Mathematical Modeling of Oak Wilt Dynamics: The Protective Role of White Oaks in Mixed Forests. A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Yorkinoy Shermatova IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Richard McGehee January, 2025 © Yorkinoy Shermatova 2025 ALL RIGHTS RESERVED Acknowledgements I would like to thank my advisor Richard McGehee for all his support during my PhD journey, as well as Jeannine Cavender- Barres for feedback and opportunities to engage in conversations from different fields. I would like to thank my academic siblings for their encouragement, constructive feedback and support. i Dedication I would like to dedicate all the work done her to my children Sunnatillo and Sarvar for being champions of a parent-student for as long as they remember themselves. ii Abstract Title: Mathematical Modeling of Oak Wilt Dynamics: The Protective Role of White Oaks in Mixed Forests. Oak wilt, caused by the fungus Bretziella fagacearum, is a devastating disease that poses a significant threat to oak forests, particularly affecting red oaks with rapid mortality upon infection. In contrast, white oaks exhibit greater resistance, leading to slower disease progression and lower mortality rates. This thesis presents an analysis of oak wilt dynamics in mixed oak forests using a compartmental disease model, emphasizing how species composition influences disease spread and mortality. We use compartmental ordinary differential equation (ODE) model that incorporates the dis- tinct biological characteristics and transmission pathways of red and white oaks. The model ac- counts for root graft transmission among red oaks, beetle-mediated transmission to both red and white oaks, and differential mortality rates. By formulating species-specific compartments and parameters, we capture the essential mechanisms driving oak wilt epidemics. Through analytical proofs, we show that increasing the proportion of white oaks in a mixed forest leads to a decrease or no increase in both the total mortality and the infection rate due to oak wilt. Specifically, we show that the total mortality Dtotal and the rate of new infections are decreasing functions of the white oak proportion ρw. These results are obtained without relying on specific parameter values, highlighting the robustness of the conclusions. It does for both simple (short term) and extended (long term) models of the disease. It also talks about climate change effects on overall mortality. Our findings are strongly supported by empirical evidence from peer-reviewed literature, con- firming the validity of the model and its assumptions. The mathematical analysis aligns with observed patterns in forest ecology, where higher proportions of red oaks compared to white oaks are most affected by disease outbreak. This underscores the critical role of species composition in influencing disease dynamics. This research provides a mathematically rigorous and ecologically meaningful examination of oak wilt dynamics, demonstrating how species composition can serve as risk assessment where red and white oaks coexist. It reinforces the importance of interdisciplinary approaches in address- ing complex environmental challenges and offers practical guidance for preserving the health and diversity of oak forests. iii Most importantly, this finding does not rely on computational tools or highly complex models, highlighting the importance of simple models in shedding light on interesting aspects of complex environments. iv Contents Acknowledgements i Dedication ii Abstract iii List of Tables vii List of Figures viii 1 Introduction 1 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Literature Review 4 2.1 Oak Wilt Pathology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Biological Differences Between Red and White Oaks . . . . . . . . . . . . . . 4 2.1.2 Transmission Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Previous Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Management Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Empirical Evidence Supporting Species Composition Effects . . . . . . . . . . . . . . 6 2.5 Gaps in the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Simple Model 8 3.1 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 v 3.3 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Detailed Explanation of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Justification of Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.6 Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.7 Analysis of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.8 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.9 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.10 Results for Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.10.1 Preliminary Definitions and Assumptions . . . . . . . . . . . . . . . . . . . . 16 3.10.2 Proof that Increasing ρw Decreases Total Mortality . . . . . . . . . . . . . . . 16 3.10.3 Proof that Increasing ρw Decreases Infection Rate . . . . . . . . . . . . . . . 19 4 Extended Model 22 4.1 Extended Model with Reproduction and background Mortality . . . . . . . . . . . . 22 4.2 Computation of the Basic Reproduction Number R0 . . . . . . . . . . . . . . . . . . 24 4.3 Total Mortality vs white oak ratio for Extended Model . . . . . . . . . . . . . . . . . 28 4.4 Sensitivity Analysis of Beetle Transmission Including Climate Change Effects . . . . 31 5 Discussion and Conclusion 36 5.1 Empirical and Theoretical Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Insights from the Decline of White Oaks . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.3 Implications for Forest Management . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 Limitations and Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . 38 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 Data and Code 39 6.1 DATA [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Bifurcation Diagram Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.3 Total Mortality VS White Oak ratio code . . . . . . . . . . . . . . . . . . . . . . . . 41 References 46 vi List of Tables 6.1 Total mortality count/percent counts for red oak and white oak group trees . . . . . 39 vii List of Figures 3.1 Schematic diagram of oak wilt transmission dynamics in the model. Solid arrows rep- resent transitions between compartments, while dashed arrows indicate transmission pathways. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Total mortality and white oak ratio relationship . . . . . . . . . . . . . . . . . . . . . 19 4.1 Diagram of the extended oak wilt model showing the dynamics of susceptible, in- fected, and dead trees for red and white oaks. Solid arrows represent transitions between compartments, and dashed arrows represent infection pathways. . . . . . . . 23 4.2 Equilibrium Infected Oaks and White Oak ratio relationship . . . . . . . . . . . . . 