
Diabetic retinopathy (DR) is one of the main leading causes of visual impairment worldwide. The current study elucidates the role of JQ1 in DR. A diabetic model was constructed by STZ injection and a high-fat diet. After establishment of the diabetic model, rats were assigned to treatment groups: 1) control, 2) diabetic model, and 3) diabetic+JQ1 model. In vitro Transwell and wound-healing assays were used to measure BV2 cell viability by stimulation with low glucose and high glucose with or without JQ1 and 740Y-P. Pathological methods were used to analyze DR, and Western blotting was used to analyze protein expression. Identification of enriched pathways in DR was performed by bioinformatics. Histopathological examination demonstrated that JQ1 rescued the loss of retinal cells and increased the thickness of retinal layers in diabetic rats. JQ1 attenuated high glucose-stimulated BV2 microglial motility and migration. The bioinformatics analysis implied that the Pl3K-Akt signaling pathway was enriched in DR. JQ1 decreased the phosphorylation of PI3K and AKT as well as the immunostaining of PI3K in BV2 cells. 740Y-P (a PI3K agonist) significantly reversed the decrease in p-PI3K and p-AK in BV2 cells. Additionally, JQ1 decreased the protein expression of p-PI3K, p-AKT, and MMP2/9 and immunostaining of PI3K in retinal tissues of rats. JQ1 suppresses the PI3K/Akt cascade by targeting MMP expression, thus decreasing the viability and invasion capacity of retinal microglia, suggesting an interesting treatment target for DR.
Citation: Ying Zhu, Lipeng Guo, Jixin Zou, Liwen Wang, He Dong, Shengbo Yu, Lijun Zhang, Jun Li, Xueling Qu. JQ1 inhibits high glucose-induced migration of retinal microglial cells by regulating the PI3K/AKT signaling pathway[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13079-13092. doi: 10.3934/mbe.2022611
[1] | Bevina D. Handari, Dipo Aldila, Bunga O. Dewi, Hanna Rosuliyana, Sarbaz H. A. Khosnaw . Analysis of yellow fever prevention strategy from the perspective of mathematical model and cost-effectiveness analysis. Mathematical Biosciences and Engineering, 2022, 19(2): 1786-1824. doi: 10.3934/mbe.2022084 |
[2] | Lili Liu, Xi Wang, Yazhi Li . Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity. Mathematical Biosciences and Engineering, 2023, 20(12): 20914-20938. doi: 10.3934/mbe.2023925 |
[3] | Vanessa Steindorf, Sergio Oliva, Jianhong Wu . Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model. Mathematical Biosciences and Engineering, 2022, 19(3): 2950-2984. doi: 10.3934/mbe.2022136 |
[4] | Haitao Song, Dan Tian, Chunhua Shan . Modeling the effect of temperature on dengue virus transmission with periodic delay differential equations. Mathematical Biosciences and Engineering, 2020, 17(4): 4147-4164. doi: 10.3934/mbe.2020230 |
[5] | Tingting Xue, Long Zhang, Xiaolin Fan . Dynamic modeling and analysis of Hepatitis B epidemic with general incidence. Mathematical Biosciences and Engineering, 2023, 20(6): 10883-10908. doi: 10.3934/mbe.2023483 |
[6] | Chandrani Banerjee, Linda J. S. Allen, Jorge Salazar-Bravo . Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission. Mathematical Biosciences and Engineering, 2008, 5(4): 617-645. doi: 10.3934/mbe.2008.5.617 |
[7] | Ling Xue, Caterina Scoglio . Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences and Engineering, 2015, 12(3): 565-584. doi: 10.3934/mbe.2015.12.565 |
[8] | Chenwei Song, Rui Xu, Ning Bai, Xiaohong Tian, Jiazhe Lin . Global dynamics and optimal control of a cholera transmission model with vaccination strategy and multiple pathways. Mathematical Biosciences and Engineering, 2020, 17(4): 4210-4224. doi: 10.3934/mbe.2020233 |
[9] | Rajanish Kumar Rai, Arvind Kumar Misra, Yasuhiro Takeuchi . Modeling the impact of sanitation and awareness on the spread of infectious diseases. Mathematical Biosciences and Engineering, 2019, 16(2): 667-700. doi: 10.3934/mbe.2019032 |
[10] | Fumin Zhang, Zhipeng Qiu, Balian Zhong, Tao Feng, Aijun Huang . Modeling Citrus Huanglongbing transmission within an orchard and its optimal control. Mathematical Biosciences and Engineering, 2020, 17(3): 2048-2069. doi: 10.3934/mbe.2020109 |
Diabetic retinopathy (DR) is one of the main leading causes of visual impairment worldwide. The current study elucidates the role of JQ1 in DR. A diabetic model was constructed by STZ injection and a high-fat diet. After establishment of the diabetic model, rats were assigned to treatment groups: 1) control, 2) diabetic model, and 3) diabetic+JQ1 model. In vitro Transwell and wound-healing assays were used to measure BV2 cell viability by stimulation with low glucose and high glucose with or without JQ1 and 740Y-P. Pathological methods were used to analyze DR, and Western blotting was used to analyze protein expression. Identification of enriched pathways in DR was performed by bioinformatics. Histopathological examination demonstrated that JQ1 rescued the loss of retinal cells and increased the thickness of retinal layers in diabetic rats. JQ1 attenuated high glucose-stimulated BV2 microglial motility and migration. The bioinformatics analysis implied that the Pl3K-Akt signaling pathway was enriched in DR. JQ1 decreased the phosphorylation of PI3K and AKT as well as the immunostaining of PI3K in BV2 cells. 740Y-P (a PI3K agonist) significantly reversed the decrease in p-PI3K and p-AK in BV2 cells. Additionally, JQ1 decreased the protein expression of p-PI3K, p-AKT, and MMP2/9 and immunostaining of PI3K in retinal tissues of rats. JQ1 suppresses the PI3K/Akt cascade by targeting MMP expression, thus decreasing the viability and invasion capacity of retinal microglia, suggesting an interesting treatment target for DR.
Imperfect or leaky vaccines are those that do not provide complete immunity to the disease in their hosts and are incapable of preventing transmission to others. It is known that leaky vaccines can reduce the symptoms of the disease but do not offer full protection against infection or the spread of the pathogen. Therefore, studying the impact of imperfect vaccines on the human population becomes crucial. Additionally, for certain infectious diseases, it has been observed that the effectiveness of the vaccine may diminish over time. In the recent COVID-19 pandemic, it has been noted that most vaccines remain effective for a specific duration before individuals become susceptible to the virus once again.
During the course of the follow-up period of six months, Stephen et al. [1] studied the vaccine efficacy of different vaccines (shown in Figure 1). Menni et al. [2] showed that in the case of COVID-19, the vaccine's efficacy remained high after five months among those under the age of 55. They also showed that the booster dose can restore the effectiveness of the vaccine. Jake et al. [3] also studied the vaccine efficacy and demonstrated that the evidence of waning immunity is overstated. They demonstrated that antibody levels can be boosted in the general population and robust clinical data are required to assess the need for additional doses. This naturally motivates us to consider a compartmental model for infectious diseases with a vaccine-waning rate.
Kermack and Mckendrick [5,6,7] developed ordinary differential equations (ODEs) based models to study the transmission of infectious diseases. These models are formulated based on the assumption that all individuals have identical vital rates regardless of age or size. Enough literature is available on ODE-based infectious disease models; more details can be found in [8,9,10,11,12]. In epidemic models, the "disease clock" is important for studying the long-term dynamics of an infectious disease. Moreover, in transmitting infectious diseases, the age of infection or the time since infection starts plays a vital role. Age-structured population models characterize individuals based on their age, which can be class-age or demographic age. Class age means the time since infection, and demographic age means the chronological age of individuals.
Iannelli [13] discussed age-structured population models and their role in describing the growth and interaction of populations. Thieme and Castilo [14] examined the role of infection age in the HIV viral dynamics. More details about age and size-structured population models and their applications in biology and epidemiology are given in [16]. For the first time, Arino et al. [17] discussed the role of leaky vaccines with a general waning function. They consider three classes S, I, and V (without age structure), corresponding to susceptible, infected, and vaccinated individuals. They showed that subthreshold equilibria are possible, which may play an important role while designing vaccine strategies. Optimal control strategies for age-structured compartmental models are discussed by [18,19]. Semi-random epidemic network models can also describe the dynamics of many infectious diseases [20]. Awel et al. [19] incorporated imperfect vaccination in their model for Malaria disease, and Wei et al. [21] studied imperfect vaccination on scale-free networks. Many authors have also considered delay terms in fractional-order population models with or without structure variables. For more details, we refer to [22,23,24,25]. There are many interesting works on age and size-structured models for infectious diseases (see, for example, [26,27,28,29,30,31,32]).
