are not identified and eliminated by the immune system. Eventually, such viruses are
reactivated leading to the development of the acquired immune deficiency syndrome (AIDS).
Cellular automata (CA) models are spatially structured and therefore more appropriate and
efficient to describe local interactions and correlations between cells that participate in the dy-
namics between the immune response and the viral particles. The model proposed in [7] was
specially designed to describe the entire course of HIV infection, from the primary stage to the
development of AIDS, by modeling the dynamic behavior of the T-cell population in lymph
nodes. In its rules of evolution, this model tests the combination of two important factors: the
high viral mutation rate and the spatial location of the infected T cells in the lymph nodes. The
results qualitatively reproduce the three stages of the dynamics of HIV infection with their com-
patible and particular time scales. Such model has inspired other authors to propose new models
to investigate other aspects of the infection, including the processes of antiretroviral therapies
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
In this work, we re-examine the model proposed in [7] from the perspective of the thermody-
namic behavior of a three-level energy system. This correspondence highlights important aspects
of the dynamics of HIV infection that are not revealed by the usual time series analysis meth-
ods. In the following subsection, we describe the CA model and its visualization as a three-level
system. In Section 2, we formulate the three-level specific model to HIV dynamics by setting
the relevant parameters for the associated energy spectrum. The description of the energy and
entropy behaviors related to HIV dynamics are analyzed and broadly discussed in Section 3, and
a summary of the findings is presented in Section 4.
1.1. Cellular automata based models
The CA model proposed in [7] considers the population of CD4+T cells located in the lymph
nodes and the local interactions between its different states, healthy, infected (types AeB) or
dead, which evolve differently in the three phases of infection. The spatial structure of the
lymph nodes designed simply by a (L×L)-square lattice of size L=700 assuming periodic
boundary conditions. The active neighborhood for the interaction between the cells considers
the first eight closest neighbors (Moore neighborhood) and the initial configuration of the system
consists of a fixed fraction of infected cells (pHI V ) randomly distributed in the network filled with
healthy cells. The temporal evolution of the cell states is done synchronously in unit time steps
corresponding to one week.
2

Four simple rules govern the dynamics of the automata, in which some characteristic param-
eters of the process play an important role. The evolution process of the infection progression
occurs by contact of a healthy cell and an infected cell. Healthy cells may become infected, at
the next instant of time, if there is at least one infected cell type A and/or Rinfected cells type B
in their active vicinity. Infected cells of type A possess a high capacity of contamination while
those of type B have their ability diminished since they have already been identified by the im-
mune system through the specific immune response. The parameter R, (1 ≤R≤8) regulates the
decrease in the transmission capacity of the infection between the cells and in this work is set in
R=4. A specific study of the effects of variation of this parameter can be found in [13]. Infected
cells of types A and B evolve differently: while the first type survives by τ=4 time steps, the
latter is eliminated, that is, it changes to dead state in the next time step.
The τparameter regulates the average time spent by the immune system to develop the adap-
tative immune response to HIV when it is at full capacity. In this way, a newly pool of infected
cell will survive τ+1 weeks on average until it starts to be is eliminated. The time interval
of time τis also associated with the peak of viral load occurring during the primary infection,
which marks the effective action of the adaptive immune response. In the model proposed in [7],
the value τ+1 corresponds to five weeks, which is the average characteristic value of clinically
observed primary adaptive immune responses in most common viral infections, including HIV
infection. The use of a fixed value for τis a crude approximation since it does not take into
account the variability of the immune response between individuals nor among different viral
strains. The consideration of a density probability distribution for τwith appropriated mean and
dispersion certainly could improve the model. However, an investigation developed in the ref-
erence [15] shown that the variation such parameter in the interval 4 ≤τ≤9 reveals that the
three-stage dynamics profile remains qualitatively invariant and no substantial change occurs in
the clinical latency period. Therefore, for the present purpose of describing the long-term evolu-
tion of HIV infection after the primary infection such improvement seems not to be crucial, since
from the technical point of view the parameter τonly calibrates or adjust the short-time scale of
the CA model.
