# Global Dynamics of Secondary DENV Infection with Diffusion

Authors: A. M. Elaiw; A. S. Alofi
Journal: Journal of Mathematics
 (2021)

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## Abstract

During the past eras, many mathematicians have paid their attentions to model the dynamics of dengue virus (DENV) infection but without taking into account the mobility of the cells and DENV particles. In this study, we develop and investigate a partial differential equations (PDEs) model that describes the dynamics of secondary DENV infection taking into account the spatial mobility of DENV particles and cells. The model includes five nonlinear PDEs describing the interaction among the target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. In the beginning, the well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive three threshold parameters which govern the existence and stability of the four equilibria of the model. We study the global stability of all equilibria based on the construction of suitable Lyapunov functions and usage of Lyapunov–LaSalle’s invariance principle (LLIP). Last, numerical simulations are carried out in order to verify the validity of our theoretical results.


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## Body

## 1. Introduction

Mathematical models and their analysis have been proven to be an efficient and significant approach to understand the within-host dynamics of viral infections such as dengue virus (DENV), human immunodeficiency virus (HIV), hepatitis C and B virus (HCV/HBV), human T lymphotropic virus type I (HTLV-I), and recently, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). DENV is the casual of dengue fever which is one of the morbidity and mortality diseases. It can transmit to humans viaAedes aegypti and Aedes albopictus mosquitoes. Annually, about 50–100 million infected individuals by DENV are reported worldwide. The most epidemic regions are sub-Saharan Africa and Southeast Asia [1]. Several symptoms of dengue that can appear on the infected individual are high fever, vomiting, nausea, joint pains, headache, and pain behind the eyes [2]. DENV aims and infects the following types of cells: monocytes, dendritic cells, hepatocytes, macrophages, and mast cells [3–6]. There are four serologically various dengue viruses DENV (1–4) that can infect the human [7]. When a DENV enters the human body first time, the immune response is enhanced [8]. Cytotoxic T lymphocytes (CTLs) and antibody immune responses are two main components of the immune system against viruses. CTLs destroy the DENV-infected cells, while antibodies kill DENV particles and clear it from the body.During the recent years, several mathematical models have been developed which describe within-host DENV primary infection [9–17]. These models are based on the virus dynamics model introduced by Nowak and Bangham [18]. The World Health Organization (WHO) [19] has reported that an infected individual by one serotype will have lifelong immunity against that serotype but only temporary and partial cross-immunity to the other three serotypes. Mathematical models of DENV dynamics pertaining to secondary infection with another serotype have been developed in [20–25]. Gujarati and Ambika [20] have formulated the following DENV infection model:(1)dKtdt=δ−μKtMt−ξKt,dLtdt=μKtMt−ϱLt,dMtdt=τLt−ηMt−ϖ1MtNt−ϖ2MtPt,dBtdt=β−γBt+φBtMt,dNtdt=ϑϵBt−ϰ1NtMt−α1Nt,dPtdt=ϵBt−ϰ2PtMt−α2Pt,where t>0 is the time, and Kt, Lt, Mt, Bt, Nt, and Pt are the concentrations of the target cells, DENV-infected cells, DENV particles, B cells, heterologous antibody previously formed on primary infection, and homologous antibody against the new virus serotype of the secondary infection, respectively. The parameter δ represents the creation rate of the target cells. The DENV particles infect the target cells at rate μKM. The DENV-infected cells produce viruses at rate τL. The B cells are created at constant rate β and proliferated at rate φBM. The death rates of the compartments K, L, M, B, N, and P are given by ξK, ϱL, ηM, γB, α1N, and α2P, respectively. The two types of antibodies N and P are generated from the B cells at rates ϑϵB and ϵB and neutralize the DENV at rates ϖ1MN and ϖ2MP. The terms ϰ1NM and ϰ1PM represent the rates at which antibody virus complex affects the antibody growth. ϑ∈0,1 is a correlation factor that quantifies the similarity between the individual serotypes. We observe that the global stability of the models presented in [20–25] is not well studied.All of the DENV infection models in the abovementioned works are given by ordinary or delay differential equations under the assumption that the cells and DENV particles are well mixed. Spatial structure plays an important role in understanding the dynamical behavior of viral infection within a host. In recent years, spatial dependence has been incorporated into mathematical models of several viral infections such as hepatitis B virus (HBV) [26], hepatitis C virus (HCV) [27, 28], human immunodeficiency virus (HIV) [29–31], and human T lymphotropic virus type I (HTLV-I) [32]. To the best of our knowledge, the DENV infection model with diffusion has not been studied before. Therefore, the aim of the present study is to focus on the dynamical behavior of DENV infection with diffusion. Following the work of Hattaf [33], our proposed model takes into account the spatial mobility of all compartments.