We propose an age-structured SEIRV model with imperfect vaccination and vaccine-waning rates. After formulation of our model, we show the existence and uniqueness of the solution. We use semigroup of operators to show the existence and uniqueness of the solution. We find steady-state solutions to our model and check the stability of the disease-free steady state. We also study the optimal control problem with vaccination rate as a control variable. Our objective is to minimize the cost of vaccination and to reduce the number of infected individuals. Using adjoint equations, we derive the form of the optimal control variable. We show that the optimal control variable depends on the vaccine waning rate, infection rate, and the cost of vaccination. We also solve our model numerically to validate our theoretical findings. Our study is mathematical, and we study the qualitative properties of the solution. We obtain an explicit form for the optimal control variable, which involves important parameters of the proposed model. Due to the complex partial differential equations-based model, it is difficult to get the data, so we do not validate our model with real data. But the model results are interesting and involve realistic assumptions.
Our work is divided in the following manner. In Section 2, we formulated our model, and using semigroup of operators, we show the existence and uniqueness of the solution. Section 3 is devoted to the stability analysis of disease-free equilibrium points. In Section 4, we study the optimal control problem. In Section 5, we solve our model numerically, and the last section is devoted to the concluding remarks.
In this section, we formulate our model. We also show the existence and uniqueness of the solution to the model under consideration. We convert our problem into the abstract Cauchy problem to show the existence of the solution.
Let S(a,t),E(a,t),I(a,t) and R(a,t) denote the age and time-dependent population densities of individuals in the susceptible, exposed, infected, and recovered class. Whenever there is no confusion in the notation, we use the notation S,E,I and R for S(a,t),E(a,t),I(a,t) and R(a,t) respectively. We denote the age-dependent transmission coefficient by c(a,b), which describes the interaction between susceptible and infected class, that is, c(a1,a2)S(a1,t)I(a2,t)da1da2 is the number of susceptible individuals aged in (a1,a1+da1) that contract the disease after interacting with infected individuals aged in (a2,a2+da2). We assume that the functional form of the force of infection is given by
φ(a,t)=∫am0c(a,σ)I(σ,t)dσ, |
where am is the maximum age which an individual can attain. Then the following system of partial differential equations (PDEs) describes the spread of disease
∂S∂t+∂S∂a=Λ−φ(a,t)S−m(a)S−v(a)S+w0(a)V∂E∂t+∂E∂a=φ(a,t)S−m1E−m(a)E∂I∂t+∂I∂a=m1E−γI−m(a)I+βσ(a)IV∂R∂t+∂R∂a=γI−m(a)R∂V∂t+∂V∂a=−(m(a)+w0(a)+βσ(a)I)VS(0,t)=S1,E(0,t)=I(0,t)=R(0,t)=0,V(0,t)=∫am0v(a)S(a,t)daS(a,0)=S0(a),E(a,0)=E0(a),I(a,0)=I0(a),R(a,0)=R0(a)andV(a,0)=V0(a), | (2.1) |
where φ(a,t)=∫am0c(a,σ)I(σ,t)dσ.
m(a) | Mortality rate of age a individuals |
m1 | Rate at which individuals move from exposure to onset of symptoms |
v(a) | Age-dependent Vaccination policy |
w0(a) | Vaccine wanes rate |
γ | Recovery rate of infected population |
σ(a) | Infection ratio of vaccinated individuals |
βσ(a) | Rate at which vaccinated individuals can be infected |
Let X=R4×L1((0,am),R)×L1((0,am),R)×L1((0,am),R)×L1((0,am),R) and let us define the linear operator
A(0000xyzw)=(x(0)y(0)z(0)w(0)−x′−(m(a)+v(a))x−y′−(m1+m(a))y−z′−(γ+m(a))z−w′−(m(a)+w0(a))w) |
with
D(A)={0}×{0}×{0}×{0}×W1,1((0,am),R)×W1,1((0,am),R)×W1,1((0,am),R)×W1,1((0,am),R). |
Here it is clear that ¯D(A) is not dense in X. Let us also define the nonlinear operator in the following manner
F(0000xyzw)=(S100∫am0v(a)x(a)daΛ−x(a)∫am0c(a)z(a)da+w0(a)wx(a)∫am0c(a)z(a)dam1y+βσ(a)z(a)w(a)−βσ(a)z(a)w(a)). |
Our model in abstract form can be written as
{ddtv(t)=Av(t)+F(v(t)),t≥0v(0)=v0, | (2.2) |
where v(t)=(0,0,0,0,S(⋅,t),E(⋅,t),I(⋅,t),V(⋅,t)) and v0=(0,0,0,0,S0,E0,I0,V0). Instead of solving (2.2), we consider the following integral equation
v(t)=v0+A∫t0v(s)ds+∫t0F(v(s))ds. | (2.3) |
We have the following assumptions:
(i) m,w0,v,σ are positive functions and lies in L∞+((0,am),R).
(ii) For 0≤a≤am, we assume that 0<σ(a)<1.
Now, our task is to prove that A is a Hille-Yosida operator and therefore it generates a C0 semigroup on the closure of D(A). Linearized system of (2.2) can be written as
{ddtv(t)=Av(t)+DF(v∗)v(t),t≥0v(0)=v0−v∗, | (2.4) |
where v∗ is steady state solution of (2.2) and
DF(v∗)(0000xyzw)=(S100∫am0v(a)x(a)da−x∗(a)∫am0c(a)z(a,t)da−x(a,t)∫am0c(a)z∗(a)da+w0(a)w(a,t)x∗(a)∫am0c(a)z(a,t)da+x(a,t)∫am0c(a)z∗(a)dam1y+βσ(a)z∗(a)w(a,t)+βσ(a)z(a,t)w∗(a)−βσ(a)z∗(a)w(a,t)−βσ(a)z(a,t)w∗(a)). |
Here, DF(v∗) is a compact bounded linear operator on X. Let
Ω={λ∈C:Reλ>−w0}, |
where
w0=min{infa(m(a)+v(a)),infa(m1+m(a)),infa(γ+m(a)),infa(m(a)+w0(a))}. |
Now, our task is to prove that (A,D(A)) is a Hille-Yosida operator. For
(ϕ1(⋅),ϕ2(⋅),ϕ3(⋅),ϕ4(⋅),ϕ5(⋅),ϕ6(⋅),ϕ7(⋅),ϕ8(⋅))∈Xand(0,0,0,0,ξ1(⋅),ξ2(⋅),ξ3(⋅),ξ4(⋅)) |
we have
(λI−A)−1(ϕ1(⋅)ϕ2(⋅)ϕ3(⋅)ϕ4(⋅)ϕ5(⋅)ϕ6(⋅)ϕ7(⋅)ϕ8(⋅))=(0000ξ1(⋅)ξ2(⋅)ξ3(⋅)ξ4(⋅)). |
This implies
ξ′1+(λ+m(a)+v(a))ξ1(a)=ϕ5(a)ξ′2+(λ+m1+m(a))ξ2(a)=ϕ6(a)ξ′3+(λ+γ+m(a))ξ3(a)=ϕ7(a)ξ′4+(λ+m(a)+w0(a))ξ4(a)=ϕ8(a)ξ1(0)=ϕ1ξ2(0)=ϕ2ξ3(0)=ϕ3ξ4(0)=ϕ4. |
Solving we get,
ξ1(a)=ϕ1e−∫a0(λ+m(s)+v(s))ds+∫a0ϕ5(s)e−∫as(λ+m(τ)+v(τ))dτdsξ2(a)=ϕ2e−∫a0(λ+m1+m(s))ds+∫a0ϕ6(s)e−∫as(λ+m1+m(τ))dτdsξ3(a)=ϕ3e−∫a0(λ+γ+m(s))ds+∫a0ϕ7(s)e−∫as(λ+γ+m(τ))dτdsξ4(a)=ϕ4e−∫a0(λ+m(s)+w0(s))ds+∫a0ϕ8(s)e−∫as(λ+m(τ)+w0(τ))dτds. |
Now, taking L1 norm we get
‖ξ1‖L1+‖ξ2‖L1+‖ξ3‖L1+‖ξ4‖L1≤1λ+w0(‖ξ1‖L1+‖ξ2‖L1+‖ξ3‖L1+‖ξ4‖L1). |
Therefore, we have
‖(λI−A)−1‖≤1λ+w0for allλ∈Ω. |
So, we conclude that (A,D(A)) is a Hille-Yosida operator.
Now, let us recall the following results.
Lemma 2.1. [Pazy [33], 1983] Let X be a Banach space. Let operator A with domain D(A) be a Hille-Yosida operator on X and B be a bounded linear operator on X. Then A+B is also a Hille-Yosida operator.
Lemma 2.2. [Pazy [33], 1983] Let (A,D(A)) be a Hille-Yosida operator, then its part (A0,D(A0)) generates a C0 semigroup {T0(t)|t≥0} on X0, where X0=(¯D(A),‖⋅‖).