Finally, the majority of dead cells are replaced by new CD4+T cells produced by the bone
marrow machinery in most of their fullness with probability pr, while a fraction (1 −pr) remains
dead. In the meanwhile a proportion piprof the new CD4+T cells are replaced by type-A in-
fected cells and (1 −pi)prby healthy cells. The probability pi, which is responsible for the
feedback of new infected cells in the arena mimics all mechanisms that enable the HIV to evade
the immune response such as the high HIV mutation rate [20] and the existence of reservoirs of
latent infected cells lodged in other extra-nodes lymphatic compartments [21, 22, 23]. An anal-
ysis of the robustness of the dynamics of the HIV infection with respect to piand prparameters
points out that qualitatively the three-stage evolution remains unchanged under the variation of
orders of magnitude of these quantities, only producing a temporal displacement of the periods
associated with the peak of the primary infection and the latency period [15].
The value parameters were establish from experimental data following [7]: pHIV =0.05 was
chosen based on the observation that one in 102or 103T cells harbor viral DNA during the pri-
mary infection; pi=10−5is due to the fact that only one in 104to 105cells in the peripheral
blood of infected individual expresses viral proteins; and pr=0.99 to represent the high ability
of the immune system to replenish the depleted cells. See [24, 25].
In Table 1, the essential parameters of the model are summarized following [7].
The dynamics of the HIV infection generated by the simulations from the CA model proposed
3

Parameters Description Value
LLattice Linear size 700
pHIV Initial fraction of infected cells 0.05
prFraction of replaced dead cell 0.99
piFraction of infected replaced cells 0.00001
RMinimum number of infected cells type B for contact propagation 4
τTime delay (weeks): infected cell type A →infected cell type B 4
Table 1: Parameters used in the original simulations in the CA model [7] to describe the HIV infection dynamics.
in [7] can be summarized in the graphs of Figure 1, which illustrates the temporal evolution of
the average fractions of CD4+Thealthy,infected(A +B) and dead cells just after the primary
infection. The 326-week instant and the time interval (260, 350), bounded by a yellow stripe,
mark essential events in HIV dynamics, which will be explicitly revealed by the three-level model
later.
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Figure 1: Fractions of CD4 T+cells healthy (blue), infected (A e B) (red) and dead (black) as a function of time (weeks).
Note that the time scale stars at week 12 after the occurrence of the primary infection stage. The error bars indicate the
standard deviation concerning the average over 100 simulations (patients). The 326-week instant and the yellow-stripe
time interval (260, 350) mark relevant events in HIV dynamics explained later.
1.2. Three level system
In a recently published paper [26], an energy model for a three-level system was proposed
to study problems related to the Gibbs entropy. Such model considers a system of Nspins,
4

whose energy states are distributed among three energy levels. Therefore, the total number of
spins Nwith total energy Ecan be viewed as distributed between such levels characterizing a
given configuration. Each configuration, which may be characterized by the number of spins
accommodated on each level (populations), corresponds to many microscopic states provide the
spins are considered as indistinguished particles and can be permutated within each level. One
of such configuration, however, should correspond to the ground state fixed as the zero energy
state.
The population Niand the respective energy Eifor the states i=1,2 and 3 are related
accordingly the following conservation equations for the total particle number and energy:
N1+N2+N3=N(1)
E1N1+E2N2+E3N3=E(2)
For a fixed total energy Ethe number of possible microstates or indistinguishable configu-
rations that meet the conditions (1) e (2) is given by N!/(N1!N2!N3!). Hence the Boltzmann’s
entropy Sof this system when N≫1 can be written, accordingly to the Stirling’s approximation,
as:
S=kB[Nlog N−N1log N1−N2log N3−N3log N3] (3)
where kBis the Boltzmann constant.
