# Dynamics and numerical simulations to predict empirical antibiotic treatment
of multi-resistant Pseudomonas aeruginosa infection

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###### Abstract.

This work discloses an epidemiological mathematical model to predict an
empirical treatment for dogs infected by Pseudomonas aeruginosa. This
dangerous pathogen is one of the leading causes of multi-resistant infections
and can be transmitted from dogs to humans. Numerical simulations and
appropriated codes were developed using Matlab software to gather information
concerning long-time dynamics of the susceptible, infected and recovered
individuals. All data compiled from the mathematical model was used to provide
an appropriated antibiotic sensitivity panel for this specific infection. In
this study, several variables have been included in this model to predict
which treatment should be prescribed in emergency cases, when there is no time
to perform an antibiogram or the cost of it could not be assumed. In
particular, we highlight the use of this model aiming to become part of the
convenient toolbox of Public Health research and decision-making in the design
of the mitigation strategy of bacterial pathogens.

###### Key words and phrases:

numerical simulations, SIRI model, multi-resistant, Pseudomonas aeruginosa

###### 2010 Mathematics Subject Classification:

Primary: 34K28, 34A12 Secondary: 92B05

∗ Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
∗∗ Both authors contributed equally to this study

This work has been partially supported by Project 2004/0001203 from Ministerio
de Educación y Ciencia (Spanish government), Project PGC2018-096540-B-I00 from
Ministerio de Ciencia, Innovación y Universidades (Spanish government) and
Project US-1254251 from Consejería de Economía y Conocimiento (Junta de
Andalucía).

Javier López-de-la-Cruz∗,∗∗

Departamento de Ecuaciones Diferenciales y Análisis Numérico

C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla

41012 Sevilla, Spain

María Pérez-Aranda∗∗

Departamento de QuÍmica Orgánica y Farmacéutica and Departamento de
Microbiología y Parasitología

C/ Profesor García González, s/n, Facultad de Farmacia. Universidad de Sevilla

41012 Sevilla, Spain

Ana Alcudia∗ and Belén Begines

Departamento de Química Orgánica y Farmacéutica

C/ Profesor García González, s/n, Facultad de Farmacia. Universidad de Sevilla

41012 Sevilla, Spain

Tomás Caraballo

Departamento de Ecuaciones Diferenciales y Análisis Numérico

C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla

41012 Sevilla, Spain

Eloísa Pajuelo

Departamento de Microbiología y Parasitología

C/ Profesor García González, s/n, Facultad de Farmacia. Universidad de Sevilla

41012 Sevilla, Spain

and Pedro J. Ginel

Departamento de Medicina y Cirugía Animal

Edificio Hospital Clínico Veterinario, Campus de Rabanales, Facultad de
Veterinaria, Universidad de Córdoba

14071 Córdoba, Spain

(Communicated by the associate editor name)

##  1\. Introduction

Multi-drug resistant infections, typically caused by a bacterium who is
resistant to more than one antibiotic, are an emerging problem and constitute
an important threat to public health (see [1, 2, 3]), in fact, more than
700,000 people die per year due to this kind of infections currently. Latest
2019 World Health Organization report claims for urgent action to avert the
antimicrobial resistance crisis and highlights that, if no action is taken,
drug-resistant diseases could cause 10 million deaths each year by 2050 and
damage the economy as catastrophic as the 2008-2009 global financial crisis.

In this sense, these types of infections are a defiant therapeutic challenge
in both human and veterinary medicine, due to the complex mechanisms that
bacteria have developed to resist to antibiotics and their capacity to
transmit this resistance both vertically (to their descendants) and
horizontally (to other bacteria) (see [1, 2]).

In particular, Pseudomonas aeruginosa is one of the most frequent pathogen
that acquires or develops antibiotic multi-resistance. This Gram-negative rod
is very ubiquitous in the environment, can persist in water and soil with
minimal nutritional presence and can tolerate several conditions of humidity
and temperature (see [4, 5]).

Figure 1. Microscopic image of an exudate smear from the ear canal of a dog
with otitis externa. The presence of numerous rods as well as inflammatory
cells can be observed

As revealed, this capacity to resist extreme situations as well as the high
capacity to develop multi-resistance after exposure to antimicrobials and
cross-resistance between agents, makes Pseudomonas aeruginosa a leading cause
of nosocomial infections, responsible for 10% of all hospital acquired
infections. In addition, these are often severe and life threatening, and very
often very difficult to treat because of the limited susceptibility to
antimicrobial agents (see, for instance [1, 3, 4]). In particular, in dogs
this pathogen is often a cause of otitis and pyoderma. Due to the close
physical contact with their owners , dogs are a potential reservoir to multi-
resistant Pseudomonas aeruginosa, which could lead to zoonotic (animal to
human) transmission (see [5, 6, 7]).