Using Lemmas 2.1 and 2.2, and proof given above, we conclude that the operator A+DF(v∗) is a Hille-Yosida operator and the part of (A,D(A) and the part of (A+DF(u∗)) generate C0 semigroups on X0. So, the proof given above and Lemma 2.1, Lemma 2.2 prove the existence and uniqueness of mild solution to our model (for more details we refer to Pazy [33]). So, we have the following theorem:
Theorem 2.3. For any v0∈D+={0}×{0}×{0}×{0}×W1,1((0,am),R+)×W1,1((0,am),R+)×W1,1((0,am),R+)×W1,1((0,am),R+) system 2.4 has a unique continuous solution with initial data in D+. Moreover, the function defined by Ψ(t,v0)=v(t,v0) is strongly continuous semigroup.
Remark 2.4. Since Ψ(t,v0) is strongly continuous semigroup, there exists M≥1,ω≥0 such that ‖Ψ(t)‖≤Meωt. So, if we assume t∈[0,T] for some finite T>0, then we have the boundedness of solution to the abstract Cauchy problem 2.2.
In this section, we discuss the stability analysis of disease-free steady state. Steady-state solutions play an important role to study the qualitative properties of the solution when the explicit form of the solution is not known. Steady-state equations for our model are given by
dSda=Λ−φ(a)IS−m(a)S−v(a)S+w0(a)VdEda=φ(a)IS−m1E−m(a)EdIda=m1E−γI−m(a)I+βσ(a)IVdRda=γI−m(a)RdVda=−(m(a)+w0(a)+βσ(a)I)VS(0)=S1,E(0)=I(0)=R(0)=0V(0)=∫am0v(a)S(a)da. | (3.1) |
Disease free steady state (S0(a),0,0,0,V0(a)) is given by
S0(a)=S1e−∫a0(m(τ)+v(τ))dτ+∫a0(Λ−w0(s)V0(s))e−∫sa(m(τ)+v(τ))dτds | (3.2) |
V0(a)=(∫am0v(τ)S0(τ)dτ)e−∫a0(m(s)+w0(s))ds. | (3.3) |
Now, our task is to derive expressions for S0 which is independent of V0 and similarly for V0. Let
Θ=∫am0v(τ)S0(τ)dτ. | (3.4) |
Now, Eqs (3.2) and (3.3) can be written as
S0(a)=S1e−∫a0(m(τ)+v(τ))dτ+∫a0(Λ−w0(s)Θe−∫s0(m(τ)+w0(τ))dτ)e−∫sa(m(τ)+v(τ))dτds | (3.5) |
V0(a)=Θe−∫a0(m(s)+w0(s))ds. | (3.6) |
Now, substituting value of S0 from (3.5) into (3.4), we get
Θ=∫a0S1v(a)e−∫a0(m(τ)+v(τ))dτda+∫a0Λe−∫sa(m(τ)+v(τ))dτda1+∫am0v(a)(∫a0w0(s)e−∫s0(m(τ)+w0(τ))dτe−∫sa(m(τ)+v(τ))dτds)da | (3.7) |
So, we have steady-state solutions in explicit form. From these solutions, it is clear that both S0 and V0 are non-negative. Let us define the threshold number in the following manner (for more details we refer to [27]):
R0=m1∫am0S0(z)φ(z)e−∫az(m1+m(τ))dτ(∫z0e−∫zs(γ+m(τ)−βσ(τ)V0(τ))dτds)dz. | (3.8) |
The above defined threshold parameter may not be biologically relevant but plays an important role to study the stability of disease-free steady state. We can state the following result which relates the stability of disease-free steady state with the threshold parameter R0.
Theorem 3.1. The disease free steady state is stable if R0<1 and unstable if R0>1.
Proof. For the local stability of the disease-free equilibrium point, firstly we will linearize our system around the disease-free steady state. Linearizing system (2.1) around the disease-free equilibrium, we get
∂S∂t+∂S∂a=−φ(a)S0(a)I−m(a)S−v(a)S+w0(a)V∂E∂t+∂E∂a=φ(a)S0(a)I−m1E−m(a)E∂I∂t+∂I∂a=m1E−γI−m(a)I+βσ(a)V0(a)I∂R∂t+∂R∂a=γI−m(a)R∂V∂t+∂V∂a=−(m(a)+w0(a))V−βσ(a)V0(a)IS(0,t)=S1,E(0,t)=I(0,t)=R(0,t)=0V(0,t)=∫am0v(a)S(a,t)daS(a,0)=S0(a),E(a,0)=E0(a),I(a,0)=I0(a)R(a,0)=R0(a)andV(a,0)=V0(a). | (3.9) |
We are looking for solutions of the form
S(a,t)=ˉS(a)eλt,E(a,t)=ˉE(a)eλt,I(a,t)=ˉI(a)eλt,R(a,t)=ˉR(a)eλt,V(a,t)=ˉV(a)eλt. |
Therefore, system (3.9) becomes
dˉSda+λˉS=−φ(a)S0(a)ˉI−m(a)ˉS−v(a)ˉS+w0(a)ˉVdˉEda+λˉE=φ(a)S0(a)ˉI−m1ˉE−m(a)ˉEdˉIda+λˉI=m1ˉE−γˉI−m(a)ˉI+βσ(a)V0(a)ˉIdˉRda+λˉR=γˉI−m(a)ˉRdˉVda+λˉV=−(m(a)+w0(a))ˉV−βσ(a)V0(a)ˉI | (3.10) |
Solving these equations, we get
ˉS(a)=S1e−∫a0(λ+m(τ)+v(τ))dτ+∫a0(w0(z)ˉV(z)−S0(z)φ(z)ˉI(z))e−∫az(λ+m(τ)+v(τ))dτdzˉE(a)=∫a0S0(z)φ(z)ˉI(z)e−∫az(λ+m1+m(τ))dτdzˉI(a)=∫a0m1ˉE(z)e−∫az(λ+γ+m(τ)−βσ(τ)V0(τ))dτdzˉR(a)=∫a0γˉI(z)e−∫az(λ+m(τ))dτdzˉV(a)=e−∫a0(λ+m(τ)+w0(τ))dτ∫am0v(a)ˉS(a)da−∫a0βσ(z)V0(z)ˉI(z)e−∫az(λ+m(τ)+w0(τ))dτdz. |
Substituting value of ˉE(a) in ˉI(a), we get
ˉI(a)=∫a0m1(∫z0(S0(η)φ(η)ˉI(η))e−∫zη(λ+m1+m(τ))dτdη)e−∫az(λ+γ+m(τ)−βσ(τ)V0(τ))dτdz |
Observe that
dˉIda+λˉI(a)=m1(∫a0S0(η)φ(η)ˉI(η)e−∫aη(λ+m1+m(τ))dτdη)−γˉI(a)−m(a)ˉI(a)+βσ(a)V0(a)ˉI(a). |
In simplified form, it can be written as
dˉIda+(λ+γ+m(a)−βσ(a)V0(a))ˉI(a)=m1(∫a0S0(z)φ(z)ˉI(z)e−∫az(λ+m1+m(τ))dτdz). | (3.11) |
Using (3.11), the characteristic equation can be written as
G(λ)=1, |
where
G(λ)=m1∫a0S0(z)φ(z)e−∫az(λ+m1+m(τ))dτ(∫z0e−∫zs(λ+γ+m(τ)−βσ(τ)V0(τ))dτds)dz | (3.12) |
It is easy to see that G(0)=R0. G can also be written as
G(λ)=m1∫a0e−λ(a−z)S0(z)φ(z)e−∫az(m1+m(τ))dτ(∫z0e−λ(z−s)e−∫zs(γ+m(τ)−βσ(τ)V0(τ))dτds)dz | (3.13) |
Taking derivative of G with respect to λ, we get
G′(λ)=−m1∫a0(a−z)e−λ(a−z)S0(z)φ(z)e−∫az(m1+m(τ))dτ(∫z0e−λ(z−s)e−∫zs(γ+m(τ)−βσ(τ)V0(τ))dτds)dz−m1∫a0e−λ(a−z)S0(z)φ(z)e−∫az(m1+m(τ))dτ(∫z0(z−s)e−λ(z−s)e−∫zs(γ+m(τ)−βσ(τ)V0(τ))dτds)dz. |
Observe that G is a decreasing function of λ as G′(λ)<0 and limλ→∞G(λ)=0. Let us assume that λ=x1+ix2 is a root of equation G(λ)=1. Then for x1≥0, we have
1=|G(λ)|≤|G(0)|≤R0. |
Thus the real part of eigenvalue λ is negative if threshold parameter R0<1. Therefore, the disease free equilibrium point is locally asymptotically stable if R0<1 and unstable if R0>1.
