programming, Epidemiological modeling.

Vector-borne viral diseases, such as Tomato Yellow Leaf Curl Virus (TYLCV) transmitted by the whitefly Bemisia tabaci, represent a major constraint to global food security, with estimated annual economic losses exceeding USD 1.5 billion in tomato production [13]. The complexity of these pathosystems arises from the nonlinear interaction between host plants, vector population dynamics, and viral transmission processes. This complexity necessitates the development of robust mathematical models and efficient control strategies to support decision-making in integrated pest management.

Mathematical modeling has become a fundamental tool for understanding and controlling infectious diseases in both human and plant epidemiology. Classical compartmental frameworks, as introduced by Hethcote [16], have provided the basis for a wide range of epidemiological extensions. In particular, optimal control theory has been extensively applied to biological systems following the foundational work of Lenhart and Workman [19], which established a systematic framework for incorporating time-dependent intervention strategies using Pontryagin’s Maximum Principle.

In the context of vector-borne and plant disease systems, several contributions have highlighted the effectiveness of optimal intervention strategies. Chitnis et al. [12] demonstrated the importance of combined control measures targeting both host and vector populations in malaria dynamics. Similarly, Cai et al. [11] investigated optimal pesticide application in plant disease models, showing that carefully designed control policies can significantly reduce infection levels while minimizing costs. Njagarah and Nyabadza [22] further emphasized the role of integrated strategies in vector-host systems, incorporating both treatment and vector reduction mechanisms. More recently, Sharomi and Malik [25] provided a comprehensive overview of optimal control applications in epidemiology, highlighting the superiority of multi-control approaches over single-intervention strategies.

Despite these advances, most classical optimal control models assume deterministic environments with fixed parameters and pre-defined control structures. However, agricultural ecosystems are inherently dynamic, influenced by seasonal variability, climate fluctuations, and the rapid evolution of vector resistance to chemical and biological control measures [18]. These limitations motivate the development of adaptive and robust control methodologies capable of responding to time-varying epidemiological conditions.

In this regard, Adaptive Dynamic Programming (ADP), a reinforcement learning–based framework, has emerged as a promising alternative to classical optimal control approaches. By enabling real-time policy adaptation, ADP addresses the limitations of static optimization strategies. Bertsekas [9] demonstrated that ADP-based methods can significantly improve performance in dynamic decision-making problems, while Benosmans [7] showed their effectiveness in vector-borne disease systems with spatial and temporal heterogeneity.

Motivated by these developments, this study aims to integrate classical optimal control theory with adaptive learning techniques in the context of TYLCV transmission dynamics in tomato crops. The main contributions of this work are summarized as follows:

I. We develop a coupled plant–whitefly epidemiological model extending previous work by M. Ido et al.

(2025) [17], incorporating four key control strategies: biological control, removal of infected plants, deployment of resistant tomato varieties, and insect-proof nets.

II. We apply Pontryagin’s Maximum Principle to derive the optimal control system and determine the optimal combinations of the four control strategies that maximize epidemiological effectiveness while minimizing economic costs.

III. We provide a comprehensive cost–benefit analysis to evaluate the trade-off between disease reduction and implementation costs, identifying the most efficient combinations of biological control, infected plant removal, resistant varieties, and insect-proof nets.

The remainder of the paper is organized as follows. Section 2 presents the mathematical model and recalls key analytical results. Section 3 formulates and analyzes the optimal control problem using Pontryagin’s Maximum Principle. Section 4 presents numerical simulations to evaluate and illustrate the effectiveness of the optimal control strategies and their combinations. Finally, Section 5 concludes the paper with key findings and future research directions.

The global dynamics of begomovirus in tomato plant populations coupled whiteflies follow the compartmental diagram reffigure1 below. For more details on the model formulate, the reader may refer to [17].