This work compiles 40 cases of Pseudomonas aeruginosa otitis in dogs from the
south of Spain in the last five years and their susceptibility to an extended
panel of antibiotics (see Table LABEL:at in Appendix). Although the best
procedure to treat these infections is to perform an antibiogram study,
unfortunately that takes five to seven days (see [2, 3, 4]) and not always
economic resources are available, specially in Third World Countries where
sometimes the sanitary system has limited funding. Significantly, there are
some critical clinical circumstances in which the treatment cannot be delayed
due to the severity of the symptoms and the possible risk that may lead to an
irreversible situation to the patient.

Figure 2. Purulent otitis externa caused by Pseudomonas aeruginosa in a dog

In view of the facts described before, it would be crucial to have a powerful
tool that could help practitioners to predict the best empirical treatment
possible for this type of infection. Therefore, a professional could prescribe
an antibiotic treatment as soon as the infection is detected, in case there is
no time available to perform the antibiogram.

In this sense, the goal of the present work is to predict, using a
mathematical model, the best empirical antibiotic treatment to avoid the
bacterial development of resistance mechanisms.

The classical reference about epidemics in Mathematics is due to Kermack and
McKendrick (see [8, 9, 10]) who proposed a compartment model based on simple
assumptions on the flow rates between different groups of members in some
population. Basically, the population is divided in three groups: susceptible,
the individuals who are not infected yet but susceptible to the disease;
infected, who are the infected individuals; and recovered, individuals who
have been infected in the past and cannot be re-infected. From now on, both
groups will be denoted by $S$, $I$ and $R$, respectively.

Since Kermanck and McKendrick proposed the so-called SIR model, many authors
have worked to develop innovations which improve the original SIR model in
order to describe more complex situations [11, 12]. Among these more
sophisticated models, it is worth mentioning the SIR model with relapse, or
the also called SIRI model, whose main difference compared to the SIR one is
based on the the fact that recovered individuals could be re-infected.

The SIRI model can be mathematically described with the following system of
differential equations

(1.1) $\displaystyle\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle\Lambda(S+I+R)-\beta SI-\mu S,$ (1.2)
$\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta
SI-(\alpha+\kappa+\mu)I+\gamma R,$ (1.3) $\displaystyle\frac{dR}{dt}$
$\displaystyle=$ $\displaystyle\kappa I-(\mu+\gamma)R.$

where $S=S(t)$, $I=I(t)$ and $R=R(t)$ denote the number of susceptible,
infected and recovered individuals, respectively. $\Lambda$ and $\mu$ are the
birth and death rates of the populations, respectively, $\beta$ is the
transmission coefficient of the disease, $\alpha$ denotes the death rate due
to the disease, $\kappa$ describes the rate at which the infected individuals
become recovered and $\gamma$ is the rate at which the recovered individuals
become infected. We also remark that every constant is positive.

The different flows between the three groups of the population can be
represented as the transfer diagram in Figure 3.

Figure 3. Transfer diagram of SIRI model

To this end, we consider a population of 192 dogs of which 40 are infected and
Matlab is used to make numerical simulations and understand the long-time
dynamics of the different groups (susceptible, infected and recovered) of the
population depending on time. In addition, we include a code in Matlab to
compute the final number of susceptible, infected and recovered individuals in
each case as well as the time at which it happens. Therefore, we display
tables and bar diagrams compiling the information to predict the most
effective antibiotics in every case.

The present work is organized as follows: Section 2compiles the materials and
methods used in this work. Section 3 shows numerical simulations of every
case. All the relevant information is depicted on tables and we use bar
diagrams to reach a better understanding of the dynamics. Section 4 provides a
list of the most suitable antibiotics to treat infected dogs. Finally, Section
5 includes conclusions Significantly, there is a strong evidence of the
utility of this type of mathematical models performed in collaboration with
other disciplines, such as health sciences.

##  2\. Materials and methods

For this work, forty Pseudomonas aeruginosa samples were aseptically collected
from dogs with purulent otitis externa and were sent to a specialized
laboratory to perform their culture and antibiogram analysis. We refer the
reader to Table LABEL:at in Appendix to check the whole list of antibiotics
that were tested and their rates of resistance.