In this section, we discuss the optimal control strategy for our problem. Optimal control theory plays an important role in epidemiological models because it is very important to control the spread of disease in an optimal manner. Here, our control variable is vaccination effort and we want to minimize the cost of vaccination and also our aim is to reduce the number of infected individuals. We assume that φ(a,t)=ψ(a,t)I(a,t). For this form of φ(a,t), it is easy to construct operators A and F to show the existence and uniqueness of the solution. Also, we assume that v depends on both age variable a and time t. We consider the following optimal control problem:
∂S∂t+∂S∂a=Λ−ψ(a,t)IS−m(a)S−v(a,t)S+w0(a)V∂E∂t+∂E∂a=ψ(a,t)IS−m1E−m(a)E∂I∂t+∂I∂a=m1E−γI−m(a)I+βσ(a)IV∂R∂t+∂R∂a=γI−m(a)R∂V∂t+∂V∂a=−(m(a)+w0(a)+βσ(a)I)VS(0,t)=S1,E(0,t)=I(0,t)=R(0,t)=0V(0,t)=∫am0v(a,t)S(a,t)daS(a,0)=S0(a),E(a,0)=E0(a),I(a,0)=I0(a)R(a,0)=R0(a)andV(a,0)=V0(a). | (4.1) |
v(a,t) is the control variable and our aim is to minimize the number of infected individuals and the cost of implementing control.
Theorem 4.1. Let (Sv1,Ev1,Iv1,Rv1,Vv1),(Sv2,Ev2,Iv2,Rv2,Vv2) be solutions of system 4.1 corresponding to the control variables v1 and v2, respectively. Then for sufficiently small T>0, the following estimates hold
∫Q(|Sv1−Sv2|+|Ev1−Ev2|+|Iv1−Iv2|+|Rv1−Rv2|+|Vv1−Vv2|)dadt≤C01∫Q(|v1−v2|)dadt, |
where C01 is a positive constant which depends on the bound of m,m1,γ,β,σ,ψ,ω0, and initial data. Moreover,
‖Sv1−Sv2‖L∞(Q)+‖Ev1−Ev2‖L∞(Q)+‖Iv1−Iv2‖L∞(Q)+‖Rv1−Rv2‖L∞(Q)+‖Vv1−Vv2‖L∞(Q)≤C02‖v1−v2‖L∞(Q), |
where C02 is a positive contant.
Proof. Using the method of characteristics, the system 4.1 can be solved as:
S(a,t)={−∫t0[m(τ+a−t)+v(τ+a−t)]S(τ+a−t,τ)dτ−∫t0ψ(τ+a−t)I(τ+a−t,τ)S(τ+a−t,τ)dτ+∫t0(Λ+ω0(τ+a−t)V(τ+a−t,τ)dτ+S0(a−t)ifa>t,−∫tt−a[m(τ+a−t)+v(τ+a−t)]S(τ+a−t,τ)dτ−∫t0ψ(τ+a−t)I(τ+a−t,τ)S(τ+a−t,τ)dτ+∫tt−a(Λ+ω0(τ+a−t)V(τ+a−t,τ)dτ+S1ifa<t | (4.2) |
E(a,t)={−∫t0[m1+m(τ+a−t)]E(τ+a−t,τ)dτ−∫t0ψ(τ+a−t)I(τ+a−t,τ)S(τ+a−t,τ)dτ+E0(a−t)ifa>t,−∫tt−a[m1+m(τ+a−t)]E(τ+a−t,τ)dτ−∫tt−aψ(τ+a−t)I(τ+a−t,τ)S(τ+a−t,τ)dτifa<t | (4.3) |
I(a,t)={−∫t0[γ+m(τ+a−t)]I(τ+a−t,τ)dτ+∫t0βσ(τ+a−t)I(τ+a−t,τ)V(τ+a−t,τ)dτ+∫t0m1E(τ+a−t,τ)dτ+I0(a−t)ifa>t,−∫tt−a[γ+m(τ+a−t)]I(τ+a−t,τ)dτ+∫tt−aβσ(τ+a−t)I(τ+a−t,τ)V(τ+a−t,τ)dτ+∫tt−am1E(τ+a−t,τ)dτifa<t | (4.4) |
R(a,t)={−∫t0m(τ+a−t)R(τ+a−t,τ)dτ+∫t0γI(τ+a−t,τ)dτ+R0(a−t)ifa>t,−∫tt−am(τ+a−t)R(τ+a−t,τ)dτ+∫tt−aγI(τ+a−t,τ)dτifa<t | (4.5) |
V(a,t)={−∫t0[m(τ+a−t)+ω0(τ+a−t)+βσ(τ+a−t)I(τ+a−t,τ)]V(τ+a−t,τ)dτ+V0(a−t)ifa>t,−∫tt−a[m(τ+a−t)+ω0(τ+a−t)+βσ(τ+a−t)I(τ+a−t,τ)]V(τ+a−t,τ)dτ+∫am0v(a)S(a,t−a)daifa<t. | (4.6) |
Let Q=(0,T)×(0,am), and assume that (Sv1,Ev1,Iv1,Rv1,Vv1),(Sv2,Ev2,Iv2,Rv2,Vv2) be solution of system (4.1) corresponding to the control variables v1 and v2 respectively. Then using the solution given by (4.2), we get
∫Q∩{a<t}|Sv1−Sv2|dadt≤C11∫Q(|v1−v2|+|Sv1−Sv2|)dadt+C22∫Q(|Sv1−Sv2|+|Iv1−Iv2|)dadt+C33∫Q|Vv1−Vv2|dadt, |
where C11,C22,C33 are constants which depends on the bound of m,ψ, and ω0. Similarly, we can consider the case when a>t, and we will obtain similar types of estimates. So, we can write
∫Q|Sv1−Sv2|dadt≤C11∫Q(|v1−v2|+|Sv1−Sv2|)dadt+C22∫Q(|Sv1−Sv2|+|Iv1−Iv2|)dadt+C33∫Q|Vv1−Vv2|dadt, |
Using the same procedure for other state variables, we get
∫Q(|Sv1−Sv2|+|Ev1−Ev2|+|Iv1−Iv2|+|Rv1−Rv2|+|Vv1−Vv2|)dadt≤C01∫Q(|v1−v2|)dadt, |
where C01 is a positive constant. Now, estimating the integral in the age variable only, we will get
‖Sv1−Sv2‖L∞(Q)+‖Ev1−Ev2‖L∞(Q)+‖Iv1−Iv2‖L∞(Q)+‖Rv1−Rv2‖L∞(Q)+‖Vv1−Vv2‖L∞(Q)≤C02‖v1−v2‖L∞(Q), |
where C02 is a positive constant that depends on model parameters and initial population distributions. More details about these types of estimates can be found in [31,32].
Let us define the following objective functional:
J(v)=∫T0∫am0(C1I(a,t)−C2S(a,t)+C3v(a,t)2)dadt. | (4.7) |
Here, C1,C2 and C3 are weight factors and our aim is to minimize J(v). We assume that the admissible control variable lies in the set V defined by
V={u:[0,am]×[0,T]↦[0,vm]}, | (4.8) |
where we assume that v∈L∞((0,am)×(0,T)) and 0≤v≤vm. Now, our next result establish the existence of optimal control.
Theorem 4.2. Let J:L1(0,T)→(−∞,∞] be defined by
J(v)={∫T0∫am0(C1I(a,t)−C2S(a,t)+C3v(a,t)2)dadtifv∈V∞otherwise. | (4.9) |
Then to the control problem (4.1), there exists an optimal control u∗ such that
minv∈VJ(v)=J(v∗). |
Proof. We have already shown the existence of unique solution to the system (4.1) which is uniformly bounded. Also control variable v is uniformly bounded in (0,am). Therefore, there exist a sequence vn∈V such that
limn→∞J(un)=lim infv∈VJ(v). |
Because the function C3v2 is a lower semi-continuous function which is also convex, we have
∫am0C3(v∗)2da≤lim infn→∞∫am0C3(vn)2da. | (4.10) |
Also, using the fact that the bound of S,E,I,R and V depends on the control variable v, we have Sn=S(vn),En=E(vn),In=I(vn),Rn=R(vn),Vn=V(vn). Because Sn,En,In,Rn and Vn are uniformly bounded for each n. We have S∗=S(v∗),E∗=E(v∗),I∗=I(v∗),R∗=R(v∗) and V∗=V(v∗). Now,
J(v∗)=∫T0∫am0[C1I∗−C2S∗+C3(v∗)2]dadt≤lim infn→∞∫T0∫am0[C1In−C2Sn+C3(vn)2]dadt=limn→∞∫T0∫am0[C1In−C2Sn+C3(vn)2]dadt=infv∈VJ(v). |
From the relation J(v∗)≤infv∈VJ(v), we conclude that v∗ is optimal control which minimizes the objective functional.
