Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 9585497, 18 pages
https://doi.org/10.1155/2019/9585497

Research Article

Global Properties of a General Latent Pathogen Dynamics Model with Delayed Pathogenic and Cellular Infections

A. M. Elaiw,¹ A. A. Almatrafi,1,2 A. D. Hobiny,¹ and K. Hattaf³

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department of Mathematics, Jamoum University College, Umm Al Qura University, Makkah, Saudi Arabia
³Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

Correspondence should be addressed to A. M. Elaiw; a_m_elaiw@yahoo.com

Received 30 January 2019; Revised 14 May 2019; Accepted 10 June 2019; Published 1 July 2019

Academic Editor: Youssef N. Raffoul

Copyright © 2019 A. M. Elaiw et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper studies the global dynamics of a general pathogenic infection model with two ways of infections. The effect of antibody immune response is analyzed. We incorporate three discrete time delays and both latently infected cells and actively infected cells. The infection rate and production and clearance/death rates of the cells and pathogens are given by general functions. We determine two threshold parameters to investigate the global stability of three equilibria. We use Lyapunov method to establish the global stability. We support our theoretical results by numerical simulations.

1. Introduction

The study of mathematical modeling and analysis of pathogen infection is helpful for understanding within-host pathogen dynamics; therefore, it has attracted the attention of many researchers during the last decades [1–19]. In [1], Nowak and Bangham proposed a simple model for the interaction between a replicating virus and host cells. In the same work, they proposed a second model that improves the first model by including the cytotoxic T lymphocytes (CTL) immune response against infected cells. Several extensions of [1] without immunity are given in [2–16] and with CTL immune response in [18, 19]. Antibodies which are produced from the B cells play a prominent role in pathogen dynamics. The function of antibodies is to attack and kill pathogens. Based on the first simple model introduced in [1] and the fact that antibody immune response is more effective than CTL in some infections [20], Murase et al. [21] proposed a mathematical model that describes the interactions between the target cells, the pathogens, and the antibody immune response. This model is given as follows:

where , , , and are the concentrations of susceptible cells, actively infected cells, pathogens and antibodies, respectively. In the literature, several modifications have been done on model (1)-(4) (see, e.g., [22–29]).

In model (1)-(4), the infection rate is given by bilinear incidence which may be not reasonable to describe the dynamics of the pathogen during the several stages of infection [4]. Pathogen dynamics models with antibodies and general incidence rate have been studied in several works (see, e.g., [24–26]), but with only pathogen-to-cell transmission. In [30], the authors considered only cell-to-cell transmission in order to model the interaction of the healthy and infected cells. Both ways of transmissions have been considered in [31–37], but without humoral immunity. In , Lin et al. [38] presented a mathematical model with humoral immunity and both ways of transmissions that are modeled by two bilinear functions.

Although antiviral drug therapies can significantly limit the level of viruses in the blood, there is still a low viral load due to ongoing latently infected cells reactivation. These cells play the role of viral reservoirs and are considered a major hurdle to virus clearance in patients under treatment. Variant models have been developed to study the dynamics of pathogens in the presence of latent reservoirs (see, e.g., [24, 39–43]). Recently, Elaiw et al. [44] have studied the global stability of a general pathogen dynamics model with delay and with both cellular and pathogenic infections by assuming that all the infected cells are active. In [45], a class of latently infected cells has been added in pathogen dynamics with two routes of infections but with specific forms of the incidence rate and production and clearance/death rates of the pathogens and cells.

The main objective of this study is to develop a general pathogenic infection model with two ways of infections, antibody immune response, three discrete time delays, and both latently infected cells and actively infected cells. This model is described by the following nonlinear system:

where the fractions and with are the proportions of infection that lead to latency and actively, respectively. The functions , , , , , , , and are continuously differentiable and satisfy the following hypotheses:

(H1) (i) there exists such that for ,

(ii) for all ,

(iii) there exists and such that for all

(H2) (i) for all and , ,

(ii) and for all , , for all ,

(iii) there are , and such that , , , and for all

(H3) and for all

Let the initial conditions for system (5)-(9) be given as

where and is the Banach space of continuous functions mapping the interval into with norm . This shows the uniqueness of the solution for .

