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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# A discrete epidemic model for bovine Babesiosis disease and tick populations

Diego F. Aranda
/ Deccy Y. Trejos
/ Jose C. Valverde
Published Online: 2017-06-16 | DOI: https://doi.org/10.1515/phys-2017-0040

## Abstract

In this paper, we provide and study a discrete model for the transmission of Babesiosis disease in bovine and tick populations. This model supposes a discretization of the continuous-time model developed by us previously. The results, here obtained by discrete methods as opposed to continuous ones, show that similar conclusions can be obtained for the discrete model subject to the assumption of some parametric constraints which were not necessary in the continuous case. We prove that these parametric constraints are not artificial and, in fact, they can be deduced from the biological significance of the model. Finally, some numerical simulations are given to validate the model and verify our theoretical study.

PACS: 05.45.Pq; 87.10.Ca; 87.10.Vg

## 1 Introduction

Bovine babesiosis is the most important arthropod-borne disease of cattle worldwide. It provokes morbidity and even mortality of cattle after a tick-borne, parasitic infection. Ticks infected due to the ingestion of parasites in the blood of infected cattle are the most relevant transmission agent of such a disease. The permanence of the infection depends on the probability of the vertical transmission in ticks population as suggested in [1] for dengue.

The most common varieties of babesiosis are Babesia bovis and Babesia bigemina which can be found throughout the majority of tropical regions. In fact, as reported in the literature [2] both varieties became endemic in the south of USA and affected to the associated industry very seriously.

Control strategies based on vaccination and antiparasitic treatments have been performed [3]. But, due to residues and other problems, some vaccines and drugs have been eliminated from these strategies [4, 5].

All these features make this disease interesting to be modeled mathematically in order to know its dynamical behavior. Actually, to know its behavior could lead to the design of new control strategies.

In the literature, one can find few deterministic mathematical models of bovine Babesiosis. In [6], we introduced a model based on ordinary differential equations for bovine Babesiosis caused by Babesia bigemina and Babesia bovis for the first time. After that, in view of this previous one, a model of partial differential equations for Babesia bovis was formulated by Friedman and Yakubu [7]. Besides, based on our work in [6], Carvalho et al. [8] presented a new version of our classical model changing the ordinary derivative by fractional Caputo derivate. Likewise, in [9], authors study a fractional-order scheme model on the disease. Our continuous model in [6] has also served as a basis to set out and study similar models, including other factors such as a two-stage in the cattle class in [10] or the effect of seasonal changes in [11]. Moreover, in [12] a study of the dynamic behavior of our model is performed using a multistage modified sinc method which is a computational algorithm for approximating solutions of the classical system in a sequence of (time) intervals.

In the theory of epidemics, there are two fundamental types of mathematical models: continuous-time models described by differential equations and discrete-time models described by difference equations. As said in [13], discretization of continuous-time models is an interesting and prominent trend. Following this idea, in the present work, we deal with a discrete-time version of the classical model for bovine Babesiosis proposed in [6]. In fact, discretization of classical continuous models could help to check the skill in the selection of the factors to the design of the model. Note that factors providing very different dynamics could be not adequate to be considered in such a design.

Discrete versions of models can be obtained by direct formulation of the evolution with difference equations, as we do in this work, or using discretization schemes of continuous models as for example the Euler scheme [14]. In the last years, the study of discrete-time (epidemic) models has increased (see for instance ([15] or [16]) due to different considerations (see [17]). First of all, statistical data on epidemics are collected in discrete time. Hence, to describe epidemics using such discrete models seems to be convenient. In addition, more accurate numerical simulation results are maybe obtained using discrete-time models, although the dynamic behaviors of such discrete models are often more complex [16]. Furthermore, numerical simulations of continuous-time models are obtained by discretizing these models. All these considerations justify the interest and appropriateness of the present work.

In this work, the discrete system is set out by considering the same parameters as in the design of the continuous case. After a simplification of the system, the model is reduced to three difference equations. For such a discrete-time model some necessary parametric constraints, which are fundamental for the rest of the results in the paper, are demonstrated. In fact, these parametric constraints are not artificial and they can be deduced from the biological significance of the model. This shows that additional care should be taken when dealing with discrete-time systems.