28 5.1 This illustration has been created by author based on information provided by Abrams [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 viii Chapter 1 Introduction Oak wilt is a devastating fungal disease that has significant ecological and economic impacts on oak forests, particularly in North America. Caused by the pathogen Bretziella fagacearum, oak wilt affects a wide range of oak species, leading to rapid decline and mortality [13]. The disease poses a substantial threat to forest biodiversity, timber production, and urban landscaping. Red oak groups are highly susceptible to oak wilt, often succumbing to the disease within weeks of infection [13], [16]. In contrast, white oak groups exhibit greater resistance, with slower disease progression and higher survival rates [19]. This dichotomy in susceptibility between red and white oaks plays a crucial role in the dynamics of oak wilt spread within mixed-species forests. Understanding the factors that influence the spread and impact of oak wilt is essential for devel- oping effective management strategies as well as understand risk factors based on the composition of sites. Mathematical modeling offers a powerful tool for capturing the complexity of disease dy- namics and predicting the outcomes of different scenarios. By integrating biological knowledge with mathematical frameworks, we can gain insights into the mechanisms driving oak wilt epidemics and identify leverage points for intervention, maybe even origins of the spread across Eastern United States [14]. This thesis presents a rigorous analysis of oak wilt dynamics in mixed oak forests using a com- partmental ordinary differential equation (ODE) model specifically focusing on the relationship of total mortality and infection rates dependence on ratio of oaks. Compartmental differential equa- tions have been used in disease modeling for decades and have been established as a foundational theoretical framework [3]. We focus on the role of species composition, specifically the proportion of white oaks, in influencing the total mortality and infection rates due to oak wilt. Through ana- lytical proofs, we demonstrate that areas with a higher proportion of white oaks, compared to red 1 2 oak group trees, experience smaller decreases or no increases in both total mortality and infection rates. Our assumptions used in proof are supported by empirical evidence from the literature, reinforcing the robustness of our conclusions. 1.1 Objectives The primary objectives of this thesis are: 1. To develop compartmental ODE models that accurately reflects the biological differences between red and white oaks in the context of oak wilt disease. 2. To analytically prove that increasing the proportion of white oaks in a mixed forest reduces or does not increase total mortality due to oak wilt. 3. To analytically prove that the infection rate decreases or remains the same as the proportion of white oaks increases. 4. To find basic reproduction number R0 for both models and its relationship to oak ratios 5. To corroborate the theoretical findings with empirical evidence from peer-reviewed literature. 6. To provide insights for forest management, we discuss the origins of the disease’s epidemic throughout the eastern United States and assess the risks associated with infection rates and mortality based on the white-to-red oak ratio in the system. 1.2 Significance of the Study This study investigates mortality and infection rates in white-red oak ecosystems by analyzing how varying species composition ratios influence oak wilt dynamics. Through mathematical modeling, we demonstrate that increasing the proportion of white oaks can significantly reduce both infec- tion rates and overall mortality caused by oak wilt. These findings provide crucial insights for forest management practices, enabling the development of strategies that optimize species com- position to mitigate disease impacts. By integrating mathematical models with ecological data, this research enhances our ability to predict disease progression and implement effective control measures, ultimately promoting the health and sustainability of oak populations. 3 1.3 Structure of the Thesis The thesis is organized as follows: • Chapter 2 provides a literature review on oak wilt pathology, the biological differences between red and white oaks, transmission mechanisms, previous and current models, and existing management strategies. • Chapter 3 details the development of the Simple (epidemic) ODE model, including assump- tions, variable definitions, and justifications based on empirical evidence. It also includes analytical results for the Simple Model. • Chapter 4 presents the development of the Extended (endemic) ODE model, including assumptions, variable definitions, and justifications based on empirical evidence. It also includes analytical results for the Extended Model including basic reproduction number and its relationship to oak ratios. • Chapter 5 This section explores the implications of our findings, tries to integrate them with empirical studies, and addresses the limitations of our model. Furthermore, it considers how alterations in oak species ratios might have contributed to the origin and onset of significant outbreaks across the eastern United States. • Chapter 6 includes one table and two code examples, the illustrations of which are presented here. Chapter 2 Literature Review 2.1 Oak Wilt Pathology Oak wilt is a vascular disease caused by the fungal pathogen Bretziella fagacearum, which invades the water-conducting vessels (xylem) of oak trees [13]. The infection leads to wilting symptoms, leaf discoloration, defoliation, and ultimately, tree death. The disease was first identified in the United States in the 1940s and has since spread to numerous states, affecting thousands of oak trees [9], [14],[13]. Symptoms and Diagnosis Infected trees exhibit a range of symptoms, including: • Wilting and bronzing of leaves from the top of the canopy downward. • Premature leaf drop and branch dieback. • Discoloration of vascular tissues under the bark. Accurate diagnosis often requires laboratory testing to confirm the presence of the pathogen [13]. 2.1.1 Biological Differences Between Red and White Oaks Red oaks and white oaks exhibit distinct physiological and anatomical characteristics that influence their susceptibility to oak wilt. Red oaks have vessels that remain open and interconnected, allowing the fungus to spread rapidly throughout the tree [17]. In contrast, white oaks possess tyloses 4 5 and gummy substances that form within their xylem vessels, effectively blocking the pathogen’s movement [9]. Red Oaks Red oaks are characterized by: • High susceptibility to oak wilt infection. • Rapid disease progression leading to mortality within weeks to months. • Efficient root graft formation among conspecifics, facilitating underground pathogen spread [13]. • Attraction to sap-feeding beetles that serve as vectors for overland transmission. White Oaks White oaks exhibit: • Partial resistance to oak wilt due to tyloses formation. • Slower disease progression, with some trees surviving infection. • Less efficient root grafting and lower susceptibility to insect vectors [19]. 