1.1. Properties of Solution

Lemma 1. Let (H1), (H2), and (H3) be valid. Then, the solutions of system (5)-(9) with initial conditions (11) are nonnegative and ultimately bounded.

Proof. System (5)-(9) can be written as , where

and

where

It follows that

Any solution of system (5)-(9) with the initial conditions (11) satisfies for all [46]. Hence, is positively invariant for system (5)-(9).
From (5), we obtain . It follows that . Let . Then

where It follows that , where , and then and Moreover, let Then

where Then, , where This yields and , where . Then, , , , , and are ultimately bounded.

Define the feasible region

Clearly, contains omega limit sets of system (5)-(9) and is also positively invariant for the system.

1.2. Equilibria

Lemma 2. Let (H1)-(H3) be satisfied; then there exist two bifurcation numbers and such that
(i) there exists only one equilibrium if and ,
(ii) there exist only two equilibria and if , and
(iii) there exist three equilibria , and if and .

Proof. Let be any equilibrium satisfying

From (24), we have either or . First, we consider ; then using (H2), we obtain . From (20)-(23), we obtain

where

Hypothesis (H2) implies that the following functions:

are strictly increasing and continuous.
Substituting (27) into (20), we get

Let

According to (H1) and (H2), we have and
Substituting in (27), we get , , and . This yields pathogen-free equilibrium Now from (29), we get

Moreover, from (25), we have

Therefore, (30) becomes

Hypothesis (H2) implies that

From (H1), we have Since and , hence, if

then and there exists such that If inequality (34) is satisfied, then from (27) and (H1)-(H2), we have

It follows that an infected equilibrium without antibodies exists when condition (34) is satisfied. Therefore, we can define the basic reproduction number for system (5)-(9) as

Let , where

The other possibility of (24) is and

Let in (20)-(22) and define function as

Based on (H1) and (H2), we have and . Hence, there exists such that . Thus, and . Now, from (23), we have

From (H2), we have if , then . Now we can define

which represents the antibody immune response activation number. We have ; then if , there exists an infected equilibrium with antibodies .

1.3. Global Properties

It should be noted that finding an appropriate Lyapunov functional for higher-order nonlinear delay differential equations is a more difficult task. For the construction of Lyapunov functional of fifth-order nonlinear differential equations with multiple deviating arguments, we refer the reader to the work [47]. Here, we study the global stability of equilibria of system (5)-(9) by means of direct Lyapunov method. We first define a function and use the notation = .

Theorem 3. For system (5)-(9), suppose that and (H1)-(H3) be held true; then is globally asymptotically stable (GAS).

Proof. Define as

where is defined by (26). It is seen that for all , while at the pathogen-free equilibrium , we get Calculating along system (5)-(9), we obtain

Collecting terms of (43), we get

From (H3), we have

It follows that

We have

Therefore, we obtain

From (H1) and (H2), we have

Therefore, if , then for all . Moreover, when , , and . The solutions of system (5)-(9) tend to the largest invariant subset of [48]. For each solution in satisfies and from (8), we get

Then, . Similarly, from (7), we get . It follows that . From LaSalle’s invariance principle, is GAS when .

Remark 4. From (H1)-(H3), we have

Thus,

Obviously, the equilibrium does not exist if , and if . When , we have the following theorem.

Theorem 5. For system (5)-(9), assume that and let (H1)-(H3) be held true; then, is GAS.

Proof. Let be given as follows:

We have for all and . Calculating , we obtain

Simplifying (54), we get

The components of satisfy

We get

Using the equalities in case of ,

Therefore, we obtain

If , then does not exist since This means , for all Thus, for all It follows that . Then according to (H1)-(H3), we have The solutions of system (5)-(9) tend to the largest invariant subset of It is easy to show that . Applying LaSalle’s invariance principle, we get that if and , then is GAS.

On the other hand, it is not hard to verify that the equilibrium does not exist if , and if . When , we have the following result.