## 2. Mathematical DENV Dynamics Model

We develop a DENV infection model with secondary infection and diffusion as(2)∂Ku,t∂t=dKΔKu,t+δ−μKu,tMu,t−ξKu,t,(3)∂Lu,t∂t=dLΔLu,t+μKu,tMu,t−ϱLu,t,(4)∂Mu,t∂t=dMΔMu,t+τLu,t−ηMu,t−ϖ1Mu,tNu,t−ϖ2Mu,tPu,t,(5)∂Nu,t∂t=dNΔNu,t+λ1Mu,tNu,t−α1Nu,t,(6)∂Pu,t∂t=dPΔPu,t+λ2Mu,tPu,t−α2Pu,t,where u∈Γ is the position. The heterologous and homologous antibodies are activated at rates λ1MN and λ2MP, respectively. Here, Δ is the Laplacian operator and dx is the diffusion coefficient, where x∈K,L,M,N,P. The spatial domain Γ⊂ℝm (where m≥1) is bounded and connected; moreover, its boundary ∂Γ is smooth.The initial conditions are given by(7)Ku,0=G1u,Lu,0=G2u,Mu,0=G3u,Nu,0=G4u,Pu,0=G5u,u∈Γ¯,where Gℓu≥0, ℓ=1,…,5, are the continuous functions. In addition, we take the following homogeneous Neumann boundary conditions:(8)∂K∂ℬ⟶=∂L∂ℬ⟶=∂M∂ℬ⟶=∂N∂ℬ⟶=∂P∂ℬ⟶=0,t>0,u∈∂Γ,where ∂/∂ℬ⟶ is the outward normal derivative on the boundary ∂Γ. These boundary conditions indicate that cells and viruses cannot cross the isolated boundary [34].


## 3. Well-Posedness of Solutions

Theorem 1.
Assume thatdK=dL=dM=dN=dP=d˜. Then, models (2)–(6) with any initial satisfying (7) has a unique, nonnegative, and bounded solution defined on Γ¯×0,+∞.Proof.
We denoteX=BUCΓ¯,ℝ5, the set of all bounded and uniformly continuous functions from Γ¯ to ℝ5, with norm θX=supu∈Γ¯θu. We define the positive cone X+=BUCΓ¯,ℝ+5⊂X which induces a partial order on X. This shows that the space X,⋅X is a Banach lattice [35, 36].
For any initial dataG=G1,G2,G3,G4,G5T∈X+, we define H=H1,H2,H3,H4,H5T:X+⟶X by(9)H1Gu=δ−μG1uG3u−ξG1u,H2Gu=μG1uG3u−ϱG2u,H3Gu=τG2u−ηG3u−ϖ1G3uG4u−ϖ2G3uG5u,H4Gu=λ1G3uG4u−α1G4u,H5Gu=ϖG3uG5u−α2G5u.
It is clear thatH is locally Lipschitz on X+. We can rewrite systems (2)–(6) with initial conditions (7) and boundary conditions (8) as the following abstract functional differential equation:(10)dU¯dt=ΘU¯+HU¯,t>0,U¯0=G∈X+,where U¯=K,L,M,N,PT and ΘU¯=dKΔK,dLΔL,dMΔM,dNΔN,dPΔPT. One can show that(11)limq⟶0+1qdistG0+qHG,X+=0,∀G∈X+.
Hence, for anyG∈X+, systems (2)–(6) with (7)-(8) has a unique nonnegative mild solution Ku,t,Lu,t,Mu,t,Nu,t,Pu,t defined on Γ¯×0,Tm, where 0,Tm is the maximal existence time interval on which the solution exists [35–37]. In addition, this solution also is a classical solution for the given problem.
We define(12)Ψu,t=Ku,t+Lu,t+ϱ2τMu,t+ϱϖ12τλ1Nu,t+ϱϖ22τλ2Pu,t.
Then, using systems (2)–(6), we obtain(13)∂Ψu,t∂t=dKΔKu,t+δ−μKu,tMu,t−ξKu,t+dLΔLu,t+μKu,tMu,t−ϱLu,t,ϱ2τdMΔMu,t+τLu,t−ηMu,t−ϖ1Mu,tNu,t−ϖ2Mu,tPu,t+ϱϖ12τλ1dNΔNu,t+λ1Mu,tNu,t−α1Nu,t+ϱϖ22τλ2dPΔPu,t+λ2Mu,tPu,t−α2Pu,t.
SincedK=dL=dM=dN=dP=d˜, then we get(14)∂Ψu,t∂t−d˜ΔΨu,t=δ−ξKu,t−ϱ2Lu,t−ϱη2τMu,t−ϱϖ1α12τλ1Nu,t−ϱϖ2α22τλ2Pu,t≤δ−σKu,t+Lu,t+ϱ2τMu,t+ϱϖ12τλ1Nu,t+ϱϖ22τλ2Pu,t=δ−σΨu,t,where σ=minξ,1/2ϱ,η,α1,α2. Thus, Ψu,t satisfies(15)∂Ψu,t∂t−d˜ΔΨu,t≤δ−σΨu,t,Ψu,0=G1u+G2u+ϱ2τG3u+ϱϖ12τλ1G4u+ϱϖ22τλ2G5u≥0,∂Ψ∂ℬ⟶=0.
LetΨ˜t be a solution of the following ODE:(16)dΨ˜tdt=δ−σΨ˜t,Ψ˜0=maxu∈Γ¯Ψu,0.
This gives thatΨ˜t≤maxδ/σ,maxu∈Γ¯Ψu,0. On the basis of comparison principle [38], we obtain Ψu,t≤Ψ˜t. Hence,(17)Ψu,t≤maxδσ,maxu∈Γ¯Ψu,0,which implies that Ku,t, Lu,t, Mu,t, Nu,t, and Pu,t are bounded on Γ¯×0,Tm. The standard theory for semilinear parabolic systems implies that Tm=+∞ [39]. This shows that solution Ku,t,Lu,t,Mu,t,Nu,t,Pu,t is defined for all u∈Γ, t>0, and also is unique and nonnegative.