1.3. Negative absolute temperatures
The condition that characterizes the thermodynamic equilibrium is that the a variation of the
entropy relative to the energy is constant and equal to the inverse of the absolute temperature,
1
T=∂S
∂E.(4)
In the year 1956, a seminal paper of Ramsey [27] opened the doors to the study of interesting
behaviors presented by nuclear and magnetic systems. When the entropy of a system presents
a monotonically decreasing behavior relative to energy, this system would have absolute nega-
tive temperatures, according to the equation (4). These circumstances where entropy is a convex
function of energy can occur in finite energy spectrum systems when the population of the state
with higher energy exceeds those of the lower energy states. In these cases, the energy distri-
bution function, characterized by the Boltzmann factor, presents an unusual behavior, that is, an
exponential growth.
P(E)∼eE/k|T|(5)
with T<0. Behaviors of this nature have recently been found in localized systems with finite
and discrete energy spectra, such as the limited spectrum quantum systems [28, 29].
In the subsequent section, we consider a three-level model to explore the corresponding sig-
nificance of negative temperature states in the process of HIV infection.
5

2. The three-level system model for HIV infection
In this Section, we design a three-level physical model to describe the dynamics of HIV
infection (without treatment) analogous to that proposed in [26], considering the CD4+T cell
population characterized by its three possible states: infected N1(t), healthy N2(t) e dead N3(t).
The first state comprising all infected cells types A and B, the second describing the cells not
reached by the viruses, and finally the third portion of cells dead. Figure 2 illustrates the energy
and transition diagram of the proposed model. This diagram represents the possible relative
Figure 2: Scheme of the energy levels and the transitions between the states of the three cell populations prescribed by
the model: E1for infected, E2for healthy and E3for dead cells respectively. The arrows indicate the direction of possible
transitions.
energy levels, whose values will be determined below, and the possible transitions between the
three characteristic states of the infection dynamics according to the rules described above. The
relative values between these energies are fixed from the populations of each state in the condition
set to the threshold of the onset of AIDS, that is, when n∗
1∼0.7, n∗
2∼0.2 and n∗
3∼0.1,
respectively, where ni=Ni/N, (i=1,2,and 3).
In terms of the variables of the HIV infection model, the equations (1-3) can be rewritten as:
n1(t)+n2(t)+n3(t)=1 (6)
E1n1(t)+E2n2(t)+E3n3(t)=E(t) (7)
S(t)=−n1(t) log n1(t)−n2(t) log n2(t)−n3(t) log n3(t) (8)
In the equations above E(t) and S(t) labels the energy (in units of kBT) and the entropy (in units
of kB) per particle at the instant t, respectively.
6

The infection process is a dynamic process. In the diagram shown in the figure 2 we indicate
the possible transitions between the three energy levels. The lowest energy state corresponds to
that of dead cells. From this, transitions can occur to the two upper levels corresponding to the
probability PD→Hof the dead cells are replaced by healthy cells produced by the immune system
or by infected cells, with probability PD→I, from infected latent cell reservoirs corresponding to
transitions 1 and 2 of the figure (2), respectively. Another transition (transition 4) can happen
between the infected state and the state of dead cells due to the action of the immune response
after a specified mean time interval, simulated in the model by τ+1 time steps.
In our correspondence between the CA model and the 3-level energy model the transition
probabilities PD→Hand PD→I, which represents the reposition of dead cells, are mimicked in the
3-level model by the respective Boltzmann weights, as indicated in the equations below:
PD→H=pr(1 −pi)∝e−(E2−E3)and PD→I=prpi∝e−(E1−E3),(9)
where Ei(i=1,2,3) are the energies in units of kBT. As can be seen in Figure 2, the direct
transition from the infected to the healthy state is absent once this process is prohibited by any
mechanism.