Now, our next task is to obtain the necessary conditions for our optimal control problem. Let us define vϵ(a,t)=v(a,t)+ϵg(a), where 0<ϵ<1,g∈L∞(0,am) and also called the variation function. Let us denote Sϵ=S(vϵ),Eϵ=E(vϵ),Iϵ=I(vϵ),Rϵ=R(vϵ) and Vϵ=V(vϵ). Then we have the following system
∂Sϵ∂t+∂Sϵ∂a=Λ−ψ(a,t)SϵIϵ−m(a)Sϵ−(v(a,t)+ϵg(a))Sϵ+w0(a)Vϵ∂Eϵ∂t+∂Eϵ∂a=ψ(a,t)SϵIϵ−m1Eϵ−m(a)Eϵ∂Iϵ∂t+∂Iϵ∂a=m1Eϵ−γIϵ−m(a)Iϵ+βσ(a)IϵVϵ∂Rϵ∂t+∂Rϵ∂a=γIϵ−m(a)Rϵ∂Vϵ∂t+∂Vϵ∂a=−(m(a)+w0(a)+βσ(a)Iϵ)VϵSϵ(0,t)=S1,Eϵ(0,t)=Iϵ(0,t)=Rϵ(0,t)=0Vϵ(0,t)=∫am0(v(a,t)+ϵg(a))S(a,t)daSϵ(a,0)=S0(a),Eϵ(a,0)=E0(a),Iϵ(a,0)=I0(a)Rϵ(a,0)=R0(a)andVϵ(a,0)=V0(a). | (4.11) |
Let us set the following notations for weak derivatives of S,E,I,R and V:
ˉS(a,t)=limϵ→0S(v+ϵg)−S(v)ϵ,ˉE(a,t)=limϵ→0E(v+ϵg)−E(v)ϵ,ˉI(a,t)=limϵ→0I(v+ϵg)−I(v)ϵ,ˉR(a,t)=limϵ→0R(v+ϵg)−R(v)ϵ,ˉV(a,t)=limϵ→0V(v+ϵg)−V(v)ϵ. Then we have the following system of differential equations
∂ˉS∂t+∂ˉS∂a=−ψ(a,t)(ˉSI+ˉIS)−m(a)ˉS−v(a,t)ˉS+w0(a)ˉV−g(a)S∂ˉE∂t+∂ˉE∂a=ψ(a,t)(ˉSI+ˉIS)−m1ˉE−m(a)ˉE∂ˉI∂t+∂ˉI∂a=m1ˉE−γˉI−m(a)ˉI+βσ(a)(ˉIV+IˉV)∂ˉR∂t+∂ˉR∂a=γˉI−m(a)ˉR∂ˉV∂t+∂ˉV∂a=−(m(a)+w0(a))ˉV−βσ(a)(ˉIV+I(a,t)ˉV)ˉS(0,t)=0,ˉE(0,t)=ˉI(0,t)=ˉR(0,t)=0ˉV(0,t)=∫am0(v(a,t)ˉS(a,t)+g(a)S(a,t))daˉS(a,0)=ˉE(a,0)=ˉI(a,0)=ˉR(a,0)=0andˉV(a,0)=0. | (4.12) |
The directional derivative of J with respect to v in the direction of g is given by
J′(v)=J(vϵ)−J(v)ϵ=∫T0∫am0[C1ˉI(a,t)−C2ˉS(a,t)+2C3v(a,t)g(a)]dadt. | (4.13) |
Now, our next task is to find the adjoint system. Let us define
⟨f,g⟩=∫T0∫∞0f(a,t)g(a,t)dadt. |
Observe that
0=⟨∂ˉS∂t+∂ˉS∂a+ψ(a,t)(ˉSI+ˉIS)+m(a)ˉS+v(a,t)ˉS−w0(a)ˉV+g(a)S,λ1(a,t)⟩0=⟨ˉS(a,t),−∂∂tλ1−∂∂aλ1+(m(a)+v(a,t))λ1+ψ(a,t)λ1I⟩+∫T0∫am0(ψ(a,t)ˉI(a,t)+g(a))S(a,t)λ1(a,t)dadt−∫T0∫am0w0(a)ˉV(a,t)λ1(a,t)dadt | (4.14) |
with the following conditions
λ1(t,am)=λ1(T,a)=0. |
Similarly we have
0=⟨∂ˉE∂t+∂ˉE∂a−ψ(a,t)(ˉSI−ˉIS)+m1ˉE+m(a)ˉE,λ2(a,t)⟩0=⟨ˉE(a,t),−∂∂tλ2−∂∂aλ2+(m1+m(a))λ2⟩−∫T0∫am0ψ(a,t)λ2(a,t)(S(a,t)ˉI(a,t)+I(a,t)ˉS(a,t))dadt | (4.15) |
0=⟨∂ˉI∂t+∂ˉI∂a−m1ˉE+γˉI+m(a)ˉI−βσ(a)(ˉIV−IˉV),λ3(a,t)⟩0=⟨ˉI(a,t),−∂∂tλ3−∂∂aλ3−m1λ3⟩−∫T0∫am0(γ+m(a)−βσ(a)V(a,t))ˉI(a,t)λ3(a,t)dadt−∫T0∫am0λ3(a,t)I(a,t)ˉI(a,t)dadt | (4.16) |
\begin{eqnarray} 0 & = & \left \langle \frac{\partial \bar{R}}{\partial t} + \frac{\partial \bar{R} }{\partial a}- \gamma \bar{I} + m (a)\bar{R} , \lambda_{4}(a,t) \right \rangle \\ 0 & = & \left \langle \bar{R}(a,t), -\frac{\partial}{\partial t}\lambda_{4} - \frac{\partial}{\partial a}\lambda_{4} -m(a) \lambda_{4} \right \rangle \\ &\; & -\; \int_{0}^{T} \int_{0}^{a_{m}} \gamma \lambda_{4}(a,t) \bar{I}(a,t) da dt \end{eqnarray} | (4.17) |
\begin{eqnarray} 0 & = & \left \langle \frac{\partial \bar{V}}{\partial t} + \frac{\partial \bar{V}}{\partial a} +(m(a)+ w_{0}(a))\bar{V} + \beta \sigma(a)(\bar{I}V + I \bar{V}) , \lambda_{5}(a,t) \right \rangle \\ 0 & = & \left \langle \bar{V}(a,t), -\frac{\partial}{\partial t}\lambda_{5} - \frac{\partial}{\partial a}\lambda_{5} +(m(a)+w_{0}(a)) \lambda_{5} + \beta \sigma(a)I\lambda_{5} \right \rangle \\ &\; & +\; \int_{0}^{T} \int_{0}^{a_{m}} \beta \sigma(a)\bar{I}(a,t) \lambda_{5}(a,t)da dt - \int_{0}^{T} \int_{0}^{a_{m}} (v(a,t)\bar{S}(a,t) + g(a)S(a,t)) \lambda_{5}(0,t) da dt \end{eqnarray} | (4.18) |
Now, we get the following adjoint system
\begin{eqnarray} \frac{\partial \lambda_{1}}{\partial t} + \frac{\partial \lambda_{1}}{\partial a} & = & (m(a)+v(a,t))\lambda_{1} + \psi(a,t)\lambda_{1}I-\psi(a,t)\lambda_{2}I-v(a,t)\lambda_{5}(0,t)-C_{2} \\ \frac{\partial \lambda_{2}}{\partial t} + \frac{\partial \lambda_{2}}{\partial a} & = & (m_{1}+m(a))\lambda_{2} \\ \frac{\partial \lambda_{3}}{\partial t} + \frac{\partial \lambda_{3}}{\partial a} & = & -m_{1}\lambda_{3} - (\gamma + m(a)-\beta \sigma(a)V - I)\lambda_{3} -\gamma \lambda_{4} \\ &\; & \; + \beta \sigma(a)\lambda_{5}- \psi(a,t)\lambda_{2}(a,t)S+C_{1} \\ \frac{\partial \lambda_{4}}{\partial t} + \frac{\partial \lambda_{4}}{\partial a} & = & -m(a)\lambda_{4} \\ \frac{\partial \lambda_{5}}{\partial t} + \frac{\partial \lambda_{5}}{\partial a} & = & (m(a)+w_{0}(a)+\beta \sigma(a)I)\lambda_{5}. \end{eqnarray} | (4.19) |
Also, we have the following conditions
\lambda_{1}(a,T) = \lambda_{2}(a,T) = \lambda_{3}(a,T) = \lambda_{4}(a,T) = \lambda_{5}(a,T) = 0. |
\lambda_{1}(a_{m},t) = \lambda_{2}(a_{m},t) = \lambda_{3}(a_{m},t) = \lambda_{4}(a_{m},t) = \lambda_{5}(a_{m},t) = 0. |
Using the same steps as we followed in 4.1, we can state the following theorem
Theorem 4.3. Let (\lambda_{11}, \lambda_{21}, \lambda_{31}, \lambda_{41}, \lambda_{51}) and (\lambda_{12}, \lambda_{22}, \lambda_{32}, \lambda_{42}, \lambda_{52}) be solution of adjoint system (4.19) with control variable v_{1} and v_{2} respectivey. Then
\begin{equation*} \begin{split} \|\lambda_{11}-\lambda_{12}\|_{L^{\infty}(\mathcal{Q})} + \|\lambda_{21}-\lambda_{22}\|_{L^{\infty}(\mathcal{Q})} + \|\lambda_{31}-\lambda_{32}\|_{L^{\infty}(\mathcal{Q})} + \|\lambda_{41}-\lambda_{42}\|_{L^{\infty}(\mathcal{Q})} &+ \|\lambda_{51}-\lambda_{52}\|_{L^{\infty}(\mathcal{Q})} \\ &\le \mathcal{C}_{11}\|v_{1}-v_{2}\|_{L^{\infty}(\mathcal{Q})}, \end{split} \end{equation*} |
where \mathcal{C}_{11} is a positive constant.