µp µp µp µp

Λ α β Sv Sg Lg Ig

β2(1 − δp) γ2(1 − δp)

SBT

β1(1 − δp) r 1 − IBT SBT µp L

µBT µBT

θBT δp θBT δp θ γ1(1 − δp)

Iv

µp

Figuer 1. Compartmental diagram for transmission dynamics TYLCV.

Remark 0.1.

I. The terms β1(1 − δp) and β2(1 − δp) represent the infection rates of tomato vegetative and generative plants, respectively, reflecting the reduction in transmission due to the effect of Verticillium lecanii.

II. The terms γ1(1 − δp) and γ2(1 − δp) describe the infection pressure on whiteflies originating from vegetative and generative tomato plants, respectively, incorporating the mitigating effect of Verticillium lecanii.

III. The term θBT δp represents the additional mortality rate of whiteflies induced by biological control application.

IV. The term θBT δpNp corresponds to the total proportion of whiteflies eliminated through spraying on tomato plants, where θBT denotes the biocontrol-induced mortality rate, δp is the efficiency of biological control, and Np is the initial total plant population.

V. Biological control simultaneously affects both transmission dynamics and vector mortality.

VI. The model is a coupled horizontal SI framework without recovery for both plant and vector populations.

By taking stock of the input-output masses in the different classes suitably to the mathematical formulation, the model of the dynamics of the propagation of the virus host-insect interaction vectors is described by the following system (0.1).

S ˙ = Λ − αS − β (1 − δ )S I − µ S ,

v v 1 p v BT p v

Lv = β1(1 − δp)SvIBT − θLv − µpLv, I˙ = θL − µ I ,  v v p v  S ˙ = αS − β (1 − δ )S I − µ S ,

g v 2 p g BT p g

L ˙ = β (1 − δ )S I − βL − µ L , (0.1)  g 2 p g BT g p g I˙ = βL − µ I ,

g g p g

 ˙ BT SBT = r 1 − SBT − γ1(1 − δp)IvSBT − γ2(1 − δp)IgSBT − (θBT δpNp + µBT ) SBT ,

I˙ = γ (1 − δ )I S + γ (1 − δ )I S − (θ δ N + µ ) I ,

BT 1 p v BT 2 p g BT BT p p BT BT

with the initial conditions

Sv(0) > 0, Lv(0) ≥ 0, Iv(0) ≥ 0, Sg(0) ≥ 0, Lg(0) ≥ 0, Ig(0) ≥ 0, SBT (0) ≥ 0, IBT (0) ≥ 0 (0.2)

The flow chart of the spread of yellow virus in tomato plants combined with the using of the insecticide Verticillium lecanii is visible in the diagram 1, with the definition of the variables and parameters used is also given by the Table 1 and Table 2.

Sv(t) Susceptible vegetative plants at time t Individual plant Lv(t) Latent vegetative plants at time t Individual plant Iv(t) Infected vegetative plants at time t Individual plant Sg(t) Susceptible generative plants at time t Individual plant Lg(t) Latent generative plants at time t Individual plant Ig(t) Infected generative plants at time t Individual plant SBT (t) Susceptible whitefly at time t Individual vector IBT (t) Infected whitefly at time t Individual vector Np Total plants at initial time Individual plant r The whitefly birth rate day Λ The number of tillers or replanting of plants day α The growth susceptible vegetative plant rate day to become generative plants θ The latent vegetative plant growth rate day to become infected vegetative plants β The latent generative plant growth rate day to become infected generative plants β1 The vegetative plant infection rate (individual × day) β2 The generative plant infection rate (individual × day) γ1 The whitefly infection rate by (individual × day) infected vegetative phase γ2 The whitefly infection rate by (individual × day) infected generative phase δp The efficacy of use of Verticillium lecanii µp The mortality rate of the plant day µBT The natural mortality rate of whitefly day θBT The fungal mortality rate of whitefly day Verticillium lecanii

The control of Tomato Yellow Leaf Curl Virus (TYLCV), a severe viral disease transmitted by the whitefly