The transition probabilities PD→Hand PD→I, corresponding to transitions 1 and 2 shown
in Figure 2, were calculated according to equation (9). Note that PD→H=pr(1 −pi) is the
probability that dead cells will be replaced by healthy (probability pr) and non-infected cells
(probability (1 −pi)), while PD→Iis the probability of dead cells being replaced by healthy
cells and infected (probability pi). Based on the values used for the parameters, pr=0.99 and
pi=10−5, energy levels were estimated at ε1(infected) =3.82939; ε2(healthy) =-7.68353
and ε3(dead) =-7.69359 in kBTunits. Notice that the high value of pr=0.99 reflects one
of the central assumptions of the base AC model, which assumes that bone marrow cd4T cell
production machinery remains fully functioning without being affected by infection. However,
the small fraction (1 −pr=0.01) that is not replaced at any given time step can be replaced at
the next step and so on.
3. Results and discussion
To represent the dynamics of HIV through a three-level system, we performed NSsimulations
of the cellular automata model proposed in [7] considering the distinct initial conditions but with
the same values of the parameters displayed in Table (1). At every time step tthe mean values
of the populations of cells N1(t), N2(t) e N3(t) are recorded, and we calculate the energy E(t) and
the entropy S(t) according to equations (7) and (8).
In figure 3 we show the histogram of the energy distribution in the interval (−7.7,0.5), which
corresponds to the dynamic process after the primary infection until the steady state, meaning
from week 12 until week ∼450. The region of the spectrum with negative energies E ∼ −1.8
corresponds to states where the infected cell population does not yet surpass that of uninfected
(healthy +dead), which occurs on average until week 320 after the peak of the primary infec-
tion. Up to E ∼ −5.4, we observe a decay of the probability density function P(E), but with a
secondary peak around ∼ −6.5. This global behavior shows the highest occupation at the most
negative energy levels due to the predominance of healthy cells at this stage. In the dynamics
of infection, this occurs during the clinical latency period until ∼231 weeks (average). The ob-
served peak around E ∼ 6.5 corresponds to the dynamics of infection at the time interval between
165 and 188 weeks. At this phase of the dynamics, the probability of the emergence of compact
7

structures becomes significant, which lead the course of the infection to the inexorable onset of
AIDS as observed in references [9, 10] and discussed further below. After values of E ∼ −5.4
the p.d.f. P(E) remains virtually constant until energies ∼ −1.8.
-8 -7 -6 -5 -4 -3 -2 -1 0
ε
0
0,1
0,2
0,3
0,4
0,5
P(ε)
-6 -4 -2 0
ε
0
0,2
0,4
0,6
0,8
1
S(ε)
S = 0.94
ε = -1.8
ε = 0.53
ε = -1.8
Figure 3: Histogram of the energy distribution P(E) for energy states with positive (E<−1.8) and negative temperatures
(E>−1.8). The doted line is guide for the eyes to separate the two regions.
The spectrum region after E ∼ −1.8 presents a histogram more homogeneous with the growth
of the number of states for higher energies. Positive energy values indicate the predominance of
infected cells (high energies) when the unwanted development of AIDS is established with the
collapse of the immune response from the week ∼350 (average). The inset of Figure 3 presents
the time-parametrized plot of entropy versus energy. We observed that the maximum entropy
value Smax =0.94 occurs for ESmax =−1.8. For energy values E<ESmax the function S(E) have
a positive slope corresponding to positive temperature values. On the other hand for E>ESmax the
derivative slope is negative, hence the corresponding three-level physical model to equilibrium
states with negative absolute temperatures, according to equation (4).
We present in the figures 4 and 5 the time-dependent behavior of the energy and the entropy
to investigate in more detail the regions of positive and negative temperatures.