After setting the constraints, we calculate the equilibria of the system and the basic reproduction number R0. This number should be regarded as the expected number of new infections from one infected individual in a fully susceptible population during its infectious period, and its value provides a insight in the designing control interventions for infectious diseases [1820]. In fact, it is a key concept in epidemiology as can be seen in [2123]. Finally, we demonstrate that the appearance of the endemic equilibrium depends on the value of the threshold parameter R0 and that such a parameter determines the local and global stability of the disease-free and the endemic equilibria. Actually, it is shown that the system undergoes a transcritical bifurcation when passing through R0.

This study represents a first attempt to model the dynamics of bovine Babesiosis by a discrete model. Due to that, it must be compared to the continuous model. Since the obtained results on the dynamics are similar, we are able to confirm the absence of contradictions between both versions. Furthermore, it opens a way to stimulate the modelization by using other discretization schemes and the comparison of the corresponding results in order to inform the debate on such new formulations.

The document is organized as follows. In Section 2, the discrete-time mathematical model is established considering the same influencing parameters as in the continuous case and some necessary conditions which will become fundamental for the next sections are proved. Section 3 is devoted to analyzing the existence and stability of equilibria, once the threshold value R0 is provided. In particular, conditions for local and global stability of the equilibria that allow us to explain the dynamics of the disease are demonstrated. As a consequence, a scheme of the bifurcation of the system is also shown. In section 4, some numerical simulations, which corroborate the previous theoretical study, are provided. Finally, in Section 5, we provide an interpretation of the results in relation to the previously published work and spell out the major conclusions and open research directions on the significance of such conclusions.

## 2 Mathematical model

In this section, we establish a discrete-time model for the dynamics of the evolution of the Babesiosis disease in bovine and tick populations, considering the same influencing parameters as in the continuous-time model proposed in [6].

According to the notation in [6], we denote the bovine population by NB(t) while the tick population is denoted by NT(t). Bovines are split into three subpopulations, namely, susceptible SB(t); infected IB(t); and controlled CB(t), i.e., treated against Babesiosis. On the other hand, ticks are naturally divided only into two subpopulations, specifically, susceptible ST(t) and infected IT(t). The birth and death rates are considered equal in each population, being denoted by μB for the bovine population and μT for the tick population.

For our purposes, susceptible bovines can become infected due to an effective transmission caused by a bite of an infected tick at a rate βB. Similarly, susceptible ticks can become infected when biting an infected bovine at a rate βT. To complete the model, we denote by λB the fraction of infected bovines which are controlled, while αB denotes the fraction of controlled ones which return to the susceptible ones. Finally, p represents the probability that a susceptible tick is born from an infected one. We assume an homogeneous-mixing for disease dynamics, that is, all the populations have same rates of disease-causing contacts.

To build our discrete-time epidemic model, we assume that population in the (t + 1)–th generation is a function of the tth generation with t ∈ ℕ. Under this assumptions, we obtain a discrete model described by the following system of difference equations: $S¯B(t+1)=S¯B(t)+(μB+αB)C¯B(t)−βBS¯B(t)I¯T(t)NT(t),I¯B(t+1)=I¯B(t)+βBS¯B(t)I¯T(t)NT(t)−λBI¯B(t),C¯B(t+1)=C¯B(t)+λBI¯B(t)−μB+αBC¯B(t),S¯T(t+1)=S¯T(t)+μTpI¯T−βTS¯T(t)I¯B(t)NB(t),I¯T(t+1)=I¯T(t)+βTS¯T(t)I¯B(t)NB(t)−μTpI¯T(t),$(1)

We shall suppose that the bovine and tick populations are constant. That is, NB(t + 1) = NB(t) and NT(t + 1) = NT(t). Besides, we shall assume that all the parameters are positive, since this is biologically logical.