Antimicrobial susceptibility was tested using Kirby-Bauer method, also known
as disk diffusion antibiotic sensitivity testing or disk diffusion test
according to Clinical Laboratory Standards Institute (CLSI) guidelines (see
[13]).

Figure 4. Antimicrobial susceptibility assay using disk diffusions test in
Pseudomonas aeruginosa culture

In addition, Matlab was used to make numerical simulations and program some
codes to obtain useful data about the behavior of the population.

##  3\. Mathematical study. Numerical simulations and data

In this section we present the mathematical study using a SIRI model
(1.1)-(1.3) predict an empirical treatment.

The complete panel of antibiotics tested in laboratory, as well as their
corresponding resistance rate is presented in Table LABEL:at in Appendix.

To organize all data obtained based on the resistance rate, the following
groups are depicted:

  * •

Group 1- Antibiotics 100% resistant.

  * •

Group 2- Antibiotics with resistance close to 100%.

  * •

Group 3- Antibiotics with resistance between 30% and 50%.

  * •

Group 4- Antibiotics with resistance between 20% and 30%.

  * •

Group 5- Antibiotics with resistance between 10% and 20%.

  * •

Group 6- Antibiotics with resistance between 1% and 10%.

  * •

Group 7- Antibiotics with resistance 0% resistant.

The percentages corresponding to the groups cited before are represented in
Figure 5.

Figure 5. Resistance rates of antimicrobials

For the implementation of the model, three tools are considered essential:

  * •

Figures (Figures 6, 8, 10, 12, 14, 16, 18) containing numerical simulations of
the SIRI model (1.1)-(1.3) with a code developed in Matlab, which will allow
to determine the dynamics of the susceptible, infected and recovered
individuals for an estimated period of time. Moreover, it allows observing how
the numbers of the different groups of our population change and its
asymptotic behavior.

  * •

Tables (Tables 1, 2, 3, 4, 5, 6, 7) where the limit number of each group of
the population is collected for different values of the transmission
coefficient of the disease, as well as the time to reach the corresponding
value. These tables are very useful to convert the dynamics information in
numerical data. This second tool was implemented due to a code programmed in
Matlab.

  * •

Bar diagrams (Figures 7, 9, 11, 13, 15, 17, 19) where the final number of each
group of the population is presented for different values of the transmission
coefficient of the disease.

Once the mathematical study is accomplished, it could provide practitioners an
appropriate and very convenient panel of the most effective antibiotics, which
will provide essential information in case the antibiogram is not available.

In order to develop the mathematical study, a population of 192 dogs were
included, where 40 of them are infected by Pseudomonas aeruginosa, 152 are
susceptible and none of them are recovered at the initial time, considering
this initial value as zero.

Concerning the parameters in the SIRI model (1.1)-(1.3) the birth and death
rates of the population were $\Lambda=0$ and $\mu=0$, assuming that no dogs
were born or died during our study. Since the average duration of this
infection is considered around 14 weeks, $\alpha=0$ since dog mortality null
in this pathology. The recovery rate $\kappa$ and the reinfection rate
$\gamma$ depend on the groups of antibiotics in study (see Section 3). We
remark that the transmission coefficient of the disease is unknown but it is a
rate. Therefore, every case will be studied using different values of this
parameter between 0.1 and 1.

###  3.1. Resistance 100%

This subsection corresponds to antibiotics with a resistance rate of 100%.

In Figure 6 we display three different panels showing the dynamics of
susceptible, infected and recovered individuals, respectively. In every panel
the number of members in each group are plotted depending on time. As the
initial number of dogs in the experiment is 192 and only 40 are infected, the
initial value of susceptible is 152 and the initial value of infected as 40.
Noticeable, there are no recovered dogs at the initial time.

Figure 6. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when using an antibiotic with a resistance rate of
100%

In Table 1 the number of susceptible, infected and recovered individuals at
the final time is depicted for different values of the transmission
coefficient of the disease $\beta$. For instance, when $\beta=0.1$ the limit
of susceptible individuals is 0.9892 and this limit is reached after 6.1075
days. The limit of infected individuals is 186.0153 and it is reached after
5.3200 days and, finally, the number of susceptible individuals remains
constant zero during all the time.

In Table 1 we can notice that most of dogs will be infected in approximately 5
days and no dogs will be recovered corresponding to a 100% resistance rate.