In the Figure 4 the temporal behavior of energy is separated into three well-marked regions
according to rate of energy growth: (i) 10 <t<200 weeks corresponding to the clinical latency
period when the density of infected cells is low and increases linearly with a time; (ii) 200 <
t<450 weeks corresponding to the onset of AIDS when the density of infected cells when the
rate of growth of infected cells approximately doubles and finally; (iii) t≥450 weeks when the
infected cell rate reaches its maximum value and becomes stationary. In the time interval where
8

Figure 4: Energy evolution from 10 to ∼200 weeks: clinical latency phase. From ∼200 to ∼450 weeks: development of
AIDS.
the AIDS onset occurs, the temperature, which was previously positive, becomes negative after
the ”crossing” between densities of healthy and infected cells , which happens in approximately
320 weeks, making prevalent the population of infected cells.
The Figure 5 shows the behavior of the entropy of the system as a function of time. Observe
the gradual growth of the entropy until reaching its maximum value in t≃326 weeks. Initially
at an approximately constant rate during the clinical latency period (10 ∼200 weeks) followed
by an increase in the growth rate up to t≃260 weeks. Next, the growth decreases continuously
to zero characterizing the instant where the temperature signal change occurs. Subsequently, the
entropy decreases steadily until reaching the steady-state equilibrium value (full establishment
of AIDS).
Comparing the graph of Figure 5 with the behavior of the entropy against the energy, shown
in the inset of Figure 3, we observe that the instant the entropy reaches its maximum value marks
the change from the regime of positive to negative temperatures as indicated in the Figure 3. This
value corresponds to negative but close to zero energy values.
From the analysis of energy and entropy behaviors associated with the dynamics of HIV
infection (no treatment), we conclude that the dominant state of infected cells is the one where the
highest energy level is populated with the majority of cells (particles) characterizing by negative
values of absolute temperature. Since this state has most of the particles in the highest excited
state hence it has the highest energy. In these circumstances, the infected state becomes the state
of equilibrium of the system.
From the temporal evolution of the entropy, we observed an almost linear monotonic growth
9

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Figure 5: Entropy evolution (weeks). The maximum point Smax =0.94 occurs in tSmax ≃326 weeks. At the point
(S ≃ 0.8,t≃260) the rate of change of entropy decreases until it cancels out at tSmax . Interval (260 −350) weeks is the
time lag for occurrence of high fluctuations on cell concentrations.
until approximately t∼200 weeks, when the appearance of partially ordered and compact spatial
structures occur that sequester cells with predominance in the number of infected cells. Such
structures originate during clinical latency and gradually grow occupying the entire network
causing the development of AIDS. Henceforth, although the mean densities, characteristic of
the steady state of each cell type, a process of spatial disorder begins to occur in the compact
structures evolving into configurations where spatial correlations are eliminated. In other words,
with the growth of entropy in the first half of clinical latency, the system is losing information
associated with the loss of dynamic correlation that happens in t≃200 weeks. This loss of
correlation can be observed through the statistical behavior of the dynamics of infection via
random matrix (RM) theory [30]. This RM methodology also shows that in the final phase of
the AIDS onset the dynamic correlation is recovered reaching its maximum value described by a
probability distribution function GOE (Gaussian Orthogonal Ensemble) [30].
In the Figure 3 we found that there is a small region of energy fluctuations still with negative
values but corresponding to negative temperatures. In the dynamics of infection, this region
corresponds to the time interval between ∼260 and ∼326 weeks, when the “crossing of the
densities of infected and uninfected cells occurs, as mentioned above. To quantify the energy
fluctuations and to establish a criterion specifying the region where absolute inversion occurs
between infected and uninfected cell populations, we define the relative deviation σE, as the
10

ratio between the mean square deviation of the energy δEand its absolute value,
σE=δE
|E| .(10)
Figure 6 illustrates the graph of this quantity as a function of time on a semi-logarithmic
scale indicating a variation of three orders of magnitude. In other words, fluctuations of order
103higher than the absolute value of the energy. It appears that these fluctuations are responsible
for the states in which energy and temperature are both negative. The instants of time for which
the relative deviation σE=1 correspond to t=231 e t=372 weeks and mark the beginning
and the end of the transition respectively. Comparing with the dynamics of infection generated
by the cellular automata model for several individuals (100), we observe that this interval (231 −
372 weeks) encompasses the interval where error bar overlaps (r.m.s.d.) occur of healthy and
infected cell densities (260 −350 weeks). Furthermore, the peak (divergence) where relative
deviation occurs (∼309 weeks) corresponds approximately to the point of maximum overlap
where infected and healthy cell densities intersect, as shown in Figure 1.