2.1.2 Transmission Mechanisms Oak wilt spreads through two primary mechanisms: Root Graft Transmission Root grafts form when the roots of adjacent trees of the same species fuse together. In red oak stands, this creates an extensive underground network that allows the pathogen to move rapidly from tree to tree [13]. Root graft transmission accounts for the majority of new infections in red oak populations. Insect Vector Transmission Sap-feeding beetles, particularly nitidulid beetles, transport fungal spores from infected trees to fresh wounds on healthy trees [9]. This overland transmission is crucial for initiating new infection centers and spreading the disease across larger distances. 6 2.2 Previous Mathematical Models We are aware of two primary models: the spatiotemporal computational model developed by Menges and Loucks [15], and the recently published machine learning model designed to enhance aerial surveillance by Guzman et al. [10]. Menges’s model [15] examines the relationship between species ratios and total mortality. We will utilize this model as part of our qualitative validation. 2.3 Management Strategies Effective management of oak wilt involves: • Disrupting root grafts through trenching to prevent underground spread. • Sanitation measures, including the removal and proper disposal of infected trees. • Chemical treatments using systemic fungicides to protect high-value trees. • Promoting species diversity, particularly increasing the proportion of white oaks, to reduce overall susceptibility [19], [12]. 2.4 Empirical Evidence Supporting Species Composition Effects Empirical studies have observed that mixed forests with higher proportions of white oaks experience lower oak wilt impacts. Wilson (2005) noted that ”the presence of white oaks in mixed stands disrupts the continuous network of susceptible hosts required for rapid disease spread,” emphasizing the role of white oaks in mitigating oak wilt dynamics [19]. Similarly, Juzwik et al. (2011) reported that stands dominated by red oaks exhibit higher infection rates and mortality compared to mixed or white oak-dominated stands [13]. 2.5 Gaps in the Literature While empirical observations suggest that species composition influences oak wilt dynamics, there is a lack of theoretical frameworks that quantitatively explain this phenomenon. Existing mathemati- cal models often overlook the role of species composition, and there is limited analytical exploration of how increasing white oak proportions affect disease spread and mortality. 7 This thesis addresses these gaps by developing a mathematical model that incorporates the biological differences between red and white oaks and analytically demonstrating the effects of species composition on oak wilt outcomes. Chapter 3 Simple Model In this chapter, we develop a mathematical model to analyze the dynamics of oak wilt disease in mixed oak forests. Our model incorporates the biological differences between red and white oaks, capturing the distinct transmission pathways and mortality rates associated with each species. By formulating an ordinary differential equation (ODE) system, we aim to understand how species composition influences the spread of the disease and the overall mortality within the forest. 3.1 Model Overview The model divides the oak population into compartments based on species (red or white oak) and disease status (susceptible, infected, or dead). We denote the following variables: • Sr(t): Number of susceptible red oaks at time t. • Ir(t): Number of infected red oaks at time t. • Dr(t): Number of dead red oaks at time t. • Sw(t): Number of susceptible white oaks at time t. • Iw(t): Number of infected white oaks at time t. • Dw(t): Number of dead white oaks at time t. The total living tree population at time t is: N(t) = Sr(t) + Ir(t) + Sw(t) + Iw(t). (3.1) 8 9 We assume that dead trees are promptly removed or no longer participate in disease transmis- sion, so they are not included in N(t). 3.2 Assumptions Our model is built upon several key assumptions, grounded in empirical observations from the literature: 1. Species Composition: The forest consists of a mixture of red and white oaks, with pro- portions ρr and ρw, respectively, such that ρr + ρw = 1. 2. Transmission Pathways: • Red oaks can transmit the disease through root grafts to other red oaks and via insect vectors (beetles) to both red and white oaks. • White oaks are less susceptible to beetle transmission and have limited root graft trans- mission (we assume it is neglible and don’t include in the model, main reason being white oaks’ capability of localizing the disease and not spreading throughout the trees body and roots). 3. Mortality Rates: Infected red oaks die at a higher rate than infected white oaks due to their lack of resistance mechanisms. 4. Homogeneous Mixing: The trees are well-mixed within the forest, allowing for uniform interaction probabilities. 5. No Recruitment or Recovery: We assume no new susceptible trees are added to the population and that infected trees do not recover. 3.3 Model Equations Based on these assumptions, we formulate the following system of ODEs: 10 Red Oak Dynamics dSr dt = −βrr SrIr N − βbr SrIr N , (3.2) dIr dt = βrr SrIr N + βbr SrIr N − µrIr, (3.3) dDr dt = µrIr. (3.4) White Oak Dynamics dSw dt = −βbw SwIr N , (3.5) dIw dt = βbw SwIr N − µwIw, (3.6) dDw dt = µwIw. (3.7) • βrr: Transmission rate among red oaks via root grafts. • βbr: Beetle transmission rate from infected red oaks to susceptible red oaks. • βbw: Beetle transmission rate from infected red oaks to susceptible white oaks. • µr: Mortality rate of infected red oaks. • µw: Mortality rate of infected white oaks. We assume βrw-transmission via roots between red and white group oaks and βww - transmission via roots among white oak groups are negligible. Observations indicate that root graft transmission among white oaks is rare due to their anatomical features that limit graft formation. Moreover, root grafts between red and white oaks are virtually nonexistent, suggesting that such pathways do not contribute significantly to the spread of oak wilt between these groups [2], [11]. 3.4 Detailed Explanation of Equations Susceptible Red Oaks (Sr) Equation (3.2) describes the rate of change of susceptible red oaks: 11 dSr dt = −βrr SrIr N − βbr SrIr N . (3.8) This equation accounts for two transmission pathways: 1. Root Graft Transmission (βrr): Susceptible red oaks become infected through root grafts with infected red oaks. The term βrr SrIr N represents the rate at which this occurs. 