For the system (1) above, we use the following proportions $SB(t)=S¯B(t)NB(t),IB(t)=I¯B(t)NB(t),CB(t)=C¯B(t)NB(t),ST(t)=S¯T(t)NT(t),IT(t)=I¯T(t)NT(t),$ and the following equalities CB(t) = 1 − SB(t) − IB(t) and ST(t) = 1 − IT(t) to obtain the next system of nonlinear difference equations: $SB(t+1)=SB(t)+μB+αB⋅1−SB(t)−IB(t)−βBIT(t)SB(t),IB(t+1)=IB(t)+βBSB(t)IT(t)−λBIB(t),IT(t+1)=IT(t)+βT1−IT(t)IB(t)−μTpIT(t).$(2)

#### Lemma 2.1

System (2) is epidemiologically meaningful if and only if $1−(μB+αB)≥0.$

#### Proof

Observe that, at any time t, the controlled bovine population CB(t) cannot be less than zero, that is, CB(t) ≥ 0. Then, when passing from time t to time t + 1, as formulated in the third equation of (1), a fraction μB CB(t) of this population dies a natural death while a fraction αBCB(t) of the controlled bovine return to susceptible state. Therefore, it is impossible that the sum of these two amounts exceed the value of the initial stock, i.e., $CB(t)−(μBCB(t)+αBCB(t))=[1−(μB+αB)]CB(t)≥0,∀CB(t)≥0.$

From the inequality above, we can deduce that $1−(μB+αB)≥0,∀CB(t)>0,$ which is also valid for CB(t) = 0, since 1 −( μB + αB) does not depend on CB(t)  □.

#### Remark 2.2

Proceeding in a similar way as in the proof of Lemma 2.1, from the rest of the equations of system (1), it can be also deduced $1−βB≥0,1−λB≥0,1−βT≥0,1−μTp≥0.$

Nevertheless, it is not necessary to proceed in such a way. Actually, since all the parameters involved in such inequalities represent rates, fractions of a normalized population or probability (see [6]) in order to have a epidemiologically meaningful model, all of them are less than or equal 1 and such inequalities hold automatically.

#### Proposition 2.3

The region $Ω=(SB,IB,IT)∈R+3:0≤SB+IB≤1,0≤IT≤1$ is a positive invariant set for system (2).

#### Proof

The previous conditions in Lemma 2.1 and Remark 2.2 can be used to prove that (2) is well posed. Effectively, suppose that (SB(t),IB(t),IT(t)) is in the region Ω at any initial time t.

First of all, looking at (1) from which these conditions come, one can easily check that SB(t + 1), IB(t + 1), IT(t + 1) in system (2) are greater than or equal to zero and consequently SB(t + 1), IB(t + 1), IT(t + 1) are greater than or equal to zero. Thus, in particular, we have $0≤SB(t+1)+IB(t+1)and0≤IT(t+1).$

Additionally, we have $SB(t+1)+IB(t+1)=SB(t)+IB(t)+μB+αB⋅1−SB(t)−IB(t)−λBIB(t)=[1−μB+αB](SB(t)+IB(t))+(μB+αB)−λBIB(t)≤1−μB+αB+(μB+αB)=1.$

Analogously, we have $IT(t+1)=IT(t)+βT1−IT(t)IB(t)−μTpIT(t)≤IT(t)+βT−βTIT(t)=(1−βT)IT(t)+βT≤1−βT+βT=1.$

Therefore, the region Ω is a positive invariant set for system (2).  □

In such a context, we shall consider the region Ω as the state space of system (2).

## 3 Results on the existence and stability of equilibria

This section is devoted to studying the existence and stability of the equilibria of model (2). In this sense, we shall assume the following threshold parameter: $R0=βBβTλBμTp.$

The value of this parameter means that each infected bovine produces $\begin{array}{}\frac{{\beta }_{T}}{{\mu }_{T}p}\end{array}$ new infected ticks over its expected infectious period, and each infected tick produces $\begin{array}{}\frac{{\beta }_{B}}{{\lambda }_{B}}\end{array}$ new infected bovines over its expected infectious period.

Contrary to the continues-time case, the (parametric) positive constraints obtained in Lemma 2.1 and Remark 2.2 become fundamental for the proofs of the results in our discrete-time case. Although such results are similar to the continuous case, the proofs need to be performed by using different techniques corresponding to discrete dynamical systems.

