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# Codimension-one bifurcation and stability analysis in an immunosuppressive infection model

- Zohreh Dadi
^{1}Email author and - Samira Alizade
^{2}

**Received:**18 September 2015**Accepted:**18 January 2016**Published:**1 February 2016

## Abstract

One of the important medical problems is infectious diseases such as HIV and hepatitis which annually causes the death of many people. So it is important to study infectious diseases parametric models. In this paper, we investigate differential equations system of HIV and hepatitis (with delay and without delay) from the stability and codimension-one bifurcation point of view. We show that their dynamical behaviour will change when the parameters vary. We prove that this model has a saddle-node bifurcation and transcritical bifurcation when the delay parameter is absent. Also by using the center manifold theory, we show that the delay model has a saddle-node bifurcation.

## Keywords

- Infectious diseases model
- Delay differential equations
- Bifurcation theory

## Background

Clinical reports have shown that drug treatment in some human pathogens such virus HIV, hepatitis B virus (HBV), and hepatitis C virus (HCV), is not effective. Therefore, designing an optimal drug treatment strategy that leads to sustained immunity has become the essential subject (Shu et al. 2014).

This is the place where mathematical modeling plays an important role as it helps understanding the interactions between viral replication and immune response, (Atangana 2015; Atangana and Alkahtani 2015; Atangana and Goufo 2014; Fenton et al. 2006; Komarova et al. 2003; Li and Shu 2010; Shu et al. 2014).

Note that the time lag should not be taken in this model, however, they proved the existence of two stable equilibrium; virus dominant equilibrium (no sustained immunity) and immune control equilibrium (with sustained immunity).

The bistability in this model leads to sustained immunity when the treatment is stopped, because a solution from the basis of the attraction of the virus dominant equilibrium can be lifted to that of the immune control equilibrium via a single phase of therapy.

After that, Shu et al. (2014) incorporated the time lag during the immune response process into Komarova et al.’s model and studied the dynamics between an immunosuppressive infection and antiviral immune response.

*y*and

*z*denote the virus population size and population size of immune cells, respectively. The virus population is assumed to grow logistically:

*r*is the viral replication rate and

*a*is clearance rate. In addition, they assumed virus is killed by immune cells at a rate

*pyz*and immune cells are assumed to be inhibited by the virus at a rate

*qyz*and died at a rate

*b*. The activation rate of immune cells at time

*t*is assumed to depend on the virus load and the number of immune cells at time \(t- \tau\). Here, \(\tau\) is the time lag accounting for the time needed for the immune system to trigger a sequence of events such as antigenic activation, selection and proliferation of immune cells to produce new immune cells. In model 1, it is important to note that

*f*(

*y*), function of immune expansion by virus load, is considered as follows (Shu et al. 2014)

They studied the local and global stability of the most of equilibria. By using bifurcation theory, they only found Hopf bifurcation in the model when \(\tau = \tau _{bif}\).

*r*and

*c*, as bifurcation parameters. The parameter

*r*is the viral replication rate and the parameter

*c*is a coefficient in the function of immune expansion by virus load. We consider

*r*and

*c*as bifurcation parameters and obtain the following result:

- (1)
(

*i*) if \(r=r_{bif}\), then the transcritical bifurcation occurs in system 4, - (2)
(

*ii*) if \(c = c_{bif}\), then the saddle-node bifurcation occurs in system 4, - (3)
(

*iii*) if \(c = c_{bif}\), then the saddle-node bifurcation occurs in system 2.

The rest of paper is organized as follows. In the next section, we obtain the necessary condition of existence of equilibria in immunosuppressive infection model. In “Dynamics of the model without delay (system 4)” section, we will consider the dynamics of model 4. The dynamical behaviour of model 2 is investigated in “Dynamics of the model with delay (system 2)” section. In “Numerical simulation” section, the validity of the main results is illustrated by numerical simulations. Finally, we state some main conclusions.