2. Beetle Transmission (βbr): Susceptible red oaks are infected by beetles carrying spores from infected red oaks. The term βbr SrIr N captures this rate. The total rate at which susceptible red oaks become infected is the sum of these two rates. Infected Red Oaks (Ir) Equation (3.3) describes the dynamics of infected red oaks: dIr dt = βrr SrIr N + βbr SrIr N − µrIr. (3.9) The first two terms represent the new infections of red oaks, as described above. The last term, −µrIr, represents the removal of infected red oaks due to mortality. Dead Red Oaks (Dr) Equation (3.4) accounts for the accumulation of dead red oaks: dDr dt = µrIr. (3.10) Infected red oaks die at a rate proportional to their number, with mortality rate µr. Susceptible White Oaks (Sw) Equation (3.5) models the susceptible white oaks: dSw dt = −βbw SwIr N . (3.11) White oaks become infected only through beetle transmission from infected red oaks. The term βbw SwIr N represents this rate, where βbw is smaller than βbr due to the lower susceptibility of white oaks. Infected White Oaks (Iw) Equation (3.6) describes infected white oaks: dIw dt = βbw SwIr N − µwIw. (3.12) 12 Infected white oaks progress to mortality at a rate µw, which is lower than µr due to their greater resistance. Dead White Oaks (Dw) Equation (3.7) accounts for dead white oaks: dDw dt = µwIw. (3.13) Infected white oaks die at a rate µw. Figure 3.1 illustrates the flow of individuals between compartments and the transmission path- ways of the disease. The diagram highlights the differences in transmission for red and white oaks. 3.5 Justification of Model Structure Our model structure reflects the key biological characteristics of oak wilt transmission: • Species-Specific Transmission: The model differentiates between red and white oaks, capturing their distinct susceptibilities and transmission routes. • Transmission Rates: The parameters βrr, βbr, and βbw are chosen to reflect the relative efficiencies of transmission pathways, with βrr > βbr and βrr > βbw. • Mortality Rates: The mortality rates µr and µw represent the rapid decline of red oaks and the slower progression in white oaks, respectively 6.1,[11]. 13 Sr Ir Dr Sw Iw Dw βrr Ir N + βbr Ir N µr βbw Ir N µw Beetle transmission Figure 3.1: Schematic diagram of oak wilt transmission dynamics in the model. Solid arrows represent transitions between compartments, while dashed arrows indicate transmission pathways. 3.6 Non-Dimensionalization To simplify the analysis, we can non-dimensionalize the model by introducing scaled variables and parameters. Let: τ = µrt, sr = Sr N , ir = Ir N , sw = Sw N , iw = Iw N . (3.14) The total proportion of living trees is then: sr + ir + sw + iw = 1. (3.15) We define the following dimensionless parameters: β̃rr = βrr µr , β̃br = βbr µr , β̃bw = βbw µr , µ̃ = µw µr . (3.16) Substituting these into the model equations, we obtain the dimensionless system: Dimensionless Red Oak Equations dsr dτ = − ( β̃rrir + β̃brir ) sr, (3.17) dir dτ = ( β̃rrir + β̃brir ) sr − ir. (3.18) Dimensionless White Oak Equations 14 dsw dτ = −β̃bwirsw, (3.19) diw dτ = β̃bwirsw − µ̃iw. (3.20) 3.7 Analysis of the Model Equilibrium Points We can analyze the equilibrium points of the system to understand the long-term behavior. An equilibrium occurs when: dsr dτ = dir dτ = dsw dτ = diw dτ = 0. (3.21) Solving these equations, we find the disease-free equilibrium (DFE): (s∗r , i ∗ r , s ∗ w, i ∗ w) = (ρr, 0, ρw, 0), (3.22) where ρr and ρw are the initial proportions of red and white oaks. Basic Reproduction Number The basic reproduction number R0 is a key threshold parameter that determines whether the disease will invade the population. It represents the expected number of secondary infections produced by a single infected individual in a completely susceptible population. For our model, R0 can be calculated separately for red and white oaks. Red Oaks The basic reproduction number for red oaks is: R0r = ( β̃rr + β̃br ) ρr. (3.23) White Oaks Since white oaks become infected only through contact with infected red oaks, their reproduction number depends on the prevalence of infection in red oaks. However, due to their lower susceptibility and mortality rate, the contribution to the overall R0 is less significant. 15 3.8 Interpretation of Parameters Transmission Rates • β̃rr: Reflects the effectiveness of root graft transmission among red oaks. • β̃br: Represents the efficiency of beetle-mediated transmission to red oaks. • β̃bw: Indicates the lower susceptibility of white oaks to beetle transmission. Mortality Rates • µr: High mortality rate for infected red oaks, consistent with their rapid decline upon infec- tion. • µw: Lower mortality rate for infected white oaks, reflecting their partial resistance. 3.9 Sensitivity Analysis To understand how changes in parameters affect the dynamics, we perform sensitivity analysis. Effect of Increasing ρw As the proportion of white oaks ρw increases, the proportion of red oaks ρr = 1− ρw decreases. This directly affects the basic reproduction number R0r, reducing the potential for the disease to spread among red oaks. Impact of Transmission Rates Changes in βrr and βbr have a significant impact on disease spread among red oaks. Reducing these rates through management interventions can lowerR0r below 1, leading to disease eradication. 3.10 Results for Simple Model In this section, we present the analytical proofs that demonstrate how increasing the proportion of white oaks in a mixed oak forest influences the dynamics of oak wilt disease both in simple and updated models. Specifically for simple models, we prove that: 1. Increasing the proportion of white oaks (ρw) decreases or does not increase the total mortality due to oak wilt. 2. Increasing ρw decreases or does not increase the infection rate in the forest. 16 Our proofs are grounded in the mathematical model developed in the previous chapter and are supported by established ecological principles and empirical observations from peer-reviewed literature. 3.10.1 Preliminary Definitions and Assumptions Before proceeding to the proofs, we summarize key definitions and assumptions from our model: • The total living tree population is N = Sr + Ir + Sw + Iw. • The proportions of red and white oaks in the forest are ρr = Nr N and ρw = Nw N , with ρr+ρw = 1. • Transmission rates satisfy βrr > βbr > βbw > 0. • Mortality rates satisfy µr > µw > 0. • We assume that βrr and βbr are significantly larger than βbw, reflecting the higher suscepti- bility and transmission among red oaks [13]. 3.10.2 Proof that Increasing ρw Decreases Total Mortality Total Mortality Expression The total mortality at time t is given by: Dtotal(t) = Dr(t) +Dw(t). (3.24) Our goal is to show that dDtotal dρw ≤ 0, meaning that increasing ρw leads to a decrease or no increase in total mortality. Derivation of Mortality Proportions We denote: αr = Dr Nr , αw = Dw Nw , (3.25) where αr and αw represent the proportions of red and white oaks that have died due to oak wilt. Based on empirical observations: • Red oaks are highly susceptible, with αr ≈ 1 [11], [13]. 