#### Proposition 3.1

System (2) has a disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ = (1,0,0)for all the values of the (positive) parameters, while only if 𝓡0 > 1, there exists a unique endemic equilibrium $\begin{array}{}{E}_{2}^{\ast }=\left({S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast },{I}_{{T}_{2}}^{\ast }\right)\end{array}$ in the region Ω.

#### Proof

A fixed point $\begin{array}{}{E}^{\ast }=\left({S}_{B}^{\ast },{I}_{B}^{\ast },{I}_{T}^{\ast }\right)\end{array}$ of model (2) can be obtained by solving the equations below $μB+αB1−SB(t)−IB(t)+1−βBIT(t)SB(t)=SB(t),βBSB(t)IT(t)+1−λBIB(t)=IB(t),βT1−IT(t)IB(t)+(1−μTp)IT(t)=IT(t).$(3)

This is equivalent to solve the following system $μB+αB1−SB(t)−IB(t)−βBIT(t)SB(t)=0,βBSB(t)IT(t)−λBIB(t)=0,βT1−IT(t)IB(t)−μTpIT(t)=0.$(4)

If IB(t) = 0 and IT(t) = 0, from the first equation of system (4), one can easily check that SB(t) = 1 independently of the values of the parameters. Therefore the disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ = (1, 0, 0) exists for any value of the parameters. This is epidemiologically meaningful since, when there is no infected individual, all the bovine population become susceptible.

Now, if we suppose that IB(t) > 0, IT(t) > 0 and consider the second and third equations of (4), we get $SB(t)=λBIB(t)βBIT(t),IT(t)=βTIB(t)βTIB(t)+μTp.$(5)

Taking into account the above equalities, we can replace SB(t) and IT(t) into the first equation of (4) to obtain the following dependent function for the subpopulation of infected bovines: $F(IB)=μB+αB1−1R0−μB+αBλBβB1+βBλB+βBμB+αBIB,$(6) where F(IB) is obviously continuous and strictly decreasing in [0,1].

If 𝓡0 ≤ 1, then $F(0)=μB+αB1−1R0≤0$ and since F is strictly decreasing in [0, 1], there is no 0 < $\begin{array}{}{I}_{B}^{\ast }<1\text{\hspace{0.17em}such that\hspace{0.17em}}F\left({I}_{B}^{\ast }\right)=0.\end{array}$ Thus, the model (2) has only an equilibrium: the disease-free one.

Now, if 𝓡0 > 1, then we have $F(0)=μB+αB1−1R0>0,$ while $F(1)=μB+αB1−1R0−μB+αBλBβB1+βBλB+βBμB+αB<μB+αB−μB+αBλBβB1+βBλB+βBμB+αB=μB+αB1−1+λBβB+λBμB+αB=μB+αB−λBβB+λBμB+αB<0$ and, since F is continuous and strictly decreasing in [0,1], there exists a unique 0 < $\begin{array}{}{I}_{B}^{\ast }\end{array}$ < 1 such that F( $\begin{array}{}{I}_{B}^{\ast }\end{array}$) = 0. Therefore, model (2) has a unique endemic equilibrium $\begin{array}{}{E}_{2}^{\ast }=\left({S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast },{I}_{{T}_{2}}^{\ast }\right)\end{array}$. In fact, one can check that $IB2∗=μB+αBβBβT−λBμTpβTαBβB+λB+μBβTλB+βTβBλB+μB.$

Substituting $\begin{array}{}{I}_{{B}_{2}}^{\ast }\end{array}$ in the second equation of (5), we obtain $IT2∗=μB+αBβBβT−λBμTpβTβBαB+μB+βBμTpαB+λB+μB.$

and, replacing $\begin{array}{}{I}_{{B}_{2}}^{\ast }\text{\hspace{0.17em}and\hspace{0.17em}}{I}_{{T}_{2}}^{\ast }\end{array}$ in the first equation of (5), we have $SB2∗=βTλBαB+μB+λBμTpαB+λB+μBβTαBβB+λB+βTλBμB+βTβBλB+μB.$

($\begin{array}{}{S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast },{I}_{{T}_{2}}^{\ast }\end{array}$) is in the interior of Ω, as is demonstrated below. To do such a demonstration, we must verify that $0