## 4. Equilibria

Theorem 2.
There exist three threshold parametersℛ0, ℛ1, and ℛ2 with ℛ0>ℛ1 and ℛ0>ℛ2, such that(i)
Ifℛ0≤1, then the system has a single equilibrium Ω0(ii)
Ifℛ1≤1<ℛ0 and ℛ2≤1<ℛ0, then the system contains only two equilibria Ω0 and Ω1(iii)
Ifℛ1>1 and ℛ2<1, then the system contains three equilibria Ω0, Ω1, and Ω2(iv)
Ifℛ2>1 and ℛ1<1, then the system contains three equilibria Ω0, Ω1, and Ω3(v)
Ifℛ2>1 and ℛ1>1, then the system contains four equilibria Ω0, Ω1, Ω2, and Ω3Proof.
To calculate the equilibria of systems (2)–(6), we let(18)0=δ−μKM−ξK,(19)0=μKM−ϱL,(20)0=τL−ηM−ϖ1MN−ϖ2MP,(21)0=λ1MN−α1N,(22)0=λ2MP−α2P.
Then, solving the system of algebraic equations (18)–(22), we get four equilibria such as the following:(i)
Infection-free equilibriumΩ0K0,0,0,0,0,0, where K0=δ/ξ(ii)
Persistent DENV infection equilibrium with ineffective antibodies isΩ1K1,L1,M1,0,0, where(23)K1=K0ℛ0,L1=ηξτμℛ0−1,M1=ξμℛ0−1.(iii)
Persistent DENV infection equilibrium with only effective heterologous antibody isΩ2K2,L2,M2,N2,0, where(24)K2=λ1δξλ1+μα1,L2=μδα1ϱξλ1+μα1,M2=α1λ1,N2=ηϖ1ℛ1−1.(iv)
Persistent DENV infection equilibrium with only effective homologous antibody isΩ3K3,L3,M3,0,P3, where(25)K3=λ2δξλ2+μα2,L3=μδα2ϱξλ2+μα2,M3=α2λ2,P3=ηϖ2ℛ2−1.Where(26)ℛ0=K0τμϱη,ℛ1=ℛ01+μα1/ξλ1,ℛ2=ℛ01+μα2/ξλ2.Clearly,ℛ0>ℛ1 and ℛ0>ℛ2(v)
Clearly from (iii) and (iv), ifℛ1>1 and ℛ2>1, then Ω0, Ω1, Ω2, and Ω3 all exist
Here,ℛ0 represents the basic infection reproduction number, ℛ1 represents the heterologous antibody immune response activation number, and ℛ2 is the homologous antibody immune response activation number.