17 • White oaks have partial resistance, with 0 ≤ αw < 1 [9], [11]. Total Mortality as a Function of ρw The total mortality can be expressed as: Dtotal = Dr +Dw = Nrαr +Nwαw = N (ρrαr + ρwαw) . (3.26) Derivative of Total Mortality with Respect to ρw Differentiating Dtotal with respect to ρw, we obtain: dDtotal dρw = d dρw [N (ρrαr + ρwαw)] = N ( dρr dρw αr + ρr dαr dρw + αw + ρw dαw dρw ) . (3.27) Since ρr = 1 − ρw, we have dρr dρw = −1. Assuming that αr and αw are approximately constant with respect to small changes in ρw, we have dαr dρw ≈ 0 and dαw dρw ≈ 0. Substituting these into equation (3.27), we get: dDtotal dρw = N (−αr + αw) = N (αw − αr) . (3.28) Given that αr ≈ 1 and 0 ≤ αw < 1, it follows that αw − αr ≤ 0. Therefore: dDtotal dρw ≤ 0. (3.29) This result shows that increasing the proportion of white oaks (ρw) decreases or does not increase the total mortality due to oak wilt. Conclusion 18 The analytical derivation confirms that promoting white oaks in a mixed forest reduces the overall mortality caused by oak wilt. Since this is a very simple model and doesn’t involve no oak species, we can conclude that forest diversity plays a big role improving forest health, particularly in this case, manage oak wilt caused death to the oak population.This is consistent with empirical findings that geographic regions with more diversity in their stands will do better compared to those that are red oak dominated [19]. It also confirms findings of Menges’s model [15] which shows that when red oaks supress 0.5 poroportion comared to white oaks, total mortality exponentially increases as it is shown below 3.2. Figure 3.2: [15] Below is the visual demonstration 3.3 of this relationship. Note that it is very difficult to get how exactly this negative correlation is since we don’t have a good estimation of the parameters, we only use parameter relationship here 19 Note: This does not imply that red oaks should be replaced by white oaks. Both species interact in diverse and complementary ways that are crucial for the ecosystem’s health. Our findings highlight the importance of maintaining biodiversity—ensuring the presence of both red and white oaks—as a strategy to mitigate disease spread. It shows that when initial spread starts, areas with higher proportion of red oaks compared to white oaks will have more disease related mortality. It is important to acknowledge that this conclusion stems from the simplicity of our model, which does not account for the myriad of interactions and complexities inherent in real-world forest ecosystems. Figure 3.3: Total mortality and white oak ratio relationship 3.10.3 Proof that Increasing ρw Decreases Infection Rate Expression for Total New Infections 20 The total number of new infections per unit time is: New Infections = βrr SrIr N + βbr SrIr N + βbw SwIr N = (βrr + βbr) SrIr N + βbw SwIr N . (3.30) Dependence on ρw Expressing Sr and Sw in terms of ρr and ρw: Sr = ρrS, Sw = ρwS, S = Sr + Sw. (3.31) Assuming that the total number of susceptible trees S is approximately constant for small changes in ρw, we substitute into equation (3.30): New Infections = (βrr + βbr) ρrSIr N + βbw ρwSIr N = SIr N [(βrr + βbr) ρr + βbwρw] . (3.32) Derivative of New Infections with Respect to ρw Differentiating with respect to ρw: d dρw (New Infections) = SIr N ( d dρw [(βrr + βbr) ρr + βbwρw] ) = SIr N (− (βrr + βbr) + βbw) . (3.33) Sign of the Derivative Since root transmission has been found to be responsible for majority spread, βbw < βrr and βbr < βrr, since all of them are positive we have: βbw < βrr < βrr + βbr (3.34) 21 which leads to βbw − (βrr + βbr) < 0. (3.35) Therefore: d dρw (New Infections) ≤ 0. (3.36) This shows that increasing ρw decreases or does not increase the total number of new infections. Which is the opposite for red oak population, meaning increasing ρr will increase the total number of new infections. Proof is analogous where you take derivative with respect to ρr. Conclusion The analytical result confirms that increasing the proportion of white oaks reduces the infection rate in the forest. This aligns with observations that white oaks, being less susceptible, contribute less to the spread of oak wilt [11]. Chapter 4 Extended Model 4.1 Extended Model with Reproduction and background Mortal- ity Assumptions • Root-to-root transmission is higher than beetle-mediated transmission: βrr > βbr, βrr > βbw • Red oaks have higher reproduction rates than white oaks: br > bw • Red oaks have higher disease-induced and background mortality rates than white oaks: µr > µw, γr > γw Model Equations Red Oak Dynamics 22 23 dSr dt = brSr ( 1− N K ) − (βrr + βbr) SrIr N − γrSr, (4.1) dIr dt = (βrr + βbr) SrIr N − µrIr, (4.2) dDr dt = µrIr + γrSr. (4.3) White Oak Dynamics dSw dt = bwSw ( 1− N K ) − βbw SwIr N − γwSw, (4.4) dIw dt = βbw SwIr N − µwIw − γwIw, (4.5) dDw dt = µwIw + γwSw + γwIw. (4.6) Sr Ir Dr Sw Iw Dw βrr Ir N + βbr Ir N µr βbw Ir N µw Beetle transmission br bw γr γw γw Figure 4.1: Diagram of the extended oak wilt model showing the dynamics of susceptible, infected, and dead trees for red and white oaks. Solid arrows represent transitions between compartments, and dashed arrows represent infection pathways. Total Population Dynamics 24 N = Sr + Ir + Sw + Iw, (4.7) dN dt = (brSr + bwSw) ( 1− N K ) − (γrSr + γwSw + γwIw)− (µrIr + µwIw) . (4.8) Parameter Descriptions • βrr: Transmission rate from infected red oaks to susceptible red oaks via root-to-root contact. • βbr: Beetle-mediated transmission rate from infected oaks to susceptible red oaks. • βbw: Beetle-mediated transmission rate from infected oaks to susceptible white oaks. • µr: Disease-induced mortality rate for infected red oaks. • µw: Disease-induced mortality rate for infected white oaks. • γr: Background mortality rate for red oaks. • γw: Background mortality rate for white oaks. • br: Reproduction rate for red oaks. • bw: Reproduction rate for white oaks. • K: Carrying capacity of the environment. • N : Total population size, N = Sr + Ir + Sw + Iw. 4.2 Computation of the Basic Reproduction Number R0 The basic reproduction numberR0 represents the expected number of secondary infections produced by a single infected individual in a completely susceptible population. We compute R0 using the next-generation matrix method based on our model. The next-generation matrix method, developed by Diekmann and colleagues, provides a system- atic approach to calculating the basic reproduction number R0 in compartmental epidemic models. This method involves constructing two key matrices: one that represents the rate of new infec- tions generated by each compartment (the ”new infection” matrix) and another that captures the transition rates between compartments (the ”transition” matrix). By analyzing these matrices, the 25 next-generation matrix is formed, typically by multiplying the new infection matrix by the inverse of the transition matrix. The basic reproduction number R0 is then determined as the spectral radius (the largest eigenvalue) of this next-generation matrix. This value quantifies the average number of secondary infections produced by a single infected individual in a completely susceptible population, providing critical insight into the potential for an epidemic to spread and informing public health interventions [7]. Infected Compartments The infected compartments are: • Ir: Infected red oaks • Iw: Infected white oaks New Infection and Transition Terms Define the new infection terms F and the transition terms V as follows: F(x) = [ F1 F2 ] =  βr SrIr N βbw SwIr N  , where βr = βrr + βbr, and V(x) = [ V1 V2 ] = [ (µr + γr)Ir (µw + γw)Iw ] . Here, x = [Ir, Iw] T represents the vector of infected compartments. Jacobian Matrices at the Disease-Free Equilibrium At the disease-free equilibrium (DFE), Ir = 0, Iw = 0, and the total population is susceptible (N = S∗ r + S∗ w). The Jacobian matrices F and V are obtained by taking the partial derivatives of F and V with respect to the infected compartments and evaluating them at the DFE: F = [ ∂Fi ∂xj ] DFE , V = [ ∂Vi ∂xj ] DFE . 