First of all, observe that, since R0 > 1, we have that λBμTpβTβB > 0. Besides, all the parameters involved in the expressions of $\begin{array}{}{S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast },{I}_{{T}_{2}}^{\ast }\end{array}$ are greater than zero. Hence, both conditions allow us to prove that $\begin{array}{}{S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast },{I}_{{T}_{2}}^{\ast }\end{array}$ > 0. In particular, we also have that $\begin{array}{}{S}_{{B}_{2}}^{\ast }+{I}_{{B}_{2}}^{\ast }>0.\end{array}$

Secondly, observe that $SB2∗+IB2∗=βTλBαB+μB+λB2μTp+βBβTμB+αBβTαBβB+λB+μBβTλB+βTβBλB+μB<1,$ if and only if $βTλBαB+μB+λB2μTp+βBβTμB+αB<βTαBβB+λB+μBβTλB+βTβBλB+μB,$ Canceling all the common terms, the inequality above becomes $λB2μTp<λBβTβB.$

But, this last inequality is equivalent to λBμTpβTβB < 0, which holds since R0 > 1.

Finally, note that $\begin{array}{}{I}_{{T}_{2}}^{\ast }\end{array}$ < 1 if and only if $μB+αBβBβT−λBμTp<βTβBαB+μB+βBμTpαB+λB+μB$

Again, canceling some common terms, the inequality above becomes the following equivalent one $μB+αB−λBμTp<βBμTpαB+λB+μB,$ which is true since its left hand side is less than zero, while its right hand side is greater than zero  □.

Now, we are going to analyze the local stability of the disease-free fixed point. In order to do that, we consider the Jacobian matrix related to system (2) given by $JSB∗,IB∗,IT∗=1−μB+αB−βBIT∗−μB+αB−βBSB∗βBIT∗1−λBβBSB∗0βT1−IT∗1−μTp−βTIB∗$(7)

As in the continuous case, the demonstration takes into account the eigenvalues of this Jacobian matrix evaluated at the equilibria. Nevertheless, the method we apply here is different from the one employed for the continuous-time case in [6]. In this case, if all the eigenvalues of J(E*) have magnitude less than one, then the equilibrium E* is locally asymptotically stable, i.e., all solutions of system (2) sufficiently close to the equilibrium point approach it. Actually, we will prove that the conditions of the Jury-Criterion (see [14] or [24]) are satisfied by the equilibria.

#### Theorem 3.2

The disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ of (2) is locally asymptotically stable if 𝓡0 < 1 and unstable if 𝓡0 > 1.

#### Proof

The characteristic polynomial at the disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ = (1,0,0) is $p(γ)=1−μB−αB−γγ2+a1γ+a2,$ where a1 = μTp + λB − 2 and a2 = 1 + μTBβTβBμTpλB.

The first root of the polynomial is γ1 = 1 −(μB + αB). By Lemma (2.1), we have that |1 −(μB + αB)| = 1 −(μB + αB) and since μB,αB are greater than zero, $|γ1|=|1−(μB+αB)|=1−(μB+αB)<1.$

Now, to analyze if the other two roots satisfy |γ2,3| < 1, we need to check the Jury conditions.

The characteristic polynomial of the Jacobian matrix J( $\begin{array}{}{E}_{1}^{\ast }\end{array}$) can be rewritten as $p(γ)=1−μB−αB−γγ2−tr(J^)γ+det(J^),$ where Ĵ is the submatrix 2 × 2 of J( $\begin{array}{}{E}_{1}^{\ast }\end{array}$) given by, $J^1,0,0=1−λBβBβT1−μTp.$(8)

In this context, Tr(Ĵ) = (1 − λB) + (1 − μTp) and det(Ĵ) = (1 − λB)(1 − μTp) − βTβB. The (simplified) Jury criterion (see [24]) states that the eigenvalues of Ĵ have magnitude less than one if and only if |Tr(Ĵ)| < det(Ĵ) + 1 < 2.