## Existence of equilibrium points

For any \(\tau > 0\), let \(C:= \lbrace \phi : [ - \tau , 0 ] \rightarrow R \; is \; continuous\rbrace\) be Banach space of continuous function on \([- \tau , 0]\) with the norm is defined as \(\Vert \phi \Vert = \sup _{- \tau \le \theta \le 0} \phi ( \theta )\). We denote the nonnegative cone of *C* by \(C^{+}\).

*y*(

*t*),

*z*(

*t*)) remains nonnegative for \(t \ge 0\) and is bounded in \(C^{+} \times C^{+}\). Furthermore, they showed that the bounded region

Now we find the equilibria of system 2. We then investigate their stability. As we said in “Background” section, we obtain an equilibrium point that it is not considered in Shu et al. (2014).

*g*(

*y*) where \(y^{*} < \bar{y}\) (Fig. 1).

Shu et al. (2014) investigated the existence of positive roots of *g*(*y*) when \(c> (\sqrt{q} + \sqrt{bd})^{2}\). We obtain new results on positive roots of *g*(*y*) when \(c= (\sqrt{q} + \sqrt{bd})^{2}\).

###
*Remark 1*

*g*(

*y*) has a double positive root that it is same vertex of parabola,

we have the following Lemma.

###
**Lemma 1**

*By considering*\(H_1\),

*the following cases occur*

- (a)
*if*$$r \le a$$(12)*holds, then the equilibrium*\(E_0 = (0,0)\)*is the only equilibrium*, - (b)
*if*$$a<r \le r_{t} \; (i.e \; a<r\; \& \; y^{*} \ge \bar{y})$$(13)*holds, then there are two equilibria:*\(E_0\) and \(E_1 =( \bar{y}, 0)\),*where*\(\bar{y}=\frac{k(r-a)}{r}\), - (c)
*if*$$r > r_{t} \; (i.e \; a<r \; \& \; y^{*} < \bar{y})$$(14)*holds, then there are three equilibrium*: \(E_0\), \(E_1\), \(E^{*} = (z^{*}, y^{*})\) where \(z^{*} =\frac{r(k-y^{*}) - ak}{pk}\) (Fig. 2).

## Dynamics of the model without delay (system 4)

In this section, we provide a complete description about dynamics of system 4. To this end, we begin with the following result on local stability of system 4.

###
**Lemma 2**

*Assume that*\(H_1\)

*is satisfied*.

###
*Proof*

*g*(

*y*), it is obvious that \(g(\bar{y}) >0\). Now, if 13 or 14 holds, then the equilibrium \(E_1\) is asymptotically stable. We suppose that 15 holds, by substituting \(E^{*}\) at Eq. 15, we have

When \(r \ge a\), the infection can not spread in body of patient, so there is no virus cell and immune response. In this case, system 4 converges to \(E_0\). We know viral cells infect the host without immune response as *r* increases from *a* to \(r_t\). In this case, system 4 converges to \(E_1\) and the equilibrium point \(E_1\) is locally asymptotically stable. By increasing *r* from \(r_t\), immune response increases and controls viral cells. In this case, \(E^{*}\) and \(E_1\) exist. Therefore, to obtain the better conditions and control of virus cells, we should converge the system to the equilibrium point \(E^{*}\).

###
**Lemma 3**

*Assume that*
\(H_1\)
*is satisfied, therefore system*
4
*has a saddle node bifurcation at equilibrium*
\(E^{*}\)
*when the parameter*
*c*
*varies*.

###
*Proof*

*A*and \(A^{T}\) at zero eigenvalue are

###
**Lemma 4**

*If*
\(H_1\)
*is satisfied, then system*
4
*has a trancscritical bifurcation at equilibrium*
\(E_0\)
*when*
\(r=r_{bif}=a\).