## 5. Global Stability

In this section, we investigate the global asymptotic stability of all equilibria by the Lyapunov method. The construction of Lyapunov functions are based on the works presented in [40–44]. To prove Theorems 1–4, we need to define a function gs=s−1−lns and the arithmetic-geometric mean inequality:(27)1s∑ℓ=1sχℓ≥n∏ℓ=1sχℓ,χℓ≥0,ℓ=1,2,…,which implies(28)13KℓK+LMℓLℓM+MKLℓMℓKℓL≥1,ℓ=1,2,3.Neumann boundary conditions (8) and divergence theorem imply that(29)0=∫∂Γ∇U⋅ℬ⟶du=∫Γdiv∇Udu=∫ΓΔUdu,0=∫∂Γ1U∇U⋅ℬ⟶du=∫Γdiv1U∇Udu=∫ΓΔUU−▽U2U2du,for U∈K,L,M,N,P. Thus, we obtain(30)∫ΓΔUdu=0,∫ΓΔUUdu=∫Γ▽U2U2du.For convenience, we drop the input notation, i.e.,K,L,M,N,P=Ku,t,Lu,t,Mu,t,Nu,t,Pu,t.Consider a functionΠℓK,L,M,N,P and define(31)Π˜ℓt=∫ΓΠℓu,tdu,ℓ=0,1,2,3.LetYℓ′ be the largest invariant subset of(32)Yℓ=K,L,M,N,P:dΠ˜ℓdt=0,ℓ=0,1,2,3.Theorem 3.
Letℛ0≤1, then Ω0 is globally asymptotically stable (GAS).Proof.
DefineΠ0u,t as(33)Π0u,t=K0gKK0+L+ϱτM+ϱϖ1τλ1N+ϱϖ2τλ2P.
Clearly,Π˜0K,L,M,N,P>0 for all K,L,M,N,P>0 and Π˜0K0,0,0,0,0=0. We calculate ∂Π0/∂t along the solutions of model (2)–(6) as(34)∂Π0∂t=1−K0KdKΔK+δ−ξK−μKM+dLΔL+μKM−ϱL+ϱτdMΔM+τL−ηM−ϖ1MN−ϖ2MP+ϱϖ1τλ1dNΔN+λ1MN−α1N+ϱϖ2τλ2dPΔP+λ2MP−α2P.
Collecting terms of equation (34), we obtain(35)∂Π0∂t=−ξK−K0K2+μK0−ϱητM−ϱϖ1α1τλ1N−ϱϖ2α2τλ2P+1−K0KdKΔK+dLΔL+ϱτdMΔM+ϱϖ1τλ1dNΔN+ϱϖ2τλ2dPΔP=−ξK−K0K2+ϱητℛ0−1M−ϱϖ1α1τλ1N−ϱϖ2α2τλ2P+1−K0KdKΔK+dLΔL+ϱτdMΔM+ϱϖ1τλ1dNΔN+ϱϖ2τλ2dPΔP.
Consequently, we calculatedΠ˜0/dt as follows:(36)dΠ˜0dt=−ξ∫ΓK−K0K2du+ϱητℛ0−1∫ΓMdu−ϱϖ1α1τλ1∫ΓNdu−ϱϖ2α2τλ2∫ΓPdu+dK∫Γ1−K0KΔKdu+dL∫ΓΔLdu+ϱτdM∫ΓΔMdu+ϱϖ1τλ1dN∫ΓΔNdu+ϱϖ2τλ2dP∫ΓΔPdu.
Using equality (30), equation (36) is reduced to the following form:(37)dΠ˜0dt=−ξ∫ΓK−K0K2du+ϱητℛ0−1∫ΓMdu−ϱϖ1α1τλ1∫ΓNdu−ϱϖ2α2τλ2∫ΓPdu−dKK0∫Γ▽K2K2du.
Therefore,dΠ˜0/dt≤0, for all K,M,N,P>0, and dΠ˜0/dt=0 with equality holding when K,M,N,P=K0,0,0,0. The solutions of models (2)–(6) converge to Y0′. The elements of Y0′ satisfy K,M,N,P=K0,0,0,0, and then, ∂M/∂t=ΔM=0. Equation (6) reduces to(38)0=∂M∂t=τL.
This yieldsL=0. Hence, Y0′=Ω0, and by applying LLIP [45–47], we get that Ω0 is GAS.Theorem 4.
Ifℛ1≤1<ℛ0 and ℛ2≤1, then Ω1 is GAS.Proof.
DefineΠ1u,t as(39)Π1u,t=K1gKK1+L1gLL1+ϱτM1gMM1+ϱϖ1τλ1N+ϱϖ2τλ2P.