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 6.1075 | 0.9892 | 5.3200 | 186.0153 | 0.0000 | 0.0000
$\beta=0.2$ | 3.0450 | 0.9997 | 2.8175 | 185.0218 | 0.0000 | 0.0000
$\beta=0.3$ | 2.0300 | 0.9911 | 1.8200 | 183.0880 | 0.0000 | 0.0000
$\beta=0.4$ | 1.5225 | 0.9821 | 1.4525 | 182.0727 | 0.0000 | 0.0000
$\beta=0.5$ | 1.2250 | 0.9368 | 1.1025 | 180.0496 | 0.0000 | 0.0000
$\beta=0.6$ | 1.0150 | 0.9635 | 0.9800 | 179.0448 | 0.0000 | 0.0000
$\beta=0.7$ | 0.8750 | 0.9183 | 0.7875 | 177.0125 | 0.0000 | 0.0000
$\beta=0.8$ | 0.7700 | 0.8748 | 0.7350 | 176.0281 | 0.0000 | 0.0000
$\beta=0.9$ | 0.6825 | 0.8819 | 0.6125 | 174.0724 | 0.0000 | 0.0000
$\beta=1$ | 0.6125 | 0.8888 | 0.5950 | 173.1766 | 0.0000 | 0.0000

Table 1. Data for different values of the transmission coefficient of the
disease $\beta$ in case of using antibiotics with a resistance rate of 100%

Based on Table 1 we present in Figure 7 a bar diagram to to illustrate the
results obtained.

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$2$$4$$6$

$6.11$

$3.05$

$1.52$

$1.02$

$0.77$

$0.61$

$5.32$

$2.82$

$1.45$

$0.98$

$0.74$

$0.60$

$0.00$

$0.00$

$0.00$

$0.00$

$0.00$

$0.00$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 7. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease in case of using antibiotics with a resistance rate of 100%

In order to avoid redundancies, in the following cases we will present the
different figures following the structure presented before.

###  3.2. Resistance close to 100%

This subsection corresponds to antibiotics with a resistance rate close to
100%. Due to the similarities of the rates among the antibiotics of this
group, Amoxicillin and Clavulanic acid has been considered to be
representative, with a resistance rate of 97.50%.

Figure 8. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when using an antibiotic with resistance 97.50%

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 6.2475 | 0.9857 | 5.5475 | 182.0012 | 1.9775 | 3.0085
$\beta=0.2$ | 3.1150 | 0.9875 | 2.9400 | 181.0199 | 1.0675 | 3.0005
$\beta=0.3$ | 2.0825 | 0.9541 | 1.8725 | 179.0799 | 0.7700 | 3.0049
$\beta=0.4$ | 1.5575 | 0.9571 | 1.4875 | 178.0629 | 0.6300 | 3.0395
$\beta=0.5$ | 1.2425 | 0.9603 | 1.1200 | 176.1075 | 0.5425 | 3.0513
$\beta=0.6$ | 1.0325 | 0.9637 | 0.9800 | 175.0559 | 0.4900 | 3.0900
$\beta=0.7$ | 0.8925 | 0.8982 | 0.7875 | 173.1549 | 0.4375 | 3.0372
$\beta=0.8$ | 0.7700 | 0.9709 | 0.7175 | 172.1004 | 0.4025 | 3.0154
$\beta=0.9$ | 0.6825 | 0.9746 | 0.5950 | 170.2078 | 0.3850 | 3.0559
$\beta=1$ | 0.6125 | 0.9783 | 0.5425 | 169.0158 | 0.3675 | 3.0651

Table 2. Data for different values of the transmission coefficient of the
disease $\beta$ in case of using antibiotics with a resistance rate of 97.50%

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$2$$4$$6$

$6.25$

$3.12$

$1.56$

$1.03$

$0.77$

$0.61$

$5.55$

$2.94$

$1.49$

$0.98$

$0.72$

$0.54$

$1.98$

$1.07$

$0.63$

$0.49$

$0.40$

$0.37$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 9. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease when an antibiotic with a resistance rate of 97.50% is prescribed

###  3.3. Resistance 30%-50%

This subsection corresponds to the group of antibiotics with a resistance rate
between 30% and 50%. Among all the antibiotics included, Enrofloxacin, with a
resistance rate of 34,21%, is used as an example.