Bemisia tabaci, represents a major challenge for tomato producers. Yield losses can be substantial, affecting

both farmers’ income and global food security. To optimize TYLCV management strategies, we propose an integrated control framework incorporating four intervention mechanisms as part of an Integrated Pest Management (IPM) strategy. These include: biological control using Verticillium lecanii, which acts as an

entomopathogenic agent against whiteflies; removal of infected plants during both vegetative and generative stages to reduce disease spread; cultivation of resistant tomato varieties to decrease host susceptibility; and installation of insect-proof nets acting as a physical barrier against whiteflies. All four control measures are modeled as time-dependent control variables.

io

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦI − ΦS ) + θBT Np (SBT + IBT ) (ΦS + ΦI ) , 1

BT BT BT BT

∗ 1 ∗ ∗

2 v v g g

1 h i

∗ ∗ ∗ ∗ ∗ ∗

U = max 0, min (I (β1S + L ) (ΦL − ΦS ) + β2S + L (ΦL − ΦS ) +

3 BT v v v v g g g g

io

∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT )(ΦI − ΦS ), 1

BT BT

1 h i

∗ ∗ ∗ ∗

U = max 0, min (I β1S (ΦL − ΦS ) + β2S (ΦL − ΦS ) +

4 BT v v v g g g

io

∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT )(ΦI − ΦS ), 1

BT BT

∗ ∗ ∗ 1 ∗ ∗ ∗ U + U + U = max 0, min (I [β1S (ΦL − ΦS ) + β2S (ΦL − ΦS )]+

1 3 4 BT v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦI − ΦS ) + θBT Np (SBT + IBT ) (ΦS + ΦI ) +

BT BT BT BT

1 h i

∗ ∗ ∗ ∗ ∗

(I (β1S + L ) (ΦL − ΦS ) + β2S + L (ΦL − ΦS ) +

BT v v v v g g g g

∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT )(ΦI − ΦS )+

BT BT

1 h i

∗ ∗ ∗

(I β1S (ΦL − ΦS ) + β2S (ΦL − ΦS ) +

BT v v v g g g

io

∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT )(ΦI − ΦS ), 1

BT BT

Φ1, Φ2, Φ3, Φ5 > 0 and where are the adjoint functions of system (0.4)

Φ = (ΦS , ΦL , ΦI , ΦS , ΦL , ΦI , ΦS , ΦS ) that satisfy the following backward in time

v v v g g g BT BT system of ordinary differential equation.

˙ ∗ ∗ ∗ ΦS (t) = α ΦS − ΦS + β1(1 − U − U − U ) (ΦS − ΦL ) + µpΦS ,

v v g 1 3 4 v v v

ΦL (t) = U (ΦL − ΦS ) + θ (ΦL − ΦI ) + µpΦL ,

v 3 v v v v v

˙ ∗ ∗ ∗ ∗ ΦI (t) = −∆1 + γ1 (1 − U − U − U ) (ΦS − ΦI ) SBT + (U + µp) ΦI ,

v 1 3 4 BT BT 2 v

˙ ∗ ∗ ∗ ΦS (t) = β2 (1 − U − U − U ) ΦS − ΦL IBT + µpΦS ,

g 1 3 4 g g g

ΦL (t) = U ΦL − ΦS + β ΦL − ΦI + µpΦL ,

g 3 g g g g g

˙ ∗ ∗ ∗ ∗ ΦI (t) = −∆2 + γ2 (1 − U − U ) (ΦS − ΦI ) S + (U + µp) ΦI ,

g 1 3 BT BT BT 2 g

Φ ˙ (t) = −∆ + r 2 − 1 Φ + (1 − U − U − U )(γ I + γ I )(Φ − Φ ) +

SBT 3 SBT 1 3 4 1 v 2 g SBT IBT

(θBT U1 N P + µI)ΦS ,

BT

˙ ∗ ∗ ∗ ∗ ∗ ΦS (t) = −∆4 + (1 − U − U − U )[β1(ΦS − ΦL )S + β2(ΦS − ΦL )S ] +

BT 1 3 4 v v v g g g

(θBT NpU1 + µI)ΦI ,

BT

with the transversality condition Φ(T) = 0.