26 Computing the derivatives, we get: F = ∂F1 ∂Ir ∂F1 ∂Iw ∂F2 ∂Ir ∂F2 ∂Iw  DFE =  βr S∗ r N∗ 0 βbw S∗ w N∗ 0  , V = ∂V1 ∂Ir ∂V1 ∂Iw ∂V2 ∂Ir ∂V2 ∂Iw  DFE = [ µr + γr 0 0 µw + γw ] . Next-Generation Matrix The next-generation matrix K is given by: K = FV −1. First, compute V −1: V −1 =  1 µr + γr 0 0 1 µw + γw  . Then, compute K: K = FV −1 =  βrS ∗ r N∗(µr + γr) 0 βbwS ∗ w N∗(µw + γw) 0  . Eigenvalues and Basic Reproduction Number The eigenvalues of K are the solutions to: det(K − λI) = 0, where I is the identity matrix. Since K is upper-triangular, the eigenvalues are on the diagonal: 27 λ1 = βrS ∗ r N∗(µr + γr) , λ2 = 0. The basic reproduction number R0 is the largest eigenvalue: R0 = λ1 = βrS ∗ r N∗(µr + γr) . (4.9) Let ρ∗r = S∗ r N∗ be the proportion of susceptible red oaks at the DFE. Then: R0 = βrρ ∗ r µr + γr . Since ρ∗r + ρ∗w = 1, where ρ∗w = S∗ w N∗ is the proportion of susceptible white oaks. Threshold Condition for Disease Invasion For the disease to invade the population, R0 > 1. Therefore, the threshold condition is: R0 = βrρ ∗ r µr + γr > 1. Solving for ρ∗r : ρ∗r > µr + γr βr . Since ρ∗r = 1− ρ∗w, we can express the threshold in terms of the proportion of white oaks: ρ∗w < 1− µr + γr βr . Critical Proportion of White Oaks To prevent disease invasion (R0 < 1), the proportion of white oaks must satisfy: ρ∗w > 1− µr + γr βr . 28 This defines the critical threshold proportion of white oaks required to minimize the disease’s impact on the population. As demonstrated, higher transmission rates among red oaks necessitate a greater proportion of white oaks to reduce the number of infected individuals. We illustrate this relationship using the bifurcation diagram below, which presents various values of βr. This scenario exhibits a trans- critical bifurcation, where altering the proportion of white oaks transitions the system from an endemic equilibrium to a disease-free state (Figure 4.2). Figure 4.2: Equilibrium Infected Oaks and White Oak ratio relationship 4.3 Total Mortality vs white oak ratio for Extended Model Statement: Under the given model and assumptions, a higher proportion of white oaks is nega- tively correlated with total mortality in the long term. Specifically, the derivative of total mortality with respect to the proportion of white oaks is negative. 29 Proof We aim to show that the total mortality M decreases as the proportion of white oaks increases, i.e., dM dPw < 0, where Pw is the proportion of white oaks in the total population. 1. Total Mortality Rate The total mortality rate M at any time t is: M = γrSr + µrIr︸ ︷︷ ︸ Red Oak Mortality + γwSw + (µw + γw)Iw︸ ︷︷ ︸ White Oak Mortality . 2. Proportion of White Oaks Define: Pw = Sw + Iw N , Pr = 1− Pw = Sr + Ir N . 3. Simplifying Assumptions To facilitate the analysis, we make the following assumptions: 1. Long-Term Equilibrium: The system reaches a steady state where all derivatives are zero. 2. Negligible Infection in White Oaks: Iw ≈ 0 due to lower transmission and mortality rates. 3. Population at Carrying Capacity: N = K. 4. Mortality at Equilibrium Under these assumptions, the total mortality simplifies to: M = γrSr + µrIr + γwSw. Since N = K and Iw ≈ 0, we have: Sr + Ir + Sw = K. 5. Express Variables in Terms of Proportions Express Sr, Ir, Sw in terms of Pw: Sw = PwK, Sr + Ir = (1− Pw)K. 30 6. Red Oak Infection Dynamics at Equilibrium From equation (2) at equilibrium (dIrdt = 0): (βrr + βbr) SrIr K = µrIr. Solving for Sr: Sr = µrK βrr + βbr . Total red oak population: Ir = (1− Pw)K − Sr = (1− Pw)K − µrK βrr + βbr . 7. Total Mortality in Terms of Pw Substitute Sr, Ir, Sw into M : M = γrSr + µrIr + γwSw = γr ( µrK βrr + βbr ) + µr [ (1− Pw)K − µrK βrr + βbr ] + γwPwK = [ γrµrK βrr + βbr − µ2 rK βrr + βbr ] + µr(1− Pw)K + γwPwK = µrK(γr − µr) βrr + βbr + µr(1− Pw)K + γwPwK. 8. Compute the Derivative of M with Respect to Pw Differentiate M with respect to Pw: dM dPw = d dPw [ µrK(γr − µr) βrr + βbr + µr(1− Pw)K + γwPwK ] = −µrK + γwK = K(γw − µr). 9. Sign of the Derivative Given µr > γw (from the assumptions), we have: γw − µr < 0 =⇒ dM dPw < 0. 10. Conclusion Since dM dPw < 0, we conclude that the total mortality M decreases as the pro- portion of white oaks Pw increases. Therefore, a higher proportion of white oaks is negatively correlated with total mortality in the long term. 31 Remarks • Interpretation: Increasing white oak proportion reduces total mortality due to their lower mortality rates compared to red oaks. • Assumption Validity: The proof relies on the assumption that Iw ≈ 0, which holds when white oaks have low infection rates. • Parameter Significance: The inequality µr > γw is crucial for the negative derivative. 4.4 Sensitivity Analysis of Beetle Transmission Including Climate Change Effects Climate change significantly heightens the risk of plant pathogen outbreaks by driving their evo- lution, expanding their geographic ranges, and intensifying host–pathogen interactions. These increases threaten global food security and environmental sustainability by reducing agricultural productivity and biodiversity. Effective monitoring and adaptive management strategies are essen- tial to mitigate these risks and ensure long-term ecosystem resilience [18] Beetle-Mediated Transmission • Beetle-mediated transmission rates (βbr, βbw) represent the rate at which beetles trans- mit the disease to red and white oaks, respectively. • Climate change can influence beetle populations by affecting their reproduction, survival, and activity patterns, potentially increasing transmission rates [18]. • Assumptions: Both βbr, βbw will increase [18] Mathematical Sensitivity Analysis Key Outputs to Analyze • Total mortality M . • Infected red oak population Ir. • Infected white oak population Iw. We calculate the partial derivatives of M, Ir, Iw with respect to βbr and βbw. 32 A. Sensitivity of Total Mortality to βbr At equilibrium, dIr dt = 0, so from equation (2): (βrr + βbr) SrIr N = µrIr =⇒ Sr N = µr βrr + βbr . Total mortality M includes: M = γrSr + µrIr + γwSw + (µw + γw)Iw. Assuming Iw ≈ 0 (white oaks are less affected), M simplifies to: M ≈ γrSr + µrIr + γwSw. Express Sr and Ir in terms of βbr: • As βbr increases, βrr + βbr increases, leading to: Sr N = µr βrr + βbr =⇒ Sr ∝ 1 βrr + βbr . • Ir = (1− Pw)N − Sr, where Pw is the proportion of white oaks. Compute the partial derivative: ∂M ∂βbr = ∂ ∂βbr (γrSr + µrIr) = γr ∂Sr ∂βbr + µr ∂Ir ∂βbr . Since Ir depends on Sr, we can express ∂Ir ∂βbr in terms of ∂Sr ∂βbr . B. Sensitivity of Ir to βbr From the equilibrium condition: Ir = ((1− Pw)N)− Sr. Compute ∂Ir ∂βbr : ∂Ir ∂βbr = − ∂Sr ∂βbr . So, ∂M ∂βbr = γr ∂Sr ∂βbr + µr ( − ∂Sr ∂βbr ) = (γr − µr) ∂Sr ∂βbr . 33 Given γr < µr (since disease-induced mortality µr is typically higher than natural mortality γr), we have γr − µr < 0. Also, ∂Sr ∂βbr = − µrN (βrr+βbr)2 < 0. Therefore, ∂M ∂βbr = (γr − µr) ( − µrN (βrr + βbr)2 ) > 0. Interpretation • Positive Sensitivity: An increase in βbr leads to an increase in total mortality M . • Magnitude of Change: The larger the value of βbr, the greater the increase in total mor- tality. Numerical Illustration Parameter Values To illustrate, we assign hypothetical values (units are arbitrary): br = 0.5, bw = 0.4, γr = 0.1, γw = 0.05, µr = 0.3, µw = 0.1, βrr = 0.4, βbr = 0.2, βbw = 0.1, K = 1000. Scenarios • Baseline Scenario: Current βbr = 0.2. • Climate Change Scenario: Increased βbr = 0.4 (doubling due to climate effects). Results Compute the equilibrium values of Sr, Ir,M for both scenarios. Baseline Scenario 1. Compute βtotal = βrr + βbr = 0.4 + 0.2 = 0.6. 