Theorem 6. For system (5)-(9), suppose that and let (H1)-(H3) be held true; then, is GAS.

Proof. Consider

We have for all , while reaches its global minimum at . Calculate as

Simplify (61) as

The components of satisfy

we get

Using equalities (58) when , we obtain

From (H1)-(H3), we have that for all and . One can show that at ; then apply LaSalle’s invariance principle to show the global asymptotic stability of .

2. Numerical Simulations

For numerical simulation purpose, we consider model (5)-(9) with the following specific functions:

First, we show that (H1)-(H3) are satisfied. We have , , , and It follows that for all Moreover, (H1)(iii) is satisfied with and Thus, (H1) is satisfied. We have for all and We also have

Hence, (H2) (i)-(ii) are satisfied. Moreover, (H2) (iii) is satisfied, where . Finally, we have

Thus, (H3) is also valid.

With the above choice, system (5)-(9) will be

where is the Holling-type II constant. The numbers and are given as

Therefore, Theorems 3–6 are valid for such choice.

Let us consider and the following initial values:

: , , , , ,

: , , , , ,

: , , , , ,

: , , , , ,

The stability of the equilibria will be investigated by varying some parameters , and and fixing the other parameters as , , , , , , , , , and .

Case (I) (effect of the parameters , and ). We take , . We choose three sets of the parameters , , and to confirm Theorems 3–6 (see Figure 1).

(a)

(b)

(c)

(d)

(e)

Figure 1: The evolution of system (69)-(73) for Case (I).

Set (1) (stability of ). , , and This yields and

Set (2) (stability of ). , , and With these values, we obtain and

Set (3) (stability of ). , , and and then and .

Case (II) (effect of the Holling-type II parameter). Let us take and choose the values , , , and . Figure 2 and Table 1 show the effect of Holling-type II parameter on the stability of the system. We observe that as is increased, the pathogen-to-susceptible and infected-to-susceptible transmission rates are decreased. Then, the concentrations of the susceptible cells are increased, while the concentrations of the infected cells and free pathogens are decreased. In addition,

Table 1: The values of and for system (69)-(73) with different values of .

Equilibrium

0.0

2.29091
2.52851

0.0007

1.34759
1.37541

0.0009

1.20574
1.19569

0.00118999

1.04608
1

0.00125

1.01818
0.966725

0.00129091

1
0.945205

0.05

0.0449198
0.0325008

1.0

0.00228862
0.0016358

(a)

(b)

(c)

(d)

(e)

Figure 2: The evolution of system (69)-(73) with different values of for Case (II).

(i) is GAS when ,

(ii) is GAS when ,

(iii) is GAS when .

Case (III) (effect of time delay parameters). For this case, we take and choose the values , , , and . Figure 3 and Table 2 show the effect of the time delay on the dynamical behavior of the pathogen. We have the following cases:

Table 2: The values of and for system (69)-(73) with different values of .

Equilibrium

0.0
= (832.667, 1.52121, 1.82545, 2.5, 16.5091)
1.69697
1.82545

0.1
= (841.463, 1.3041, 1.56491, 2.5, 8.31986)
1.43111
1.41599

0.2412
= (851.503, 1.06065, 1.27278, 2.50001, 0)
1.12918
1

0.29
= (946.997, 0.360549, 0.432659, 0.809357, 0)
1.04146
0.889019

0.314637
= (1000, 0, 0, 0, 0)
1
0.838138

0.5
= (1000, 0, 0, 0, 0)
0.73999
0.542645

1
= (1000, 0, 0, 0, 0)
0.342409
0.177726

(a)

(b)

(c)

(d)

(e)

Figure 3: The evolution of system (69)-(73) with different values of for Case (III).

(i) is GAS when ,

(ii) is GAS when ,

(iii) is GAS when .

Therefore, if the time delay is long enough, then the system can be stabilized around , where the pathogens are cleared

Data Availability

There is no data available supporting this paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under Grant no. KEP-MSc-13-130-38. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, the authors would like to thank the anonymous reviewers for their useful suggestions that greatly improved the quality of this study.

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