Figure 10. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when using an antibiotic with a resistance rate of
34.21% is prescribed

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 16.9575 | 0.9994 | 12.8275 | 64.0028 | 14.5075 | 124.0053
$\beta=0.2$ | 8.1025 | 0.9991 | 6.0375 | 64.0011 | 7.3850 | 124.0110
$\beta=0.3$ | 5.1975 | 0.9824 | 3.1325 | 63.0067 | 4.4975 | 123.0316
$\beta=0.4$ | 3.7450 | 0.9942 | 2.0475 | 63.0236 | 3.5525 | 123.0152
$\beta=0.5$ | 2.9050 | 0.9703 | 1.2250 | 63.0482 | 2.6425 | 122.0149
$\beta=0.6$ | 2.3450 | 0.9633 | 0.5250 | 62.2112 | 2.0825 | 121.0022
$\beta=0.7$ | 1.9425 | 0.9752 | 0.3150 | 62.8271 | 1.9775 | 121.0016
$\beta=0.8$ | 1.6450 | 0.9842 | 0.2100 | 62.7526 | 1.6625 | 120.0108
$\beta=0.9$ | 1.4175 | 0.9882 | 1.5225 | 62.9793 | 1.4525 | 119.0726
$\beta=1$ | 1.2425 | 0.9731 | 0.1225 | 62.8659 | 1.2775 | 118.0144

Table 3. Data for different values of the transmission coefficient of the
disease $\beta$ when an antibiotic with a resistance rate of 34.21% is used

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$5$$10$$15$$20$

$16.96$

$8.10$

$3.75$

$2.35$

$1.65$

$1.24$

$12.83$

$6.04$

$2.05$

$0.53$

$0.21$

$0.12$

$14.51$

$7.39$

$3.55$

$2.08$

$1.66$

$1.28$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 11. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease when an antibiotic with a resistance rate of 34.21% is prescribed

###  3.4. Resistance 20%-30%

This subsection is focused in the group of antibiotics with a resistance rate
between 20 and 30 Gentamicin, with a resistance rate of 23.68%, has been
chosen as an example.

Figure 12. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when an antibiotic with a resistance rate of
23.68% is used

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 17.4475 | 5.9890 | 13.8425 | 42.0097 | 16.5375 | 139.0219
$\beta=0.2$ | 11.6025 | 0.9932 | 6.8075 | 43.0015 | 10.2375 | 143.0048
$\beta=0.3$ | 7.3500 | 0.9962 | 3.9025 | 43.0159 | 6.9300 | 143.0074
$\beta=0.4$ | 5.2675 | 0.9891 | 2.3450 | 43.0155 | 4.6900 | 142.0205
$\beta=0.5$ | 4.0425 | 0.9787 | 1.1025 | 43.0038 | 3.9900 | 142.0091
$\beta=0.6$ | 3.2200 | 0.9987 | 0.1925 | 43.2377 | 3.0625 | 141.0019
$\beta=0.7$ | 2.6600 | 0.9864 | 0.0525 | 42.0490 | 2.4675 | 140.0001
$\beta=0.8$ | 2.2400 | 0.9866 | 0.0350 | 42.6969 | 2.0650 | 139.0233
$\beta=0.9$ | 1.9250 | 0.9699 | 0.0175 | 42.2403 | 1.7675 | 138.0169
$\beta=1$ | 1.6625 | 0.9865 | 0.0175 | 43.3302 | 1.5575 | 137.0543

Table 4. Data for different values of the transmission coefficient of the
disease $\beta$ when an antibiotic with a resistance rate of 23.68% is
evaluated

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$5$$10$$15$$20$

$17.45$

$11.60$

$5.27$

$3.22$

$2.24$

$1.66$

$13.84$

$6.81$

$2.35$

$0.19$

$3.5\cdot 10^{-2}$

$1.75\cdot 10^{-2}$

$16.54$

$10.24$

$4.69$

$3.06$

$2.07$

$1.56$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 13. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease in case of using antibiotics with a resistance rate of 23.68%

###  3.5. Resistance 10%-20%

In this subsection the antibiotics with a resistance rate between 10% and 20%
are studied. In this case, Tobramycin, with a resistance rate of 15,15% has
been considered as an example.