Figure 6: Semi-log plot of Energy relative error dynamics, same parameters used in Figure 1
4. Concluding Remarks
In analogy with a three-level energetic thermodynamic system, the dynamics of HIV infec-
tion described by a model of cellular automata [7] was analyzed revealing relevant aspects not
seen before. For example, equilibrium states with positive temperatures are identified with states
of negative energies E ≤ −1.8 associated with states where the population of infected cells does
not supplant non-infected (healthy and dead), which occurs, on average, until the 320 weeks after
11

the primary infection. Moreover, for values of the energy spectrum E ≤ −5.4 there is a marked
global decay of the probability density function P(E), however, displaying a local maximum
nearby E ≃ −6.5. This peak marks an essential moment in the dynamics of CD4+T cell popu-
lations when it is observed the emergence of compact spatial structures that grow monotonically
while sequestering a large number of infected cells. This situation can occur, on average, for
the values of the parameters adopted in the original model [7], between 165 and 180 weeks after
the primary infection and is determinant to establish the clinical latency period of 200 weeks, on
average.
From the onset of compact spatial structures, the population of infected cells grows linearly at
a rate well above that observed during clinical latency, accompanied by the concomitant decrease
in healthy cells. In this period, known as the AIDS development, the energy grows appreciably
over time, finally reaching steady state around 450 weeks. The continuous inversion in the
infected cell population in this period is marked by the moment when the energy E ≃ −1.8
and the entropy reaches its value maximum, at week 326 on average. Hence, the temperature
signal is reversed thereafter when the rate of growth of the infected cells reduces reaching the
steady state at week 450. The time interval between observation of the decay of P(ε) (∼week
180) and the steady-state onset (∼week 450) corresponds in HIV dynamics to the interval where
fluctuations in infected and healthy cell densities occur when the behavior of different individuals
is put into comparison, as illustrated by the error bars in Figure 1. From the clinical point of view,
the beginning of this interval marks the onset of AIDS and its concomitant development resulting
in the appearance of opportunistic diseases that lead patients to death. The present study of the
energy-entropic model reveals that the time interval where such fluctuations in the density of
infected and healthy cells are most significant corresponds to the time interval where the mean
squared deviation of energy relative to its absolute value exceeds 1 reaching a peak of three
orders of magnitude in value (week ∼309), associated with crossing the average densities of
healthy and infected cells, and very close to the time of occurrence of maximum entropy (week
326).
To summarize we conclude that this simple entropic-energetic analogy of the dynamics of
HIV infection, presented in this study, unravel important moments in the course of the HIV
infection that can not be observed and described by cellular automata models. The inclusion
of antiretroviral therapy (ARV) mechanisms, as considered by the authors in references [18] is
the subject of current studies. Such mechanisms of ARV therapies introduce a rapid and intense
change in infected and healthy cell densities, almost instantaneous when compared to the model
timescales leading the HIV dynamics probably occurring with transitions between states out
of equilibrium making the correspondence with the thermodynamic energy-entropic structure
conceptually tricky.
References
[1] A. S. Perelson, P. W. Nelson, Mathematical Analysis of HIV-1 Dynamics in Vivo, SIAM Rev. 41 (1999) 3?44.
[2] Martin A. Nowak and Robert M. May, Virus Dynamics Oxford Univ. Press, New York, (2000)
[3] Ch. F. Kougias and J. Schulte. Simulating the immune response to the HIV-1 virus with cellular automata. Journal
of Statistical Physics, 60:263–273, 1990. 10.1007/BF01013677.