 ˙ H ˙ H ˙ H ΦS = − , ΦL = − , ΦI = − ,

 v ∂Sv v ∂Lv v ∂Iv

˙ H ˙ H ˙ H ΦS = − , ΦL = − , ΦI = − ,

 g ∂Sg g ∂Lg g ∂Ig

  ˙ H ˙ H ΦS = − , ΦI = − ,

BT ∂SBT BT ∂IBT

we obtain the adjoint functions.

Proof 2 In characterizing each control, we consider three cases concerning the control bounds.

And we show this in detail for the characterization of the first control U1

∗ max

I. First case, on the set {t | 0 < U1 < U1 }, we have

∂H ∗ ∗ ∗ ∗ ∗

0 = = Φ1U + (IBT∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )] + (γ1I S +

∂U ∗ 1 v v v g g g v BT ∗ ∗ ∗ ∗

γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

∗ ∗ ∗

U = (IBT ∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )]+

1 v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

II. Second case one the set {t | U1 = 0}, we have

∂H ∗ ∗ ∗ ∗ ∗

0 ≤ = Φ1U + (IBT∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )] + (γ1I S +

∂U ∗ 1 v v v g g g v BT ∗ ∗ ∗ ∗

γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

0 ≥ (IBT ∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )]+

v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

holds on this set.

∗ max

III. Third case one the set {t | U1 = U1 }, we have

∂H ∗ ∗ ∗ ∗ ∗

0 ≥ = Φ1U + (IBT∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )] + (γ1I S +

∂U max 1 v v v g g g v BT ∗ ∗ ∗ ∗

γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

max ∗ ∗

−ΦU ≥ (IBT∗ [β1S (ΦL − ΦS ) + β2S (ΦL − ΦS )]+

1 v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦI − ΦS ) + θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

and dividing both sides by the negative quantity −Φ1, it yields

max ∗ ∗

U ≤ (IBT ∗ [β1S (ΦS − ΦL ) + β2S (ΦS − ΦL )]+

1 v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦS − ΦI ) − θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

Therefore,

∗ 1 ∗ ∗ U = min (IBT ∗ [β1S (ΦL − ΦS ) + β2S (ΦL − ΦS )]+

1 v v v g g g

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦI − ΦS ) + θBT Np (SBT + IBT ) (ΦS + ΦI )

BT BT BT BT

∗ 1 ∗ ∗ U = max 0, min (IBT ∗ [β1S (ΦL − ΦS ) + β2S (ΦL − ΦS )]+

1 v v v g g g

io

∗ ∗ ∗ ∗ ∗ ∗

(γ1Iv SBT + γ2Ig SBT ) (ΦI − ΦS ) + θBT Np (SBT + IBT ) (ΦS + ΦI ) , 1 .

BT BT BT BT

∗ ∗ ∗

Similarly by going through the three-step arguments the remaining optimal controls U2 , U3 , U4

can also be obtained explicitly.

2 2 2 2

∂ H ∂ H ∂ H ∂ H = Φ1 > 0, = Φ2 > 0, = Φ3 > 0, = Φ4 > 0.

2 2 2 2

∂ U1 ∂ U2 ∂ U3 ∂ U4

indicating that the optimal controls minimize the Hamiltonian.

r 0.5 [17] γ2 0.2 [4, 17] δp 0.001 [17] µp 0.3 [4, 17] α 0.1 [4, 17] µBT 0.07 [4, 17] θ 0.01 [4, 17] θBT 0.05 [17] β 0.1 [4, 17] T 150 [17]

(a) (b)

(c) (d)

Figure (11)-a illustrates the evolution of the optimal control variable U1.

Figure (11)-b illustrates the evolution of the optimal control variable U2.