Figure 14. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when using an antibiotic with a resistance rate of
15.15%

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 33.2850 | 1.9970 | 23.6075 | 28.0021 | 30.7300 | 157.0074
$\beta=0.2$ | 17.6050 | 0.9952 | 10.1500 | 28.0028 | 16.3275 | 158.0060
$\beta=0.3$ | 11.0775 | 0.9917 | 5.6175 | 28.0040 | 9.5200 | 157.0011
$\beta=0.4$ | 7.8575 | 0.9982 | 3.2025 | 28.0095 | 7.1575 | 157.0100
$\beta=0.5$ | 5.9850 | 0.9881 | 0.6300 | 28.9433 | 5.1625 | 156.0244
$\beta=0.6$ | 4.7425 | 0.9984 | 2.8350 | 28.9999 | 4.5325 | 156.0066
$\beta=0.7$ | 3.8850 | 0.9942 | 2.5200 | 28.9953 | 3.6050 | 155.0242
$\beta=0.8$ | 3.2550 | 0.9880 | 2.1175 | 28.9973 | 2.9575 | 154.0025
$\beta=0.9$ | 2.7650 | 0.9933 | 1.8025 | 28.9944 | 2.5200 | 153.0369
$\beta=1$ | 2.3800 | 0.9956 | 1.5750 | 28.9719 | 2.1875 | 152.0432

Table 5. Data for different values of the transmission coefficient of the
disease $\beta$ in case of using antibiotics with a resistance rate of 15.15%

$\beta=0.1$$\beta=0.2$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$0$$10$$20$$30$$40$

$33.29$

$17.61$

$7.86$

$4.74$

$3.26$

$2.38$

$23.61$

$10.15$

$3.20$

$2.84$

$2.12$

$1.58$

$30.73$

$16.33$

$7.16$

$4.53$

$2.96$

$2.19$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 15. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease in case when an antibiotic with resistance rate of 15.15% is used

###  3.6. Resistance 1%-10%

This subsection studies the group of antibiotics with a resistance rate
between 1% and 10%. Imipenem, with a resistance rate of 7,14% is considered
representative.

Figure 16. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when an antibiotic with a resistance rate of 7.14%
is assayed

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 34.7375 | 30.9964 | 25.6550 | 10.0024 | 34.2650 | 144.0183
$\beta=0.2$ | 33.4250 | 1.9991 | 15.4875 | 12.0004 | 30.4500 | 171.0015
$\beta=0.3$ | 23.6775 | 0.9966 | 7.5425 | 12.0045 | 21.0700 | 172.0027
$\beta=0.4$ | 16.6600 | 0.9966 | 3.3425 | 12.0015 | 15.4700 | 172.0035
$\beta=0.5$ | 12.5475 | 0.9978 | 0.9975 | 13.9876 | 10.7975 | 171.0122
$\beta=0.6$ | 9.8875 | 0.9940 | 1.6275 | 13.9938 | 9.2575 | 171.0008
$\beta=0.7$ | 8.0325 | 0.9916 | 1.9950 | 13.9923 | 7.1750 | 170.0007
$\beta=0.8$ | 6.6675 | 0.9927 | 1.9600 | 13.9912 | 5.8100 | 169.0100
$\beta=0.9$ | 5.6350 | 0.9884 | 1.8025 | 13.9858 | 4.8475 | 168.0026
$\beta=1$ | 4.8125 | 0.9920 | 5.9500 | 12.9997 | 4.1650 | 167.0077

Table 6. Data for different values of the transmission coefficient of the
disease $\beta$ in case of using antibiotics with a resistance rate of 7.14%

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$10$$20$$30$$40$

$34.74$

$33.43$

$16.66$

$9.89$

$6.67$

$4.81$

$25.66$

$15.49$

$3.34$

$1.63$

$1.96$

$5.95$

$34.27$

$30.45$

$15.47$

$9.26$

$5.81$

$4.17$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 17. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease in case of using antibiotics with a resistance rate of 7.14%

###  3.7. Resistance 0%

This subsection corresponds to the antibiotics with total effectiveness.

Figure 18. Dynamics of susceptible, infected and recovered individuals
depending on time (in days) when an antibiotic with a resistance rate of 0% is
used