Note that, the inequality det(Ĵ) + 1 < 2 is equivalent to det(Ĵ) < 1. This is the one we are going to prove next. Since 1 − λB ≤ 1 and 1 − μTp ≤ 1, we also have that (1 − λB)(1 − μTp) ≤ 1 and, taking into account that βT,βB > 0, we have that $det(J^)=(1−λB)(1−μTp)−βTβB<1$ is always satisfied.

On the other hand, observe that in this case Tr(Ĵ) ≥ 0, since 1 − λB ≥ 0 and 1 − μTp ≥ 0. Hence, |Tr(Ĵ)| = Tr(Ĵ) and we must demonstrate that $(1−λB)+(1−μTp)<1+(1−λB)(1−μTp)−βTβB$ which is equivalent to $2−λB−μTp<2−λB−μTp+λBμTp−βTβB.$

After removing all the common terms in both sides, the inequality Tr(Ĵ) < 1 + det(Ĵ) becomes $0<μTpλB−βTβB$ which is true if and only if 𝓡0 < 1  □.

In the next theorem, we prove the global stability of the disease-free equilibrium when 𝓡0 ≤ 1, using the LaSalle Invariance Principle for discrete-time systems given in [25].

#### Theorem 3.3

The disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ of system (2) is globally asymptotically stable if 𝓡0 ≤ 1.

#### Proof

First of all, we relocate our disease-free equilibrium to the origin of coordinates. That is, we perform the change of coordinates $XB(t)=1−SB(t)$ in the system (2) and it becomes $XB(t+1)=1−(μB+αB)−βBIT(t)XB(t)+(μB+αB)IB(t)+βBIT(t),IB(t+1)=βB1−XB(t)IT(t)+1−λBIB(t),IT(t+1)=βT1−IT(t)IB(t)+(1−μTp)IT(t).$(9)

We consider the following Lyapunov function V : Ω → ℝ+ defined by $VXB(t),IB(t),IT(t)=βTIB(t)+λBIT(t),$ where VC(Ω,ℝR+), such that V(0, 0, 0) = 0 and V(XB(t),IB(t),IT(t)) ≥ 0 in Ω − (0, 0, 0).

Then, the difference $ΔVXB,IB,IT=VXB(t+1),IB(t+1),IT(t+1)−VXB(t),IB(t),IT(t)$ is given by $βTβB1−XB(t)IT(t)+1−λBIB(t)+λBβT1−IT(t)IB(t)+1−μTpIT(t)−βTIB(t)−λBIT(t).$

Simplifying and grouping terms, we have $ΔVXB(t),IB(t),IT(t)=βTβB−λBμTp−βTβBXB(t)−λBβTIB(t)IT(t).$(10)

At this point, we shall prove that, if 𝓡0 ≤ 1, this last expression is non-positive for every (XB(t),IB(t),IT(t)) in Ω.

As by hypotheses 𝓡0 ≤ 1, equivalently, we have that $βTβB−λBμTp≤0.$

Additionally, as all the parameters are greater than zero and XB(t), IB(t), IT(t) are non-negative in Ω, we can deduce that $ΔVXB(t),IB(t),IT(t)≤0,for R0≤1.$

Now, we need to obtain the maximal positively invariant set G* contained in the subset GΩ given by $G={XB,IB,IT∈Ω:ΔVXB,IB,IT=0}.$

We shall distinguish two cases, depending on the values of the threshold parameter R0:

• 𝓡0 < 1: In this case, expression (10) equals zero if and only if IT(t) = 0. The system (9) in such points becomes $XB(t+1)=1−(μB+αB)XB(t)+(μB+αB)IB(t),IB(t+1)=(1−λB)IB(t),IT(t+1)=βTIB(t).$(11)

Observe that, for every initial state of the form (XB(t), IB(t),0), with IB(t) > 0, the following state in its orbit verifies that IT(t + 1) = βTIB(t) > 0. Thus, no orbit of a point of the form (XB(t), IB(t),0), with IB(t) > 0 is contained in such a set of points GΩ for which Δ V(XB,IB,IT) = 0. Nevertheless, if we avoid this problem considering only the points of G for which IB(t) = 0, one can easily check that, for any point in such a subset, the iteration of the system (9) reduces to $XB(t+1)=1−(μB+αB)XB(t),IB(t+1)=0,IT(t+1)=0.$(12)