Figure (11)-c illustrates the evolution of the optimal control variable U3, which represents the rate of planting

resistant tomato varieties aimed at reducing the spread of TYLCV within the crop.

Figure (11)-d illustrates the evolution of the optimal control variable U4, representing the installation of

insect-proof nets to control flying insects.

Figure (12)-a to (12)-h show respectively the population dynamics of vegetative susceptible Sv, latent Lv, infected Iv, generative susceptible Sg, latent Lg, infected Ig, as well as the whitefly populations SBT and IBT .

The blue curves represent the dynamics of susceptible, latent, and infected populations (both vegetative and generative), as well as whitefly populations, in the absence of any control strategy.

The yellow curves illustrate the effect of applying all four control strategies simultaneously.

The green curves show the dynamics under a combined strategy of infected plant removal and biological control. However, this approach does not achieve the optimal outcome.

The red curves describe the effect of using only biological control on tomato plant populations, which proves insufficient on its own.

The brown curves highlight the impact of resistant varieties on the overall plant population. Although not optimal, this strategy remains the second most effective control approach for field management.

Finally, the purple curves represent the dynamics when only insect-proof nets are used, which alone do not provide sufficient protection against whitefly infestation.

Our analysis demonstrates that the simultaneous implementation of all four control strategies achieves: - the

maximum increase in healthy vegetative and generative plant populations (see Figure (12)-a and (12)-d), - the minimum levels of latent and infected states (see Figure (12)-b, (12)-c, (12)-e, and (12)-f), - a significant reduction in infected plant populations through removal strategies (see Figure (12)-c, (12)-f, and (12)-h), -

and improved overall crop performance through the use of resistant varieties.

Exclusive reliance on insect-proof nets is insufficient to control whitefly infestations. An integrated approach combining all four strategies is therefore necessary to maximize agricultural productivity, optimize disease control, and ensure sustainable pest management.

Certain insect species, such as whiteflies, aphids, and leafminers, rapidly migrate to nearby healthy plants after host removal. Consequently, the removal of infected plants may induce dispersal and colonization of additional plants, sometimes increasing infestation in the short term. This explains: - the increase in latent vegetative plants

compared to the no-control scenario (Figure (12)-b), - the increase in latent generative plants (Figure (12)-e), - the increase in infected generative plants (Figure (12)-f), - and the rise in whitefly populations compared to the uncontrolled case (Figure (12)-g).

Overall, the combined implementation of all four control strategies remains the most effective management policy.

It leads to: - the highest increase in healthy vegetative and generative populations (Figure (12)-a and (12)-d), - the strongest reduction in latent and infected populations (Figure (12)-b, (12)-c, (12)-e, and (12)-f), - and a substantial decrease in both healthy and infected whitefly populations (Figure (12)-g and (12)-h).

``fi Michel and Moussa |Opt. 3(1) (2026) 38-61

(a) (b)

(c) (d)

(e) (f)

(g) (h)

This study highlights the effectiveness of integrated management strategies for controlling the spread of Tomato Yellow Leaf Curl Virus (TYLCV) in tomato crops. By combining epidemiological modeling with optimal control theory based on Pontryagin’s Maximum Principle, we design intervention strategies that minimize both economic losses and disease prevalence while accounting for the coupled dynamics of tomato plants and whitefly vectors.

Numerical simulations show that the coordinated application of biological control agents, removal of infected plants, resistant tomato varieties, and insect-proof nets provides the most efficient and cost-effective strategy, particularly under high infection pressure. The optimal combined strategy significantly reduces virus incidence, improves plant survival, and decreases reliance on chemical pesticides, thereby promoting more sustainable agricultural practices.

These results underline the importance of mathematically grounded optimal control approaches in plant disease management and provide useful insights for improving crop resilience and enhancing food security.

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

M.IDO: methodology, software, writing and editing. M.BARRO: conceptualize, supervise and editing. All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.

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