| Susceptibles | Infected | Recovered
---|---|---|---
| Time (days) | Individuals | Time (days) | Individuals | Time (days) | Individuals
$\beta=0.1$ | 0.4375 | 145.9988 | 0.7000 | 0.9824 | 0.5950 | 39.0306
$\beta=0.2$ | 0.8225 | 136.9863 | 0.8400 | 0.9809 | 0.8050 | 48.0938
$\beta=0.3$ | 0.9100 | 125.9459 | 1.0150 | 0.9564 | 0.9450 | 59.0773
$\beta=0.4$ | 1.4875 | 110.9878 | 1.1900 | 0.9695 | 1.3125 | 74.0443
$\beta=0.5$ | 1.3825 | 94.9995 | 1.3300 | 0.9970 | 1.4525 | 90.0654
$\beta=0.6$ | 1.6800 | 77.9836 | 1.4175 | 0.9886 | 1.4875 | 106.0461
$\beta=0.7$ | 1.6275 | 62.9900 | 1.4350 | 0.9982 | 1.4175 | 120.0633
$\beta=0.8$ | 2.6425 | 49.9998 | 1.4175 | 0.9869 | 1.4175 | 132.0735
$\beta=0.9$ | 1.7850 | 39.9951 | 1.3825 | 0.9673 | 1.3300 | 141.0661
$\beta=1$ | 1.6975 | 31.9943 | 1.3300 | 0.9899 | 1.2775 | 148.0393

Table 7. Data for different values of the transmission coefficient of the
disease $\beta$ in case of using antibiotics with resistance 0%

$\beta=0.1$$\beta=0.2$$\beta=0.4$$\beta=0.6$$\beta=0.8$$\beta=1$$0$$1$$2$$3$

$0.44$

$0.82$

$1.49$

$1.68$

$2.64$

$1.70$

$0.70$

$0.84$

$1.19$

$1.42$

$1.42$

$1.33$

$0.60$

$0.81$

$1.31$

$1.49$

$1.42$

$1.28$

Transmission coefficient of the diseaseTime (days)Susceptibles Infected
Recovered

Figure 19. Time (days) to reach the limit number of susceptible, infected and
recovered individuals for different values of the transmission coefficient of
the disease in case of using antibiotics with a resistance rate of 0%

##  4\. Results and discussions

This study includes a group of 192 dogs affected by otitis externa living in
the south of Spain, in which 40 individuals were infected by Pseudomonas
aeruginosa. All samples were aseptically collected and sent to a laboratory to
to perform an antibiogram that included 52 different antibiotics.

We recall that the goal of this work was to develop an antibiotic panel that
could be used as an empirical treatment for dogs infected with Pseudomonas
aeruginosa, a pathogen transmissible to humans, with high multi-resistance
capacities and that is considerate the leading cause of nosocomial infections
in human medicine. To this end, the well-known SIRI model (1.1)-(1.3) was used
which helped us to provide a priority antibiotic treatment.

In this way, the behavior of the different groups of individuals of the
population of dogs infected by Pseudomonas aeruginosa was studied using the
SIRI model to observe the dynamics of every group. In addition a code was
programmed in Matlab to obtain the most relevant information from these
simulations and then, they were represented in a table where the limit value
of susceptible, infected and recovered individuals were included as well as
the time at which the corresponding limits were reached.

The SIRI model (1.1)-(1.3) reunited all the necessary characteristics and,
after comparing with the possible application of other mathematical models,
this is the one that was the most suitable and accurate relating to all the
epidemiological data compiled as well as to clinical behavior of the
infection. To our knowledge, this is the firs time this specific application
of the model is described relating to pathogenic microorganisms dynamics.

Especially, it is worth mentioning that this work is quite relevant for
mathematical science, since it proves that mathematical modelling should
become part of the convenient toolbox of Public Health research and decision-
making, specially in the design of models of empirical treatment choices based
on epidemiological data.

Interestingly, results show that a significant number of antibiotics tested
(27,45%) have null action to Pseudomonas aeuruginosa infection. In addition,
it is remarkable to add that some of them are third generation Cephalosporins
(more specific and expensive, with a more complex development). Moreover,
approximately 12% of the antibiotics included had a resistance rate of almost
100% and 8% of them between 50% and 30%. Hence, when an empirical antibiotic
treatment is provided, the chance of ineffectiveness or partial effect (which
is one of the causes that increases the resistance rate) is, at least, 30%.

This fact highlights the potential health problem that presents this multi-
resistant microorganisms and the necessity of an accurate tool to prescribe
the most specific and effective treatment. On the other hand, almost a third
part of the total of antibiotic tested were always effective, however the use
of a high number of them are hospital restricted.

Regarding the results obtained, we consider that the SIRI model implemented in
the present work can be a very useful tool to predict the dynamic of
Pseudomonas aeruginosa infection in dogs, since it provides an antibiotic
panel that in case is empirically prescribed, it has an accuracy of efficacy
of more than 90% (Figure 20) . Subsequently, it can be used as an efficient
model to formulate potential personalized treatment in both human and
veterinary medicine, in order not only to save cost, but also to avoid future
problems of development of multi-resistance by this bacteria in both humans
and animals, because its high level of specificity and efficacy against the
infection.