That is, the orbit of any initial state in the subset of G given by IB(t) = 0, IT(t) = 0 remains in such a subset, i.e., the largest positively invariant set contained in G is $G∗={XB,IB,IT∈Ω:IB=IT=0}.$

Moreover, note that, since 0 ≤ [1 −(μB + αB)] < 1, the (disease-free) equilibrium (0, 0, 0) is G*–globally asymptotically stable. At this point, since all the orbits of the system remain in Ω, all of them are bounded. Therefore, applying the LaSalle Invariance Principle for discrete dynamical systems given in Theorem 3.3 of [25], we can conclude that the disease-free equilibrium is globally asymptotically stable in Ω.

• 𝓡0 = 1: In this case, expression (10) equals zero if and only if IT(t) = 0 or XB(t) = IB(t) = 0. Therefore, the set GΩ for which Δ V(XB,IB,IT) = 0 is the following: $G={XB,IB,IT∈Ω:IT=0}∪{XB,IB,IT∈Ω:XBIB=0}.$

Observe that for any initial point of the form (0, 0, IT(t)) with IT(t) > 0, the system (9) becomes $XB(t+1)=βBIT(t),IB(t+1)=βBIT(t),IT(t+1)=(1−μTp)IT(t).$(13)

This proves that no orbit of an initial state in the subset {(XB,IB,IT) ∈ Ω: XB= IB = 0} with IT(t) > 0 remains in such a subset of G. In fact, only the orbit originated by (0, 0, 0) remains in this subset.

At this point, proceeding as in the previous case, the largest positively invariant set contained in G is $G∗={XB,IB,IT∈Ω:IB=IT=0}.$

Moreover, note that, since 0 ≤ [1 −(μB + αB)] < 1, the (disease-free) equilibrium (0, 0, 0) is G*-globally asymptotically stable, and, since all the orbits of the system remain in Ω, all of them are bounded.

Therefore, applying the LaSalle Invariance Principle for discrete dynamical systems again, we can conclude that the disease-free equilibrium is globally asymptotically stable in Ω, also in this case.

□

This last result is epidemiologically significant, because it indicates that if a small number of infective individuals is introduced in a susceptible population, then the disease vanishes.

For 𝓡0 > 1, the endemic equilibrium $\begin{array}{}{E}_{2}^{\ast }\end{array}$ is locally asymptotically stable, as shown by numerical simulations. Actually, it can be seen that $\begin{array}{}{E}_{2}^{\ast }\end{array}$ is globally asymptotically stable in Ω − {(1, 0, 0)}. That is, every initial condition (SB(0),IB(0),IT(0)) ∈ Ω produces a trajectory (SB(t),IB(t),IT(t)) ∈ Ω which converges to the unique interior fixed point $\begin{array}{}{E}_{2}^{\ast }\end{array}$. However, $\begin{array}{}{E}_{2}^{\ast }\end{array}$ is not globally asymptotically stable in Ω, because the disease-free point is also in Ω.

Besides, when 𝓡0 < 1, the system has also the two equilibria, $\begin{array}{}{E}_{1}^{\ast }\end{array}$ asymptotically stable in Ω and $\begin{array}{}{E}_{2}^{\ast }\end{array}$ unstable, being $\begin{array}{}{E}_{1}^{\ast }\end{array}$ in the outside of Ω.

Moreover, when 𝓡0 = 1, $\begin{array}{}{E}_{1}^{\ast }\end{array}$ is the unique point of the system being a non-hyperbolic fixed point which is asymptotically stable in Ω.

Such issues allow us to infer the following corollary.

#### Corollary 3.4

System (2) undergoes a transcritical bifurcation at the parameter value 𝓡0 = 1.

## 4 Experimental procedures

In [6], we show through numerical simulations that when 𝓡0 > 1, the endemic equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ is locally asymptotically stable. However, as the disease-free equilibrium $\begin{array}{}{E}_{1}^{\ast }\end{array}$ is in Ω, the endemic point is not globally asymptotically stable in this region. But, it can be observed (numerically) that it is in the region Ω – { $\begin{array}{}{E}_{1}^{\ast }\end{array}$}. As in the continuous case, our numerical simulations in this work proved that it is still verified.