2. Sr N = µr βtotal = 0.3 0.6 = 0.5. 3. Sr = 0.5N = 0.5× 1000 = 500. 34 4. Assuming Pr = 0.6, N = K = 1000: Ir = (PrN)− Sr = (0.6× 1000)− 500 = 600− 500 = 100. 5. Total mortality: M = γrSr + µrIr + γwSw = 0.1× 500 + 0.3× 100 + 0.05× 400 = 50 + 30 + 20 = 100. Climate Change Scenario 1. New βtotal = βrr + βbr = 0.4 + 0.4 = 0.8. 2. Sr N = µr βtotal = 0.3 0.8 = 0.375. 3. Sr = 0.375N = 0.375× 1000 = 375. 4. Ir = (0.6× 1000)− 375 = 600− 375 = 225. 5. Total mortality: M = 0.1× 375 + 0.3× 225 + 0.05× 400 = 37.5 + 67.5 + 20 = 125. Analysis of Results • Increase in Total Mortality: M increased from 100 to 125 due to the increase in βbr. • Increase in Infected Red Oaks: Ir increased from 100 to 225. • Decrease in Susceptible Red Oaks: Sr decreased from 500 to 375. This demonstrates that an increase in beetle-mediated transmission due to climate change can significantly increase total mortality and the number of infected red oaks. Implications for Oak Populations Red Oaks • Higher Vulnerability: Increased βbr leads to higher infection rates and mortality in red oaks. • Population Decline: May result in a significant reduction of red oak populations. 35 White Oaks • Indirect Effects: Although βbw may also increase, white oaks have lower susceptibility and mortality rates. • Potential Increase in Proportion: As red oaks decline, white oaks may become more dominant in the population. Conclusion The sensitivity analysis indicates that beetle-mediated transmission parameters (βbr, βbw) have a considerable impact on the dynamics of the oak population model. Climate change, by increasing beetle activity and transmission rates, can exacerbate disease spread and lead to higher mortality, particularly in red oaks. Chapter 5 Discussion and Conclusion This chapter presents a mathematical modeling analysis that illuminates how the ratio of white to red oaks within a forest influences the dynamics of oak wilt disease. Our central finding is that increasing the proportion of white oaks effectively curbs both the spread and overall mortality associated with oak wilt. Whereas red oaks are highly susceptible and facilitate rapid transmis- sion, white oaks possess structural and physiological defenses—such as tyloses—that limit fungal spread within their vascular system. As a result, forests with greater white oak representation are inherently more resilient to this pathogen, showcasing the protective role of species diversity in forest ecosystems. This work also emphasizes importance of climate change disturbances likelihood of increasing beetle mediated transmission which is a main contributor of creating new pockets of disease. Disease made it is way to Canada just a year ago and threatens oak communities if not understood and controlled effectively [8]. 5.1 Empirical and Theoretical Support The model’s conclusions align closely with empirical research. Wilson (2005) [19] and Juzwik et al. (2011) [13] both reported lower disease severity in stands richer in white oaks, demonstrating how a higher proportion of less susceptible species can break the continuous chain of vulnerable hosts. Additionally, Cavender-Bares and colleagues have shown that evolutionary adaptations, trait dif- ferentiation, and niche partitioning among oak species enhance forest stability and resilience. These findings affirm that the diversity-driven resistance observed in our model is not only theoretically sound but also reflected in real-world ecological patterns [5], [4],[6]. 36 37 5.2 Insights from the Decline of White Oaks Historical shifts in forest composition provide a vital context for understanding current oak wilt dynamics. Abrams (2003) documents a significant decline in white oak regeneration across North American forests due to factors such as fire suppression, changes in land use, and competition from shade-tolerant species like maples and beeches [1] . Prior to widespread fire suppression—especially early in the 20th century—periodic burns maintained oak dominance by limiting these competitors. The subsequent reduction in white oaks and relative increase in red oaks may have set the stage for more severe oak wilt epidemics, as a higher abundance of susceptible hosts can accelerate pathogen spread. This historical lens suggests a preliminary hypothesis: oak wilt’s current prominence may partially stem from shifts in species composition triggered by earlier management and land-use decisions that reduced the natural, protective role of white oaks 5.1. Figure 5.1: This illustration has been created by author based on information provided by Abrams [1] 5.3 Implications for Forest Management Recognizing the critical influence of species composition on oak wilt dynamics offers practical guidance for forest managers. By prioritizing white oak regeneration—through selective planting, controlled burns to reduce competition, and careful thinning—managers can create conditions that lower overall susceptibility to oak wilt. Such strategies align with existing recommendations from forest health authorities, which emphasize species diversification to enhance ecological resilience. In areas recovering from oak wilt or vulnerable to future outbreaks, embedding these insights into reforestation efforts can help restore stability and mitigate ongoing disease impacts. There 38 is already work to be carried out in Wisconsin as part of climate adaptation cites and they are planting white oak seedlings where oak wilt pockets are located [12]. 5.4 Limitations and Future Research Directions While our model provides valuable insights, it simplifies certain complexities. It treats the forest as a homogeneously mixed community and does not account for spatial heterogeneity—such as localized host clustering or variable environmental conditions—that can significantly influence dis- ease dynamics. Future work should incorporate spatially explicit models, leveraging emerging data sources like high-resolution satellite imagery and remote sensing tools. Such data could reveal fine- scale patterns of tree distribution, canopy structure, and species composition, enabling researchers to simulate oak wilt spread with greater ecological realism. Moreover, introducing stochastic factors and integrating climate variables could yield more nuanced predictions. Long-term studies could examine how compositional changes, influenced by both natural succession and management practices, affect forest productivity, habitat quality, and resilience to environmental stressors over time. By combining robust ecological modeling, spatial analyses, and evolutionary insights, future research can deepen our understanding of complex forest disease dynamics in a rapidly changing world. 