Calculate∂Π1/∂t as(40)∂Π1∂t=1−K1KdKΔK+δ−ξK−μKM+1−L1LdLΔL+μKM−ϱL+ϱτ1−M1MdMΔM+τL−ηM−ϖ1MN−ϖ2MP+ϱϖ1τλ1dNΔN+λ1MN−α1N+ϱϖ2τλ2dPΔP+λ2MP−α2P=1−K1Kδ−ξK+μK1M−μKML1L+ϱL1−ϱητM−ϱM1LM+ϱητM1+ϱϖ1τM1N+ϱϖ2τM1P−ϱϖ1α1τλ1N−ϱϖ2α2τλ2P+dK1−K1KΔK+dL1−L1LΔL+ϱdMτ1−M1MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ2ΔP.
The equilibrium conditions ofΩ1 imply that(41)δ=ξK1+μK1M1,ϱL1=μK1M1,τL1=ηM1,
We getμK1M−ϱη/τM=0 and(42)∂Π1∂t=−ξK−K12K+ϱL11−K1K−ϱL1KML1K1M1L+2ϱL1−ϱL1M1LML1+ϱϖ1τM1−α1λ1N+ϱϖ2τM1−α2λ2P+dK1−K1KΔK+dL1−L1LΔL+ϱdMτ1−M1MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ2ΔP=−ξK−K12K+ϱL13−K1K−KML1K1M1L−M1LML1+ϱϖ1ξλ1+μα1τλ1μℛ1−1N+ϱϖ2ξλ2+μα2τλ2μℛ2−1P+dK1−K1KΔK+dL1−L1LΔL+ϱdMτ1−M1MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ2ΔP.
Calculate the time derivative ofΠ˜1t and use equality (30) to get(43)dΠ˜1dt=−ξ∫ΓK−K12Kdu+ϱL1∫Γ3−K1K−KML1K1M1L−M1LML1du+ϱϖ1ξλ1+μα1τλ1μℛ1−1∫ΓNdu+ϱϖ2ξλ2+μα2τλ2μℛ2−1∫ΓPdu+dK∫Γ1−K1KΔKdu+dL∫Γ1−L1LΔLdu+ϱdMτ∫Γ1−M1MΔMdu+ϱϖ1dNτλ1∫ΓΔNdu+ϱϖ2dPτλ2∫ΓΔPdu=−ξ∫ΓK−K12Kdu+ϱL1∫Γ3−K1K−KML1K1M1L−M1LML1du+ϱϖ1ξλ1+μα1τλ1μℛ1−1∫ΓNdu+ϱϖ2ξλ2+μα2τλ2μℛ2−1∫ΓPdu−dKK1∫Γ▽K2K2du−dLL1∫Γ▽L2L2du−ϱdMτM1∫Γ▽M2M2du.
Sinceℛ1≤1 and ℛ2≤1, then utilizing inequality (28), we obtain dΠ˜1/dt≤0 for all K,L,M,N,P>0. Furthermore, dΠ˜1/dt=0 at K,L,M,N,P=K1,L1,M1,0,0. The solutions of (2)–(6) tend to Y1′=Ω1. It follows that Ω1 is GAS by using LLIP.Theorem 5.
Letℛ1>1 and ℛ2≤ℛ1 then Ω2 is GAS.Proof.
DefineΠ2u,t as(44)Π2u,t=K2gKK2+L2gLL2+ϱτM2gMM2+ϱϖ1τλ1N2gNN2+ϱϖ2τλ2P.
Calculate∂Π2/∂t as(45)∂Π2∂t=1−K2KdKΔK+δ−ξK−μKM+1−L2LdLΔL+μKM−ϱL+ϱτ1−M2MdMΔM+τL−ηM−ϖ1MN−ϖ2MP+ϱϖ1τλ11−N2NdNΔN+λ1MN−α1N+ϱϖ2τλ2dPΔP+λ2MP−α2P.
Collecting terms of equation (45), we get(46)∂Π2∂t=1−K2Kδ−ξK+μK2M−μKML2L+ϱL2−ϱητM−ϱM2LM+ϱητM2+ϱϖ1τM2N+ϱϖ2τM2P−ϱϖ1α1τλ1N−ϱϖ1τN2M+ϱϖ1α1τλ1N2−ϱϖ2α2τλ2P+dK1−K2KΔK+dL1−L2LΔL+ϱdMτ1−M2MΔM+ϱϖ1dNτλ11−N2NΔN+ϱϖ2dPτλ2ΔP=1−K2Kδ−ξK+μK2−ϱητ−ϱϖ1τN2M+ϱϖ1τM2−α1λ1N−μKML2L+ϱL2−ϱM2LM+ϱητM2+ϱϖ2τM2P+ϱϖ1α1τλ1N2−ϱϖ2α2τλ2P+dK1−K2KΔK+dL1−L2LΔL+ϱdMτ1−M2MΔM+ϱϖ1dNτλ11−N2NΔN+ϱϖ2dPτλ2ΔP.
Applying the equilibrium conditions,(47)δ=ξK2+μK2M2,ϱL2=μK2M2,τL2=ηM2+ϖ1M2N2,M2=α1λ1,we get(48)∂Π2∂t=−ξK−K22K+ϱL21−K2K−ϱL2KML2K2M2L−ϱL2M2LML2+2ϱL2+ϱϖ2τM2−M3P+dK1−K2KΔK+dL1−L2LΔL+ϱdMτ1−M2MΔM+ϱϖ1dNτλ11−N2NΔN+ϱϖ2dPτλ2ΔP=−ξK−K22K+ϱL23−K2K−KML2K2M2L−M2LML2+ϱϖ2ξλ2+μα2τμλ2ℛ1ℛ2−ℛ1P+dK1−K2KΔK+dL1−L2LΔL+ϱdMτ1−M2MΔM+ϱϖ1dNτλ11−N2NΔN+ϱϖ2dPτλ2ΔP.