[4] R. B. Pandey and D. Stauffer. Metastability with probabilistic cellular automata in an HIV infection. Journal of
Statistical Physics, 61(1/2):235–240, 1990.
[5] R. B. Pandey. Cellular automata approach to interacting cellular network models for the dynamics of cell population
in an early HIV infection. Physica A: Statistical Mechanics and its Applications, 179(3):442–470, 1991.
[6] A T Haase. Population Biology of HVI-1 Infection: Viral and CD4 +T Cell Demographics and Dynamics in
Lymphatic Tissues. Annual Review of Immunology, 17:625–656, 1999.
12

[7] Rita Maria Zorzenon dos Santos and S´ergio Coutinho. Dynamics of HIV Infection: A Cellular Automata Approach.
Physical Review Letters, 87, 168102, (2001)
[8] Bernaschi, M., and F. Castiglione. Selection of escape mutants from immune recognition during HIV infection.
Immunology and Cell Biology, 80, 307-313, (2002)
[9] Zdenek, Hel. Jerry, R. McGhee, and Jiri Mestecky. HIV infection: first battle decides the war. Trens in Immunology,
27(6):274-280, 2006.
[10] Peter M. A. Sloot, Fan Chen, and Charles Boucher. Cellular automata model of drug therapy for HIV infection.
In ACRI ’01: Proceedings of the 5th International Conference on Cellular Automata for Research and Industry,
pages 282–293, London, UK, 2002. Springer-Verlag.
[11] A. Benyoussef, N. El HafidAllh, A. ElKenz, and H. Ez-Zahraouyand M. Loulidi. Dynamics of HIV infection on
2D cellular automata. Physica A, 322:506–520, 2003.
[12] M. A. Peer, N. A. Shan, and K. A. Khan. Cellular automata and its advances to drug therapy for HIV infection.
Indian Journal of Experimental Biology, 42(2):131–137, 2004.
[13] Guillermo Solovey, Fernando Peruani, Silvina Ponce Dawson, and Rita Maria Zorzenon dos Santos. On cell
resistance and immune response time lag in a model for the hiv infection. Physica A: Statistical Mechanics and its
Applications, 343:543 – 556, 2004.
[14] Veronica Shi, Abdessamad Tridane, and Yang Kuang. A viral load-based cellular automata approach to modeling
HIV dynamics and drug treatment. Journal of Theoretical Biology, 253(1):24–35, july 2008.
[15] P. H. Figueirˆedo, S. Coutinho, and R. M. Zorzenon dos Santos. Robustness of a cellular automata model for the
HIV infection. Physica A: Statistical Mechanics and its Applications, 387(26):6545 – 6552, 2008.
[16] E. Burkhead, J. Hawkins, and D. Molinek. A dynamical study of a cellular automata model of the spread of HIV
in a lymph node. Bulletin of Mathematical Biology, 71(1):25–74, January 2009.
[17] Monamorn Precharattana, Arthorn Nokkeaw, Wannapong Triampo, Darapond Triampo, and Yongwimon Lenbury.
Stochastic cellular automata model and Monte Carlo simulations of CD4+T cell dynamics with a proposed alter-
native leukapheresis treatment for HIV/AIDS. Computers in Biology and Medicine, 41(7):546 – 558, 2011.
[18] Ram´on E. R. Gonz´alez, S´ergio Coutinho, Rita Maria Zorzenon dos Santos, Pedro Hugo de Figueirˆedo. Dynamics
of the HIV infection under antiretroviral therapy: A cellular automata approach. Physica A, 392(2013)4701-4716.
[19] Ram´on E. R. Gonz´alez, Pedro Hugo de Figueirˆedo, S´ergio Coutinho. Cellular automata approach for the dynamics
of HIV infection under antiretroviral therapies: The role of the virus difussion. Physica A, 392(2013)4717-4725.
[20] Cuevas, Jos M. AND Geller, Ron AND Garijo, Raquel AND Lpez-Aldeguer, Jos AND Sanjun, Rafael. Extremely
High Mutation Rate of HIV-1 In Vivo PLOS Biology 13 (09), 1–19, (2015).