We consider the same parameter values as in the continuous-time model and the parameter constraints given by Lemma (2.1), with initial conditions SB = 0.3756, IB = 0.5184, IT = 0, 6000. In Figure 1 panel (a), we show that all the trajectories of SB(blue curve), IB(green curve) and IT (red curve) converges to the disease-free equilibrium point (1, 0, 0), if the reproduction number 𝓡0 < 1, as was demonstrated in Theorem 3.2. In this case, all the eigenvalues of its Jacobian matrix are less than 1, (0.9987, 0.9735, 0.9992). Note that, if 𝓡0 ≤ 1, $\begin{array}{}{E}_{1}^{\ast }\end{array}$ is still globally asymptotically stable, as shown in Figure 1, panel (b) in the state space.

Figure 1

Parameter values: μB = 0.0002999, μT = 0.001609, αB= 0.001, βT = 0.00048. (a) for 𝓡0 = 0.0068 with βB= 0.003, λB = 0.0265 and p = 0.5. (b) for 𝓡0 < 1 with βB = 0.006, p = 0.1 and λB = 0.000265

In Figure 2, we can check that {(0,0,0)} is the largest positively invariant set contained in Ω, therefore of disease-free equilibrium point is globally asymptotically stable as shown in Theorem 3.3.

Figure 2

The red point is an initial value condition for 𝓡0 < 1 and 𝓡0 = 1, panel (a) and (b) respectively

Figure 3

For panel (a) which initial condition SB = 0.3756, IB = 0.5184, IT = 0, 6000 and panel (b), (c) and (d) which SB = 0.9, IB = 0.15, IT = 0,1

The following scenarios show that 𝓡0 > 1 (see Table 1) when having high transmission of the disease in the bovine population, high vertical transmissibility in the ticks population and low control of infected cattle. In this context, we show that every trajectory converges to the endemic equilibrium $\begin{array}{}{E}_{2}^{\ast }=\left({S}_{{B}_{2}}^{\ast },{I}_{{B}_{2}}^{\ast }.{I}_{{T}_{2}}^{\ast }\right)\end{array}$ except, of course, the one starting at (0, 0, 0)

Table 1

μB = 0.0002999, μT = 0.001609, αB = 0.001, p = 0.1 and λB = 0.000265

## 5 Discussion

In the present work, we provide and study a discrete model for the transmission of Babesiosis disease in bovine and tick populations. It supposes a discretization of the continuous-time model developed by us previously which has served as a base for other works on this disease.

The results, here obtained by discrete methods as opposed to continuous ones, show that similar conclusions can be obtained for the discrete model subject to the assumption of some parametric constraints which were not necessary in the continuous case. We prove that these parametric constraints are not artificial and, in fact, they can be deduced from the biological significance of the model.

This study represents a first attempt to model the dynamics of bovine Babesiosis by a discrete model. Due to this novelty in the modelization, it must be compared to the continuous model. Since the obtained results on the dynamics are similar, we are able to confirm the absence of contradictions between both versions and the skill in the selection of the factors to design the model.

Furthermore, it opens a new research line by using other discretization schemes and the comparison of the corresponding results in order to debate on such new formulations.

## Acknowledgement

Deccy Y. Trejos has performed this work within the specific educational cooperation agreement for fellowships in doctoral programs and short research stays for professors with doctorates between the Carolina Foundation, Spain and the University Francisco José de Caldas, Colombia. She has also been supported by the Doctoral Study Commission Grant from the University Francisco José de Caldas, by Superior Resolution 038 of 2016.

Jose C. Valverde was supported by FEDER OP2014-2020 of Castilla-La Mancha (Spain) under the Grant GI20173946 and by the Ministry of Economy and Competitiveness of Spain under the Grant MTM2014-51891-P.

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Accepted: 2017-03-28

Published Online: 2017-06-16

Citation Information: Open Physics, Volume 15, Issue 1, Pages 360–369, ISSN (Online) 2391-5471,

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