Calculate the time derivative ofΠ˜2t and use equality (30) to get(49)dΠ˜2dt=−ξ∫ΓK−K22Kdu+ϱL2∫Γ3−K2K−KML2K2M2L−M2LML2du+ϱϖ2ξλ2+μα2τμλ2ℛ1ℛ2−ℛ1∫ΓPdu+dK∫Γ1−K2KΔKdu+dL∫Γ1−L2LΔLdu+ϱdMτ∫Γ1−M2MΔMdu+ϱϖ1dNτλ1∫Γ1−N2NΔNdu+ϱϖ2dPτλ2∫ΓΔPdu=−ξ∫ΓK−K22Kdu+ϱL2∫Γ3−K2K−KML2K2M2L−M2LML2du+ϱϖ2ξλ2+μα2τμλ2ℛ1ℛ2−ℛ1∫ΓPdu−dKK2∫Γ▽K2K2du−dLL2∫Γ▽L2L2du−ϱdMτM2∫Γ▽M2M2du−ϱϖ1dNτλ1N2∫Γ▽N2N2du.
Sinceℛ2≤ℛ1, then using inequality (28), we obtain dΠ˜2/dt≤0 for all K,L,M,N,P>0. Furthermore, dΠ˜2/dt≤0 at K,L,M,N,P=K2,L2,M2,N2,0. The solutions of (2)–(6) tend to Y2′=Ω2. It follows that Ω2 is GAS by using LLIP.Theorem 6.
Letℛ2>1 and ℛ1≤ℛ2, then Ω3 is GAS.Proof.
DefineΠ3u,t as(50)Π3u,t=K3gKK3+L3gLL3+ϱτM3gMM3+ϱϖ1τλ1N+ϱϖ2τλ2P3gPP3.
Calculating∂Π3/∂t along the solutions of models (2)–(6), we get(51)∂Π3∂t=1−K3KdKΔK+δ−ξK−μKM+1−L3LdLΔL+μKM−ϱL+ϱτ1−M3MdMΔM+τL−ηM−ϖ1MN−ϖ2MP+ϱϖ1τλ1dNΔN+λ1MN−α1N+ϱϖ2τλ21−P3PdPΔP+λ2MP−α2P.
Collecting terms, we get(52)∂Π3∂t=1−K3Kδ−ξK+μK3M−μKML3L+ϱL3−ϱητM−ϱM3LM+ϱητM3+ϱϖ1τM3N+ϱϖ2τM3P−ϱϖ1α1τλ1N−ϱϖ2α2τλ2P−ϱϖ2τP3M+ϱϖ2α2τλ2P3+dK1−K3KΔK+dL1−L3LΔL+ϱdMτ1−M3MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ21−P3PΔP.
Applying the equilibrium conditions ofΩ3,(53)δ=ξK3+μK3M3,ϱL3=μK3M3,τL3=ηM3+ϖ2M3P3,α2P3=λ2M3P3,we get(54)∂Π3∂t=−ξK−K32K+ϱL31−K3K−ϱL3KML3K3M3L−ϱL3M3LML3+2ϱL3+ϱϖ1τM3−M2N+dK1−K3KΔK+dL1−L3LΔL+ϱdMτ1−M3MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ21−P3PΔP=−ξK−K32K+ϱL33−K3K−KML3K3M3L−M3LML3+ϱϖ1ξλ2+μα2τμλ2ℛ1ℛ1−ℛ2N+dK1−K3KΔK+dL1−L3LΔL+ϱdMτ1−M3MΔM+ϱϖ1dNτλ1ΔN+ϱϖ2dPτλ21−P3PΔP.
Calculating the time derivative ofΠ˜3t and using equality (30), we obtain(55)dΠ˜3dt=−ξ∫ΓK−K32Kdu+ϱL3∫Γ3−K3K−KML3K3M3L−M3LML3du+ϱϖ1ξλ2+μα2τμλ2ℛ1ℛ1−ℛ2∫ΓNdu+dK∫Γ1−K3KΔKdu+dL∫Γ1−L3LΔLdu+ϱdMτ∫Γ1−M3MΔMdu+ϱϖ1dNτλ1∫ΓΔNdu+ϱϖ2dPτλ2∫Γ1−P3PΔPdu=−ξ∫ΓK−K32Kdu+ϱL3∫Γ3−K3K−KML3K3M3L−M3LML3du+ϱϖ1ξλ2+μα2τμλ2ℛ1ℛ1−ℛ2∫ΓNdu−dKK3∫Γ▽K2K2du−dLL3∫Γ▽L2L2du−ϱdMM3τ∫Γ▽M2M2du−ϱϖ2dPP3τλ2∫Γ▽P2P2du.
Sinceℛ1≤ℛ2, then using inequality (28), we obtain dΠ˜3/dt≤0, for all K,L,M,N,P>0. Furthermore, dΠ˜3/dt≤0 at K,L,M,N,P=K3,L3,M3,0,P3. The solutions of (2)–(6) tend to Y3′=Ω3. It follows that Ω3 is GAS by using LLIP.