Now, we state and prove the following result, which gives the explicit form of optimal control.
Theorem 4.4. Let (S^{*}, E^{*}, I^{*}, R^{*}, V^{*}) and (\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}, \lambda_{5}) be state solutions and solutions to corresponding adjoint system respectively. Also, let v^{*} \in \mathcal{V} be optimal control which minimizes (4.7). Then optimal control will be of the form
v^{*} = \max \left\{0, \min \left\{ v_{m}, \frac{\lambda_{1}(a,t)(w_{0}(a)V^{*}(a,t)-\psi(a,t)I^{*}(a,t))}{2C_{3}g(a))} -\frac{(\lambda_{1}(a,t)-\lambda_{5}(0,t))S^{*}(a,t)}{2C_{3}} \right\} \right\}. |
Proof. We know that
\begin{equation} J'(v) = \int_{0}^{T} \int_{0}^{a_{m}} \left[ C_{1}\bar{I}(a,t) - C_{2}\bar{S}(a,t) + 2C_{3}v(a,t)g(a) \right]da dt. \end{equation} | (4.20) |
Now using the adjoint system, we have
\begin{eqnarray} J'(v) & = & \int_{0}^{T} \int_{0}^{a_{m}} \bar{I}(a,t) \left[ \frac{\partial \lambda_{3}}{\partial t} + \frac{\partial \lambda_{3}}{\partial a} + m_{1}\lambda_{3}(a,t) + (\gamma + m(a)-\beta \sigma(a)V(a,t) - I(a,t))\lambda_{3}(a,t) \right]dat dt \\ &\; & +\int_{0}^{T} \int_{0}^{a_{m}} \bar{I}(a,t)\left[ \gamma \lambda_{4}(a,t) - \beta \sigma(a)\lambda_{5}(a,t) + \psi(a,t)\lambda_{2}(a,t)S(a,t) \right]da dt \\ &\; & - \int_{0}^{T} \int_{0}^{a_{m}}\bar{S}(a,t) \left[ -\frac{\partial \lambda_{1}}{\partial t} - \frac{\partial \lambda_{1}}{\partial a} \right] da dt +\int_{0}^{T}\int_{0}^{a_{m}} \bar{S}(a,t)\left[ (m(a)+v(a,t))\lambda_{1}(a,t) \right]dadt \\ &\; & +\int_{0}^{T}\int_{0}^{a_{m}} \bar{S}(a,t)\left[ \psi(a,t)\lambda_{1}(a,t)I(a,t)-\psi(a,t)\lambda_{2}(a,t)I(a,t)-v(a,t)\lambda_{5}(0,t)\right]da dt \\ &\; & - \int_{0}^{T} \int_{0}^{a_{m}} \bar{E}(a,t) \left[\frac{\partial \lambda_{2}}{\partial t} + \frac{\partial \lambda_{2}}{\partial a} - (m_{1}+m(a))\lambda_{2}(a,t) \right]da dt \\ &\; & - \int_{0}^{T} \int_{0}^{a_{m}}\bar{R}(a,t) \left[\frac{\partial \lambda_{4}}{\partial t} + \frac{\partial \lambda_{4}}{\partial a} + m(a) \lambda_{4}(a,t) \right]da dt \\ &\; & - \int_{0}^{T} \int_{0}^{a_{m}} \bar{V}(a,t) \left[ -\frac{\partial \lambda_{5}}{\partial t} - \frac{\partial \lambda_{5}}{\partial a} + (m(a)+w_{0}(a)+\beta \sigma(a)I(a,t))\lambda_{5}(a,t) \right]da dt \end{eqnarray} |
Using (4.14, 4.15, 4.16, 4.17) and (4.18), we have
\begin{eqnarray} J'(v) & = & \int_{0}^{T} \int_{0}^{a_{m}}2C_{3}v(a,t)g(a)da dt \\ &\; & +\int_{0}^{T}\int_{0}^{a_{m}}(\lambda_{1}(a,t)-\lambda_{5}(0,t))S^{*}(a,t) da dt - \int_{0}^{T}\int_{0}^{a_{m}}\lambda_{1}(a,t)(w_{0}(a)V^{*}(a,t)-\psi(a,t)I^{*}(a,t)) da dt. \end{eqnarray} | (4.21) |
Because J'(v) \ge 0 , so this implies in this case
\begin{equation*} v(a,t) = \frac{\lambda_{1}(a,t)(w_{0}(a)V^{*}(a,t)-\psi(a,t)I^{*}(a,t))}{2C_{3}g(a))} -\frac{(\lambda_{1}(a,t)-\lambda_{5}(0,t))S^{*}(a,t)}{2C_{3}} \end{equation*} |
By using the upper and lower bounds of control variable, we have
v^{*} = \max \left\{0, \min \left\{ v_{m}, \frac{\lambda_{1}(a,t)(w_{0}(a)V^{*}(a,t)-\psi(a,t)I^{*}(a,t))}{2C_{3}g(a))} -\frac{(\lambda_{1}(a,t)-\lambda_{5}(0,t))S^{*}(a,t)}{2C_{3}} \right\} \right\}. |
Remark 4.5. It is clear from the characterization of optimal control that it depends on the susceptible, infected, and vaccinated class. Although it depends on function g , we can always fix a particular choice of g .
Theorem 4.6. Let M be a generic positive constant that depends on \mathcal{C}_{02}, \mathcal{C}_{11} and the bound of vaccine vaning rate \omega_{0} , and transmission coefficient \psi . Then for sufficiently small \frac{M}{2C_{3}} , there exists a unique optimal controller v^{*} .
Proof. Let (S^{v_{1}}, E^{v_{1}}, I^{v_{1}}, R^{v_{1}}, V^{v_{1}}), (S^{v_{2}}, E^{v_{2}}, I^{v_{2}}, R^{v_{2}}, V^{v_{2}}) be solution of system (4.1) corresponding to the control variables v_{1} and v_{2} respectively, and (\lambda_{11}, \lambda_{21}, \lambda_{31}, \lambda_{41}, \lambda_{51}), (\lambda_{12}, \lambda_{22}, \lambda_{32}, \lambda_{42}, \lambda_{52}) be solution of adjoint system 4.19 with control variable v_{1} and v_{2} respectivey. Define the map L: \mathcal{U} \to \mathcal{U} by
L(v) = \max \left\{0, \min \left\{ v_{m}, \frac{\lambda_{1}(a,t)(w_{0}(a)V(a,t)-\psi(a,t)I(a,t))}{2C_{3}g(a))} -\frac{(\lambda_{1}(a,t)-\lambda_{5}(0,t))S(a,t)}{2C_{3}} \right\} \right\}. |
Then
\begin{equation*} \begin{split} \| L(v_{1})-L(v_{2}) \| &\le \left\| \frac{\lambda_{11}(a,t)(w_{0}(a)V^{v_{1}}(a,t)-\psi(a,t)I^{v_{1}}(a,t))}{2C_{3}g(a))} -\frac{(\lambda_{11}(a,t)-\lambda_{51}(0,t))S^{v_{1}}(a,t)}{2C_{3}} \right. \\ &- \left. \frac{\lambda_{12}(a,t)(w_{0}(a)V^{v_{2}}(a,t)-\psi(a,t)I^{v_{2}}(a,t))}{2C_{3}g(a))} +\frac{(\lambda_{12}(a,t)-\lambda_{52}(0,t))S^{v_{2}}(a,t)}{2C_{3}} \right\|_{L^{\infty}(\mathcal{Q})} \\ &\le \frac{1}{2C_{3}} \left[ \frac{\omega_{0}(a)}{g(a)} \| \lambda_{11}V^{v_{1}}-\lambda_{12}V^{v_{2}}\|_{L^{\infty}(\mathcal{Q})} + \frac{\psi(a,t)}{g(a)} \| \lambda_{11}I^{v_{1}}-\lambda_{12}I^{v_{2}}\|_{L^{\infty}(\mathcal{Q})} \right. \\ &+ \left. \|\lambda_{11}S^{v_{1}}-\lambda_{12}S^{v_{2}} \|_{L^{\infty}(\mathcal{Q})} + \| \lambda_{51}(0,t)S^{v_{1}}-\lambda_{52}(0,t)S^{v_{2}}\|_{L^{\infty}(\mathcal{Q})}\right] \\ &\le \frac{M}{2C_{3}} \left[ \| \lambda_{11}V^{v_{1}}-\lambda_{12}V^{v_{2}}\|_{L^{\infty}(\mathcal{Q})} + \| \lambda_{11}I^{v_{1}}-\lambda_{12}I^{v_{2}}\|_{L^{\infty}(\mathcal{Q})} \right. \\ &+ \|\lambda_{11}S^{v_{1}}-\lambda_{12}S^{v_{2}} \|_{L^{\infty}(\mathcal{Q})} + \| \lambda_{51}(0,t)S^{v_{1}}-\lambda_{52}(0,t)S^{v_{2}}\|_{L^{\infty}(\mathcal{Q})} \left. \right], \end{split} \end{equation*} |
where M is a positive generic constant which depends on the bound of \omega_{0}, g and \psi . Therefore, we have
\begin{equation*} \begin{split} \| L(v_{1})-L(v_{2}) \| &\le \frac{M}{2C_{3}} \|v_{1}-v_{2}\|_{L^{\infty}(\mathcal{Q})}, \end{split} \end{equation*} |
where M is a positive generic constant which also depends on \mathcal{C}_{01}, \mathcal{C}_{02} and T . So, if \frac{M}{2C_{3}} < 1 , then we have the existence of unique fixed point v^{*} .