###
*Proof*

*A*and \(A^{T}\) at zero eigenvalue are

According to Lemma 4, we know that system 4 has a transcritical bifurcation at \(E_0\), when \(r= r_{bif}\). For \(r \le r_{bif}\), only equilibrium point \(E_0\) is stable. In this case, the patient\(^{,}\)s body does not have virus cells and immune response. Also, with increasing *r* (\(r > r_{bif}\)), the equilibrium \(E_1\) occurs; in this case the system has a branch of stable equilibrium \(E_1\) and a branch of the unstable equilibrum \(E_0\) that express the transcritical bifurcation. In the branch of the stable equilibirum \(E_1\), the patient has a viral cells without any immune response. Therefore as shown if the viral replication rate *r* is greater than the threshold \(r_t\), then the two equilibrium points \(E_1\) and \(E^{*}\) at the same time are stable and the bistability phenomenon occurs. Also, we know that for \(c < c_{bif}\), there is no equilibrium \(E^{*}\) and according to assumption \(H_1\) at \(c= c_{bif}\), the equilibrium \(E^{*}\) will be found. After passing through \(c_{bif}\) (\(c > c_{bif}\)); according to Shu et al. (2014), the system has two equilibrium \(E_1 ^{*}\) and \(E_2 ^{*}\). This means that there is a saddle-node bifurcation. With finding quantity of bifurcation parameter and rising it, we should try the patient’s condition set in the stable branch of saddle-node bifurcation. In this case virus cells are controlled and patient is in the path of recuperation.

## Dynamics of the model with delay (system 2)

In this section, we would like to investigate dynamics of system 2 with \(\tau > 0\).

### Stability of equilibria

The first, we study the equilibrium \(E_0\) in following theorem.

###
**Theorem 1**

*if*
\(r\le a\), *then*
\(E_0\)
*is locally stable; while if*
\(r > a\)
*then*
\(E_0\)
*is unstable*.

###
*Proof*

###
**Theorem 2**

*The equilibrium point*
\(E_1\)
*is locally asymptotically stable*.

###
*Proof*

###
**Theorem 3**

*Roots of characteristic equation*33

*have negative real parts other than*\(\xi =0\),

*if*

- (1)
\(a_0 >0\)

- (2)
\(a_1^{2} - 2a_0 >0\).

*Hence*, \(E^{*}\)

*is locally stable*.

###
*Proof*

### Saddle-node bifurcation of system 2

*c*as bifurcation parameter. By Remark 1, we know that \(E^{*}\) exists if \(c= (\sqrt{q} + \sqrt{bd})^{2}\). Also, we know that \(E^{*}\) is locally stable by Theorem 3, and codimension-one bifurcation can occur in system 2 at \(E^{*}\). Define \(c_{bif} = (\sqrt{q} + \sqrt{bd})^{2}\). Now, we assume \(\mu = c - c_{bif}\) as bifurcation parameter and rewrite system 2 as follows

###
**Theorem 4**

*System*
36
*has a saddle-node bifurcation at*
\(E_{new}^{*} = (y^{*} , z^{*} , 0)\)
*and*
\(\mu =0\), *if*
\(qpkdy^{*} (r+(qy^{*}+b - pqkz^{*})\tau )\ne 0\).

###
*Proof*

*B*are

## Numerical simulation

## Conclusion

An immunosuppressive infection model with discrete delays and without delay is considered. We have analyzed this model without delay in this paper and showed that the model has transcritical and saddle-node bifurcation at different parameters. We obtained a new equilibrium in our model with delay. Then, we have shown that this model undergoes saddle node bifurcation at this equilibrium. We then compute its normal form. Finally, the presented numerical simulations have demonstrated the correctness of the theoretical analysis.

## Declarations

### Authors’ contributions

ZD and SA have been involved in studying models and writing and revising the manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

The authors are also grateful to SpringerPlus giving us the opportunity to consider this work. We would like to thank referees for reading the manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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