<p>Tomato bacterial wilt is a serious disease that spreads through soil and is caused by the bacterium <i>Ralstonia solanacearum</i>. This virus affects various plants, namely eggplants, peppers, potatoes, and tomatoes. The disease is particularly harmful in warm and humid environments and can result in significant yield losses in tomatoes. In this article, we have proposed a mathematical model and analyzed it to study the transmission dynamics of bacterial wilt disease in tomato plantations. We have derived the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1689_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> of the system and established the existence of a disease-free equilibrium point that is stable if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1689_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0 <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We have also seen an endemic equilibrium that exists when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1689_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0 >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Stability changes occur through the occurrence of the Hopf bifurcation. Finally, we adopted the optimal control theory to minimize bacterial infection and maximize tomato yield. Consequently, an optimal control problem is formulated and solved by maximum principle and Hamiltonian formulation. Numerical simulations of the model systems determine the impact of important parameters of the system, and simulations of the optimal control problem give cost-effective control measures of the disease.</p>

Stability and Hopf bifurcation in the dynamics of tomato bacterial wilt: an optimal control approach

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发表日期：

2025年4月29日

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Fahad Al Basir,
Aeshah A. Raezah,
Jahangir Chowdhury,
Tarak Nath Halder

摘要

Tomato bacterial wilt is a serious disease that spreads through soil and is caused by the bacterium Ralstonia solanacearum. This virus affects various plants, namely eggplants, peppers, potatoes, and tomatoes. The disease is particularly harmful in warm and humid environments and can result in significant yield losses in tomatoes. In this article, we have proposed a mathematical model and analyzed it to study the transmission dynamics of bacterial wilt disease in tomato plantations. We have derived the basic reproduction number $R_0$ $R_{0}$ of the system and established the existence of a disease-free equilibrium point that is stable if $R_0 <1$ $R_{0} < 1$ . We have also seen an endemic equilibrium that exists when $R_0 >1$ $R_{0} > 1$ . Stability changes occur through the occurrence of the Hopf bifurcation. Finally, we adopted the optimal control theory to minimize bacterial infection and maximize tomato yield. Consequently, an optimal control problem is formulated and solved by maximum principle and Hamiltonian formulation. Numerical simulations of the model systems determine the impact of important parameters of the system, and simulations of the optimal control problem give cost-effective control measures of the disease.