Although all the antibiotics included in the recommended panel have an
efficacy of more than 90%, the use of most of them are hospital-restricted,
which could present, in this concrete infection and microorganism, a problem
in prescribing the treatment in some clinical situations.

The results obtained are in consonance with the epidemiological data and the
information that the World Health Organization, as well as numerous
Epidemiological Health Services, in human and veterinary medicine, are
providing. Consequently, the present work highlights the severity of the
multi-resistance development in pathogens and reinforces the necessity of new
tools to implement in the strategies that have to be applied.

Figure 20 provides a selective antibiotic panel in case is necessary to
prescribe an efficient empirical treatment to Pseudomonas aeruginosa
infection.

Figure 20. Antibiotic panel recommended to prescribe an empirica treatment
against Pseudomonas aeruginosa infection in Spain

##  5\. Conclusions

Multi-resistant infections are an unequivocal public health problem and are
responsible for an increasing number of deceased patients in hospitals. In
this sense, companion animals can act as a reservoir of this type of
microorganisms that can be transmitted to humans.

At this point, since no studies of these characteristics have been previously
described, it is urgent and critical to develop an efficient tool to minimize
the antimicrobial resistance process. Therefore, we have applied an adapted
SIRI model as a compatible and potential tool based on Matlab software.
Remarkably, this model allows predicting the behavior of Pseudomonas
aeruginosa infection and is essential to select an effective empirical
treatment when culture and antibiogram techniques are no available or the
severity of the infection obliges an imminent treatment, providing an
antibiotic panel with an accuracy of efficacy of more than 90%.

Our study suggest that this method could be suitable to be used in other
infections therapies in other microorganisms and in other countries based on
their epidemiological data. Furthermore, it is likely to reinforce the
accuracy to provide a more effective treatment of choice, using a model that
behaves as a very useful tool to be implemented in Public Health research and
decision-making strategies.

## Appendix

In Appendix we provide the table of antibiotics tested as a treatment.

Group | Antibiotic | Resistance (rate)
---|---|---
Beta lactam | Cephalosporins | Cefalotin | 100%
antibiotics |  | Ceftazidime | 16,67%
|  | Cefotaxime | 47,37%
|  | Cefepime | 21,05%
|  | Cefoxitin | 100%
|  | Cefuroxime | 100%
|  | Cefazolin | 100%
|  | Ceftazidime/CA | 82,35%
|  | Cefovecin | 95,24%
|  | Cefalexin | 100%
|  | Cefotaxime/CA | 100%
|  | Ceftiofur | 100%
|  | Cefpodoxime | 100%
|  | Cefoperazone | 0%
|  | Cefquinome | 0%
| Beta lactamase | Amoxicilin clav. | 97,50%
| inhibitors | Ampicilin | 100%
|  | Piperacillin-tazobactam | 5,88%
| Carbapenem | Imipenem | 7,14%
|  | Ertapenem | 53,85%
|  | Meropenem | 0%
| Monobactams | Aztreonam | 20,69%
| Penicilins | Piperacillin | 0%
|  | Ticarcillin | 0%
|  | Penicillin | 0%
Aminoglycosides |  | Tobramycin | 15,15%
|  | Gentamicin | 23,68%
|  | Amikacin | 6,25%
|  | Neomycin | 50%
|  | Kanamycin | 0%
|  | Sisomicin | 0%
|  | Streptomycin | 0%
Quinolones |  | Ciprofloxacin | 28,57%
|  | Enrofloxacin | 34,21%
|  | Marbofloxacin | 23,81%
|  | Nalidixic acid | 93,33%
|  | Pradofloxacin | 28,57%
|  | Levofloxacin | 0%
|  | Oxolinic acid | 0%
|  | Maxifloxacin | 0%
|  | Lomefloxacin | 0%
Tetracyclines |  | Doxycycline | 92,86%
|  | Tetracycline | 83,33%
|  | Tigecycline | 0%
Afenicol |  | Chloramphenicol | 100%
|  | Florfenicol | 100%
Miscellaneous |  | Polymyxin B | 15,79%
|  | Colistin | 0%
|  | Fusidic acid | 100%
|  | Fosfomycin | 100%
Macrolide |  | Erythromycin | 33,33%
Sulfamide |  | Sulfonamides+Trimethoprim | 100%
Table 8. Table of all the antibiotics assayed in the antibiograms included in
the study

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