In this section, we discuss numerical simulation results. We applied an explicit finite difference scheme to solve our model numerically. Since the explicit finite difference scheme is highly sensitive to the discretization size of independent variables, the discretization size is chosen in such a way that the scheme is convergent. A finite difference scheme is applied to solve the system of differential equations and for the adjoint system, we applied a backward finite difference scheme. {For the optimal control problem, we applied the forward-backward sweep method. The forward-backward sweep method for optimal control problems in PDEs combines the forward integration of the state equations with the backward integration of the adjoint equations. By iteratively adjusting the control variables based on the computed gradient, the method aims to find the optimal control that minimizes the cost functional while satisfying the given system of PDEs with initial and boundary data.} We use the following initial age distribution in our model:
\begin{eqnarray*} S_{0}(a) & = & \exp(-0.09a) + 0.7\sin(0.05a^{2}) \\ E_{0}(a) & = & 0.0001\exp(-a^2) + 0.05\sin(0.05a^{2}) \\ I_{0}(a) & = & 0.0001\exp(-a^2) + 0.02\sin(0.05a^{2}) \\ R_{0}(a) & = & 0.0001\exp(-0.5a) + 0.01\sin(0.05a^{2}) \\ V_{0}(a) & = & 0.0001\exp(-0.5a) + 0.001\sin(0.05a^{2}). \end{eqnarray*} |
Mortality rate is taken \mu(a) = \frac{0.01}{a+0.0001}, \psi(a, t) = 0.02e^{-a}. Other parameters taken are m_{1} = 0.2, q_{1} = 0.4, \gamma_{1} = 0.4, \gamma_{2} = 0.6, \gamma = 0.02, \beta = 1, \Lambda = 0.9 and g = 1 .
For the first two plots Figures 2(a), (b), we assume that a fixed proportion of individuals are vaccinated and observe the dynamics of infected individuals. So, for these figures we do not use the optimal vaccination strategy.
From Figure 2(a), (b), it is clear that increasing the vaccination rate will reduce the proportion of infected individuals. It is clear that increasing the vaccination waning rate will increase the population density of individuals who again become susceptible to disease. In Figure 3(a), (b), we have taken vaccine waning rates as 0 and 0.3 respectively. So, even if we are vaccinating a population, still there will be an increase in the population density of susceptible individuals due to imperfect vaccines.
Figure 4(a), (b) and Figure 5 show the dynamics of control variable for vaccine waning rate 0.1 and 0.01. As time progresses, the profile of the control variable attains a stable behavior. Initially, the control will be high and as time progresses, the control will take its minimum value.
In Section 3, we already prove that if threshold parameter R_{0} is less than 1, then the steady state in which there are no individuals in the exposed or infected class is stable. In Figures 6 and 7, we have taken age between 10 and 50 and studied the stability of disease-free equilibrium point. We have chosen the parameters in such a way that the threshold number R_{0} < 1 . These figures show that if threshold parameter R_{0} < 1 , then the disease-free equilibrium point is stable. So, numerical simulations validate our theoretical results.
Age-structured modeling is important when studying imperfect vaccination because it allows a more realistic representation of how infectious diseases spread in a population. Imperfect vaccination refers to situations where not everyone in a population is fully protected by a vaccine due to vaccine effectiveness or a decline in immunity over time.
In this work, we proposed a new age-structured SEIRV model, where we incorporate vaccine waning rate and imperfect vaccination in our model. The aim was to study the effect of leaky vaccines and to investigate the optimal approach to contain the disease spread. The Well-posedness of the proposed model is shown using the semigroup of operators. We convert our model into an abstract Cauchy problem and then show that the operator in a homogeneous problem is the Hille-Yosida operator. Finally, the basic reproduction number is defined as a threshold parameter to study the stability analysis of the disease-free equilibrium point.
Vaccination strategy and optimal control problems with vaccination effort as a control variable are also discussed. The optimality conditions are derived for the optimal control problem. With the help of the adjoint system, optimal control is characterized. Numerical simulations are also added to establish theoretical results. An explicit finite difference scheme is used to solve the model numerically. The discretization size is chosen so that the scheme is convergent. The stability of disease-free steady state is shown using numerical simulation. Using the forward-backward sweep method, we plot the optimal control variable with respect to age and time variables.
Age-structured compartmental models are a valuable tool for understanding and analyzing the effects of imperfect vaccination. Although getting real data for verification is difficult, the model assumptions and outcomes are quite realistic. This study can help the public health policymakers optimally vaccinate a population and reduce the impact of the disease with a low cost of vaccination. The threshold number R_0 involves some parameters, and by increasing or decreasing these parameters, accordingly, we can eradicate the infectious disease. By choosing different vaccine waning rates, the dynamics of individuals in different compartments can be studied using the numerical simulation results. We also encourage other researchers to explore this area. {Moreover, the age-structured SEIR model with stochastic perturbation can be investigated. More general size-structured variables can also be considered for epidemiological models. Finally, It is also natural to consider the delay in age and size-structured population models as it represents the maturation period. Therefore, age-structured compartmental models with delay can also be explored.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve this manuscript's quality.
A. Tridane is supported by UAEU UPAR grant number 12S125.
The authors declare there is no conflict of interest.
[1] |
S. Babapoor-Farrokhran, K. Jee, B. Puchner, S. J. Hassan, X. Xin, M. Rodrigues, et al., Angiopoietin-like 4 is a potent angiogenic factor and a novel therapeutic target for patients with proliferative diabetic retinopathy, Proc. Natl. Acad. Sci., 112 (2015), 3030–3039. https://doi.org/10.1073/pnas.1423765112 doi: 10.1073/pnas.1423765112
![]() |
[2] |
Y. C. Chang, W. C. Wu, Dyslipidemia and diabetic retinopathy, Rev. Diabetic Stud., 10 (2012), 121–132. https://doi.org/10.1900/RDS.2013.10.121 doi: 10.1900/RDS.2013.10.121
![]() |
[3] |
L. Chen, C. Y. Cheng, H. Choi, M. K. Ikram, C. Sabanayagam, G. S. Tan, et al., Plasma metabonomic profiling of diabetic retinopathy, J. Am. Diabetes Assoc., 65 (2016), 1099–1108. https://doi.org/10.2337/db15-0661 doi: 10.2337/db15-0661
![]() |
[4] |
F. Semeraro, F. Morescalchi, A. Cancarini, A. Russo, S. Rezzola, C. Costagliola, Diabetic retinopathy, a vascular and inflammatory disease: Therapeutic implications, Diabetes Metab., 45 (2019), 517–527. https://doi.org/10.1016/j.diabet.2019.04.002 doi: 10.1016/j.diabet.2019.04.002
![]() |
[5] | I. N. Mohamed, S. A. Soliman, A. Alhusban, S. Matragoon, B. A. Pillai, A. A. Elmarkaby, A. B. El-Remessy, Diabetes exacerbates retinal oxidative stress, inflammation, and microvascular degeneration in spontaneously hypertensive rats, Mol. Vision, 18 (2012), 1457–1466. |