## 6. Numerical Simulations

In this section, we numerically illustrate the global stability of equilibria by choosing the domainΓ as Γ=0,2 with a step size 0.02. The step size for time is given by 0.1. Following the works presented in [31, 48–50], we consider the following initial conditions for systems (2)–(6):(56)Ku,0=5001+0.4cos2πu,Lu,0=301+0.5cos2πu,Mu,0=41+0.5cos2πu,Nu,0=21+0.5cos2πu,Pu,0=21+0.5cos2πu,u∈0,2.The initial values are arbitrarily chosen as the global stability of the equilibria presented in Theorems3–6 guarantees the convergence regardless of the selected initial conditions.In addition, we consider the following homogeneous Neumann boundary conditions:(57)∂K∂ℬ⟶=∂L∂ℬ⟶=∂M∂ℬ⟶=∂N∂ℬ⟶=∂P∂ℬ⟶=0,t>0,u=0,2.We are using the following values of the parameters:δ=10, ξ=0.01, ϱ=0.3, τ=5, η=3, ϖ1=0.3, ϖ2=0.1, α1=0.1, and α2=0.1. The parameters μ, λ1, and λ2 will be selected according to the following strategies:Strategy 1.
(Stability ofΩ0) μ=0.00004,λ1=0.0005, and λ2=0.001. For this set of parameters, we have ℛ0=0.2778<1, ℛ1=0.1389<1, and ℛ2=0.1852<1. Figure 1 shows that the solution of systems (2)–(6) converges to the equilibrium Ω0=1000,0,0,0,0. This shows that Ω0 is GAS according to Theorem 3. In this case, the DENV will be cleared.Figure 1
TakingStrategy 1 (ℛ0≤1), the equilibrium Ω0=1000,0,0,0,0 is asymptotically stable. (a) Target cells. (b) DENV-infected cells. (c) DENV particles. (d) Heterologous antibodies. (e) Homologous antibodies.
(a)(b)(c)(d)(e)Strategy 2.
(Stability ofΩ1) μ=0.0004,λ1=0.0005, and λ2=0.001. With such choice, we get ℛ0=2.2222>1, ℛ1=0.2469<1, and ℛ2=0.4444<1. Theorem 4 implies that Ω1=450,18.333,30.556,0,0 is GAS which is displayed in Figure 2. This will lead to the situation of persistent DENV infection but with an ineffective antibody immune response.Figure 2
TakingStrategy 1 (ℛ1≤1<ℛ0 and ℛ2≤1), the equilibrium Ω1=450,18.333,30.556,0,0 is asymptotically stable. (a) Target cells. (b) DENV-infected cells. (c) DENV particles. (d) Heterologous antibodies. (e) Homologous antibodies.
(a)(b)(c)(d)(e)Strategy 3.
(Stability ofΩ2) μ=0.0004,λ1=0.005, and λ2=0.001. Then, we compute ℛ0=2.2222>1, ℛ1=1.2346>1, and ℛ2=0.4444<ℛ1. Numerical results show that Ω2=555.556,14.815,20,2.346,0 exists. Moreover, Figure 3 shows that Ω2 is GAS which supports Theorem 5. Therefore, a persistent DENV infection with only effective heterologous antibody immune response is reached.Figure 3
TakingStrategy 1 (ℛ1>1 and ℛ2≤ℛ1), the equilibrium Ω2=555.556,14.815,20,2.346,0 is asymptotically stable. (a) Target cells. (b) DENV-infected cells. (c) DENV particles. (d) Heterologous antibodies. (e) Homologous antibodies.
(a)(b)(c)(d)(e)Strategy 4.
(Stability ofΩ3) μ=0.0004,λ1=0.01, and λ2=0.02. Then, we compute ℛ0=2.2222>1, ℛ1=1.5873>1, and ℛ2=0.8519>ℛ1. According to these data, Ω3 exists with Ω3=833.333,5.556,5,0,25.556. In Figure 4, we show that Ω3 is GAS which is consistent with Theorem 6. In this case, a persistent DENV single infection with only effective homologous antibody immune response is reached.Figure 4
TakingStrategy 1 (ℛ2>1 and ℛ1≤ℛ2), the equilibrium Ω3=833.333,5.556,5,0,25.556 is asymptotically stable. (a) Target cells. (b) DENV-infected cells. (c) DENV particles. (d) Heterologous antibodies. (e) Homologous antibodies.
(a)(b)(c)(d)(e)