[21] Cao, Youfang and Lei, Xue and Ribeiro, Ruy M. and Perelson, Alan S. and Liang, Jie Probabilistic control of
HIV latency and transactivation by the Tat gene circuit Proceedings of the National Academy of Sciences 115 (49),
12453–12458, (2018).
[22] Bruner, Katherine M. and Wang, Zheng and Simonetti, Francesco R. and Bender, Alexandra M. and Kwon,
Kyungyoon J. and Sengupta, Srona and Fray, Emily J. and Beg, Subul A. and Antar, Annukka A. R. and Jenike,
Katharine M. and Bertagnolli, Lynn N. and Capoferri, Adam A. and Kufera, Joshua T. and Timmons, Andrew and
Nobles, Christopher and Gregg, John and Wada, Nikolas and Ho, Ya-Chi and Zhang, Hao and Margolick, Joseph B.
and Blankson, Joel N. and Deeks, Steven G. and Bushman, Frederic D. and Siliciano, Janet D. and Laird, Gregory
M. and Siliciano, Robert F. A quantitative approach for measuring the reservoir of latent HIV-1 proviruses, Nature,
566 (7742), 120–125, (2019).
[23] Szu-Han Huang AND Yanqin Ren AND Allison S. Thomas AND Dora Chan AND Stefanie Mueller AND Adam
R. Ward AND Shabnum Patel AND Catherine M. Bollard AND Conrad Russell Cruz AND Sara Karandish AND
Ronald Truong AND Amanda B. Macedo AND Alberto Bosque AND Colin Kovacs AND Erika Benko AND
Alicja Piechocka-Trocha AND Hing Wong AND Emily Jeng AND Douglas F. Nixon AND Ya-Chi Ho AND
Robert F. Siliciano AND Bruce D. Walker AND R. Brad Jones, Latent HIV reservoirs exhibit inherent resistance
to elimination by CD8+T cells The Journal of Clinical Investigation, 128 (2), 876–889, (2018).
[24] Schnittman, S. M and Psallidopoulos, M. C and Lane, H. C and Thompson, L. and Baseler, M. and Massari, F. and
Fox, C. H and Salzman, N. P and Fauci, A. S. The reservoir for HIV-1 in human peripheral blood is a T cell that
maintains expression of CD4, Science, 245, (4915), 305 – 308 (1989);
[25] Schnittman, Steven M. and Greenhouse, Jack J. and Psallidopoulos, Miltiades C. and Baseler, Michael and Salz-
man, Norman P. and Fauci, Anthony S. and Lane, H. Clifford. Increasing Viral Burden in CD4+T Cells from
Patients with Human Immunodeficiency Virus (HIV) Infection Reflects Rapidly Progressive Immunosuppression
and Clinical Disease. Annals of Internal Medicine, 113 (6), 438–443, (1990).
[26] Daan Frenkel and Patrick B. Warren. Gibbs, Boltzmann, and negative temperatures. Am. J. Phys. 83, 163(2015).
[27] Norman F. Ramsey. Thermodynamics ans Statistical Mechanics at Negative Absolute Temperatures. Physical
Review, 103, 1(1956).
[28] S. Braun, J. P. Ronzheimer, M. Schreiber, S. S. Hodgman, T. Rom, I. Bloch, U. Schneider. Negative Absolute
Temperature for Motional Degrees of Freedom. Science 339, (2013).
13

[29] Jorn Dunkel and Stefan Hilbert. Consistent thermostatistics forbids negative absolute temperatures. Nature Physics.
10, (2014).
[30] Ram´on E. R. Gonz´alez, Iury A. X. Santos, Marcos G. P. Nunes, Viviane M. de Oliveira and Anderson L. R. Barbosa.
Statistical behavior of time dynamics evolution of HIV infection. Physics Letters A, 381: 2912-2916 (2017).
14