[6] | Z. Zhuang, H. Hu, S. Y. Tian, Z. J. Lu, T. Z. Zhang, Y. L. Bai, Down-regulation of microRNA-155 attenuates retinal neovascularization via the PI3K/Akt pathway, Mol. Vision, 21 (2015), 1173–1184. |
[7] |
B. Bahrami, M. Zhu, T. Hong, A. Chang, Diabetic macular oedema: pathophysiology, management challenges and treatment resistance, Diabetologia, 59 (2016), 1594–1608. https://doi.org/10.1007/s00125-016-3974-8 doi: 10.1007/s00125-016-3974-8
![]() |
[8] |
M. Henricsson, A. Nilsson, L. Janzon, L. Groop, The effect of glycaemic control and the introduction of insulin therapy on retinopathy in non-insulin-dependent diabetes mellitus, Diabetic Med., 14 (1997), 123–131. https://doi.org/10.1002/(SICI)1096-9136(199702)14:2<123::AID-DIA306>3.0.CO;2-U doi: 10.1002/(SICI)1096-9136(199702)14:2<123::AID-DIA306>3.0.CO;2-U
![]() |
[9] |
J. Yin, W. Q. Xu, M. X. Ye, Y. Zhang, H. Y. Wang, J. Zhang, et al., Up-regulated basigin-2 in microglia induced by hypoxia promotes retinal angiogenesis, J. Cell. Mol. Med., 7 (2017), 14925. https://doi.org/10.1111/jcmm.13256 doi: 10.1111/jcmm.13256
![]() |
[10] |
E. Kermorvant-Duchemin, A. C. Pinel, S. Lavalette, D. Lenne, W. Raoul, B. Calippe, et al., Neonatal hyperglycemia inhibits angiogenesis and induces inflammation and neuronal degeneration in the retina, PLoS ONE, 8 (2013), e79545. https://doi.org/10.1371/journal.pone.0079545 doi: 10.1371/journal.pone.0079545
![]() |
[11] |
X. Chen, H. Zhou, Y. Gong, S. Wei, M. Zhang, Early spatiotemporal characterization of microglial activation in the retinas of rats with streptozotocin-induced diabetes, Graefe's Arch. Clin. Exp. Ophthalmol., 253 (2015), 519–525. https://doi.org/10.1007/s00417-014-2727-y doi: 10.1007/s00417-014-2727-y
![]() |
[12] |
H. Qiu, A. L. Jackson, J. E. Kilgore, Y. Zhong, L. L. Y. Chan, P. A. Gehrig, et al., JQ1 suppresses tumor growth through downregulating LDHA in ovarian cancer, Oncotarget, 6 (2015), 6915. https://doi.org/10.18632/oncotarget.3126 doi: 10.18632/oncotarget.3126
![]() |
[13] |
H. Qiu, J. Li, L. H. Clark, A. L. Jackson, L. Zhang, H. Guo, et al., JQ1 suppresses tumor growth via PTEN/PI3K/AKT pathway in endometrial cancer, Oncotarget, 7 (2016), 66809–66821. https://doi.org/10.18632/oncotarget.11631 doi: 10.18632/oncotarget.11631
![]() |
[14] |
S. Bakshi, C. McKee, K. Walker, C. Brown, G. R. Chaudhry, Toxicity of JQ1 in neuronal derivatives of human umbilical cord mesenchymal stem cells, Oncotarget, 9 (2018), 33853–33864. https://doi.org/10.18632/oncotarget.26127 doi: 10.18632/oncotarget.26127
![]() |
[15] |
J. Mu, D. Zhang, Y. Tian, Z. Xie, M. H. Zou, et al., BRD4 inhibition by JQ1 prevents high-fat diet-induced diabetic cardiomyopathy by activating PINK1/Parkin-mediated mitophagy in vivo, J. Mol. Cell. Cardiol., 149 (2020). https://doi.org/10.1016/j.yjmcc.2020.09.003 doi: 10.1016/j.yjmcc.2020.09.003
![]() |
[16] |
Y. Wu, M. Zhang, C. Xu, D. Chai, F. Peng, J. Lin, Anti-diabetic atherosclerosis by inhibiting high glucose-induced vascular smooth muscle cell proliferation via Pin1/BRD4 pathway, Oxid. Med. Cell. Longevity, 2020 (2020), 1–13. https://doi.org/10.1155/2020/4196482 doi: 10.1155/2020/4196482
![]() |
[17] |
E. Liang, M. Ma, L. Wang, X. Liu, J. Xu, M. Zhang, The BET/BRD inhibitor JQ1 attenuates diabetes-induced cognitive impairment in rats by targeting Nox4-Nrf2 redox imbalance, Biochem. Biophys. Res. Commun., 2017, S0006291X17321976. https://doi.org/10.1016/j.bbrc.2017.11.020 doi: 10.1016/j.bbrc.2017.11.020
![]() |
[18] |
J. M. Zhou, S. S. Gu, W. H. Mei, J. Zhou, Z. Z. Wang, W. Xiao, Ginkgolides and bilobalide protect BV2 microglia cells against OGD/reoxygenation injury by inhibiting TLR2/4 signaling pathways, Cell Stress Chaperones, 21 (2016), 1037–1053. https://doi.org/10.1007/s12192-016-0728-y doi: 10.1007/s12192-016-0728-y
![]() |
[19] |
H. Shen, J. Zhang, Y. Zhang, Q. Feng, H. Wang, G. Li, et al., Knockdown of tripartite motif 59 (TRIM59) inhibits proliferation in cholangiocarcinoma via the PI3K/AKT/mTOR signalling pathway, Gene, 698 (2019), 50–60. https://doi.org/10.1016/j.gene.2019.02.044 doi: 10.1016/j.gene.2019.02.044
![]() |
[20] |
Z. Y. Sun, Y. K. Jian, H. Y. Zhu, B. Li, lncRNAPVT1 targets miR-152 to enhance chemoresistance of osteosarcoma to gemcitabine through activating c-MET/PI3K/AKT pathway, Pathol. Res. Pract., 215 (2019), 555–563. https://doi.org/10.1016/j.prp.2018.12.013 doi: 10.1016/j.prp.2018.12.013
![]() |
[21] |
D. S. Boyer, J. J. Hopkins, J. Sorof, J. S. Ehrlich, Anti-vascular endothelial growth factor therapy for diabetic macular edema, Ther. Adv. Endocrinol. Metab., 4 (2013), 151–169. https://doi.org/10.1177/2042018813512360 doi: 10.1177/2042018813512360
![]() |
[22] |
M. Ahmed, A. Asrar, Role of inflammation in the pathogenesis of diabetic retinopathy, Middle East Afr. J. Ophthalmol., 19 (2012), 70. https://doi.org/10.4103/0974-9233.92118 doi: 10.4103/0974-9233.92118
![]() |
[23] |
T. Bagratuni, N. Mavrianou, N. G. Gavalas, K. Tzannis, C. Arapinis, M. Liontos, et al., JQ1 inhibits tumour growth in combination with cisplatin and suppresses JAK/STAT signalling pathway in ovarian cancer, Eur. J. Cancer, 126 (2020), 125–135. https://doi.org/10.1016/j.ejca.2019.11.017 doi: 10.1016/j.ejca.2019.11.017
![]() |
[24] |
X. Xu, Y. Zong, Y. Gao, X. Sun, H. Zhao, W. Luo, et al., VEGF induce vasculogenic mimicry of choroidal melanoma through the PI3k signal pathway, BioMed Res. Int., 2019 (2019), 1–13. https://doi.org/10.1155/2019/3909102 doi: 10.1155/2019/3909102
![]() |
[25] |
N. Akeno, J. Robins, M. Zhang, M. F. Czyzyk-Krzeska, T. L. Clemens, Induction of vascular endothelial growth factor by IGF-I in osteoblast-like cells is mediated by the PI3K signaling pathway through the hypoxia-inducible factor-2alpha, Endocrinology, 143 (2002), 420–425. https://doi.org/10.1210/endo.143.2.8639 doi: 10.1210/endo.143.2.8639
![]() |
[26] |
M. Jafari, E. Ghadami, T. Dadkhah, H. Akhavan-Niaki, PI3k/AKT signaling pathway: Erythropoiesis and beyond, J. Cell. Physiol., 234 (2019), 2373–2385. https://doi.org/10.1002/jcp.27262 doi: 10.1002/jcp.27262
![]() |
1. | Afeez Abidemi, Kolade M. Owolabi, Edson Pindza, Assessing the dynamics of Lassa fever with impact of environmental sanitation: optimal control and cost-effectiveness analysis, 2022, 2363-6203, 10.1007/s40808-022-01624-y | |
2. | Mohammed M. Al-Shomrani, Salihu S. Musa, Abdullahi Yusuf, Unfolding the Transmission Dynamics of Monkeypox Virus: An Epidemiological Modelling Analysis, 2023, 11, 2227-7390, 1121, 10.3390/math11051121 | |
3. | Rubayyi T. Alqahtani, Salihu S. Musa, Mustafa Inc, Modeling the role of public health intervention measures in halting the transmission of monkeypox virus, 2023, 8, 2473-6988, 14142, 10.3934/math.2023723 | |
4. | Y.O. Afolabi, B.A. Wade, Dynamics of transmission of a Monkeypox epidemic in the presence of an Imperfect Vaccination, 2023, 19, 25900374, 100391, 10.1016/j.rinam.2023.100391 | |
5. | Ibrahim Olalekan Abiola, Abimbola Samuel Oyewole, Tunde Tajudeen Yusuf, Mathematical modelling of Lassa-fever transmission dynamics with optimal control of selected control measures, 2024, 2363-6203, 10.1007/s40808-024-02168-z | |
6. | Alhassan Ibrahim, Usa Wannasingha Humphries, Ibrahim Mohammed, Rahat Zarin, Understanding the spread of typhoid fever: Combining vaccination and sanitation methods for better public health policies, 2024, 14, 2158-3226, 10.1063/5.0201916 | |
7. | Chinwendu Emilian Madubueze, Saheed Ajao, John Olajide Akanni, Fatmawati Fatmawati, Zviiteyi Chazuka, Impact of environmental contamination on Lassa fever transmission dynamics: a mathematical modelling approach, 2025, 140, 2190-5444, 10.1140/epjp/s13360-025-06123-4 |