## 7. Conclusion

A dynamical model to capture the behavior of secondary DENV infection was studied. The model incorporated the spatial mobility of DENV particles and cells. The model was given by five PDEs to describe the interaction between five compartments, target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. We first established that the model is biologically relevant by showing that the key variables of the model are nonnegative and bounded. We found that the model has four equilibriaΩi, i=1,2,3,4, and their existence and global stability are governed by three threshold parameters, ℛi, i=1,2,3. We performed the global stability analysis of the four equilibria by constructing suitable Lyapunov functions. We conducted some numerical simulations and found that the numerical results are fully aligned with the theoretical results. We summarize the obtained results in the following:(i)
The infection-free equilibriumΩ0 always exists, and it is GAS if ℛ0≤1. This case corresponds to the healthy state where the DENV particles are cleared.(ii)
The persistent DENV infection equilibrium with ineffective antibodiesΩ1 exists if ℛ0>1, and it is GAS if ℛ1≤1 and ℛ2≤1. At this point, the DENV infection exists while the immune response is not active.(iii)
The persistent DENV infection equilibrium with only effective heterologous antibodyΩ2 exists if ℛ1>1, and it is GAS if ℛ2≤ℛ1. This case leads to the situation where the DENV infection is chronic, while only heterologous antibody immune response is working.(iv)
The persistent DENV infection equilibrium with only effective homologous antibodyΩ3 exists if ℛ2>1, and it is GAS if ℛ1≤ℛ2. At this point, the DENV infection is chronic with only an active homologous antibody immune response.Our model can be extended to take into account the time delays during the DENV infection as(58)∂Ku,t∂t=dKΔKu,t+δ−μKu,tMu,t−ξKu,t,∂Lu,t∂t=dLΔLu,t+μe−k1ψ1Ku,t−ψ1Mu,t−ψ1−ϱLu,t,∂Mu,t∂t=dMΔMu,t+τe−k2ψ2Lu,t−ψ2−ηMu,t−ϖ1Mu,tNu,t−ϖ2Mu,tPu,t,∂Nu,t∂t=dNΔNu,t+λ1Mu,tNu,t−α1Nu,t,∂Pu,t∂t=dPΔPu,t+λ2Mu,tPu,t−α2Pu,t.Here, it is assumed that a DENV contacts a target cell at timet−ψ1, and the cell becomes infected at time t. Moreover, an infected cell at t−ψ2 produces new infectious DENV particles at time t. The factors e−kiψi, i=1,2, represent the survival rates of the cells during the delay periods.



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*Source: 5585175-2021-06-28.xml*