5.5 Conclusions This thesis integrates mathematical modeling, ecological evidence, and evolutionary perspectives to underscore the importance of species composition in determining oak wilt outcomes. We show that increasing the proportion of white oaks can reduce disease transmission and mortality, thereby buffering forests against widespread pathogen impacts. These results reinforce the broader principle that biodiversity, far from being a passive attribute, actively contributes to ecosystem stability. Looking ahead, leveraging spatial data and advanced modeling techniques will further refine our understanding of oak wilt dynamics and guide evidence-based management strategies. By acknowledging historical changes in species composition and proactively fostering tree diversity, we can enhance the resilience of oak forests. Ultimately, this research supports a comprehensive approach—rooted in both past lessons and modern technologies—that will help maintain healthy, stable forest ecosystems in the face of ongoing environmental challenges. Chapter 6 Data and Code 6.1 DATA [11] Table 6.1: Total mortality count/percent counts for red oak and white oak group trees Tree Species Average DBH (cm) Number of Trees Percentage of Total (%) Black oaks (Red Oak Group) 11.5 780 66.8 Red oaks (Red Oak Group) 14.9 254 21.8 Northern pin oaks (Red Oak Group) 10.4 56 4.8 Shingle oaks (Red Oak Group) 4.7 3 0.3 White oaks (White Oak Group) 11.1 53 4.5 Bur oaks (White Oak Group) 12.1 21 1.8 Total — 1,167 100.0 6.2 Bifurcation Diagram Code The following Python script generates a bifurcation diagram illustrating the equilibrium number of infected red oaks (I∗r ) as the proportion of white oaks (ρw) varies for different transmission rates (βr): import numpy as np import matplotlib.pyplot as plt 39 40 # Parameters beta_r_values = [0.7, 0.8, 0.9] mu_r = 0.4 gamma_r = 0.05 N_total = 1000 colors = [’blue’, ’green’, ’red’] # Range of rho_w values rho_w_values = np.linspace(0, 1, 500) # Function to compute equilibrium I_r* def compute_Ir_star(rho_w, beta_r): rho_r = 1 - rho_w R0 = (beta_r * rho_r) / (mu_r + gamma_r) if R0 <= 1: return 0 # Disease cannot invade else: # Endemic equilibrium Ir_star = (N_total * (R0 - 1)) / R0 return Ir_star # Initialize plot plt.figure(figsize=(12, 8)) for beta_r, color in zip(beta_r_values, colors): rho_w_critical = 1 - (mu_r + gamma_r) / beta_r Ir_stars = [compute_Ir_star(rho_w, beta_r) for rho_w in rho_w_values] # Determine stability: assume disease-free is stable below critical, endemic is stable above stability = [’stable’ if rho_w <= rho_w_critical else ’stable’ for rho_w in rho_w_values] # Plot equilibrium plt.plot(rho_w_values, Ir_stars, label=f’$\beta_r$ = {beta_r}’, color=color) # Mark bifurcation point 41 plt.axvline(x=rho_w_critical, color=color, linestyle=’--’, alpha=0.7, label=f’Bifurcation $\rho_w^*$ = {rho_w_critical:.2f}’) # Plot aesthetics plt.xlabel(’Proportion of White Oaks ($\rho_w$)’, fontsize=14) plt.ylabel(’Equilibrium Infected Red Oaks ($I_r^*$)’, fontsize=14) plt.title(’Bifurcation Diagram of $I_r^*$ vs. $\rho_w$ for Different $\beta_r$’, fontsize=16) plt.legend(fontsize=12) plt.grid(True) plt.ylim(0, N_total * 0.6) # Adjust y-axis limit for better visibility plt.xlim(0, 1) plt.tight_layout() plt.show() 6.3 Total Mortality VS White Oak ratio code The following Python script simulates the oak wilt disease model, analyzing the impact of varying proportions of white oaks on total mortality and total infections within the forest ecosystem: # Import necessary libraries import numpy as np import matplotlib.pyplot as plt from scipy.integrate import solve_ivp # ------------------------------- # Model Parameters with Justifications # ------------------------------- # Transmission Rates (beta) # beta_rr: Transmission rate among red oaks via root grafts # Set to 0.3, reflecting moderate root graft transmission efficiency among red oaks. beta_rr = 0.2 # beta_br: Beetle transmission rate to red oaks # Set to 0.01, indicating that beetle transmission to red oaks is relatively low compared to root graft transmission. 42 beta_br = 0.02 # beta_bw: Beetle transmission rate to white oaks # Set to 0.1, representing moderate beetle transmission to white oaks, acknowledging their susceptibility via beetle vectors. beta_bw = 0.05 # Mortality Rates (mu) # mu_r: Mortality rate of infected red oaks # Set to 0.3, indicating that infected red oaks die at a moderate rate. mu_r = 0.04 # mu_w: Mortality rate of infected white oaks # Set to 0.05, reflecting a low mortality rate for infected white oaks due to their partial resistance. mu_w = 0.01 # Total number of trees in the forest N = 10000 # ------------------------------- # Function to Simulate the Model # ------------------------------- def simulate_model(rho_w): """ Simulate the oak wilt disease model for a given proportion of white oaks (rho_w). Parameters: - rho_w: Proportion of white oaks in the forest (between 0 and 1). Returns: - D_total: Total mortality at the end of the simulation. - total_infections: Total number of infections during the simulation. """ rho_r = 1 - rho_w # Proportion of red oaks 43 # Initial conditions # Start with all susceptible trees except for one infected red oak S_r0 = rho_r * (N - 1) I_r0 = 20 # One initial infected red oak D_r0 = 0 # No dead red oaks initially S_w0 = rho_w * N # All white oaks are initially susceptible I_w0 = 5 # No initial infected white oaks D_w0 = 0 # No dead white oaks initially y0 = [S_r0, I_r0, D_r0, S_w0, I_w0, D_w0] # Time span for simulation (e.g., 0 to 50 time units) t_span = [0, 120] t_eval = np.linspace(t_span[0], t_span[1], 500) # Define the ODE system def oakwilt_model(t, y): S_r, I_r, D_r, S_w, I_w, D_w = y N_alive = S_r + I_r + S_w + I_w # Total living trees # Avoid division by zero if N_alive == 0: N_alive = 1e-6 # Transmission terms # Lambda_r: Force of infection for red oaks # Combines root graft and beetle transmission to red oaks lambda_r = ((beta_rr * I_r) + (beta_br * I_r)) / N_alive # Lambda_w: Force of infection for white oaks # Beetle transmission to white oaks lambda_w = (beta_bw * I_r) / N_alive # ODEs for red oaks dS_r_dt = -lambda_r * S_r dI_r_dt = lambda_r * S_r - mu_r * I_r 44 dD_r_dt = mu_r * I_r # ODEs for white oaks dS_w_dt = -lambda_w * S_w dI_w_dt = lambda_w * S_w - mu_w * I_w dD_w_dt = mu_w * I_w return [dS_r_dt, dI_r_dt, dD_r_dt, dS_w_dt, dI_w_dt, dD_w_dt] # Solve the ODE system sol = solve_ivp(oakwilt_model, t_span, y0, t_eval=t_eval) # Extract the final total mortality and total infections D_total = sol.y[2, -1] + sol.y[5, -1] # Dead red oaks + Dead white oaks total_infections = (sol.y[1, -1] + sol.y[2, -1] + sol.y[4, -1] + sol.y[5, -1]) - I_r0 # Total infections = Final infected and dead trees minus initial infections return D_total, total_infections # ------------------------------- # Run Simulations for Different White Oak Proportions # ------------------------------- # Create an array of white oak proportions from 0 to 1 rho_w_values = np.linspace(0, 1, 20) # Lists to store results D_total_values = [] New_Infections_values = [] # Loop over each white oak proportion and simulate the model for rho_w in rho_w_values: D_total, total_infections = simulate_model(rho_w) D_total_values.append(D_total) New_Infections_values.append(total_infections) # ------------------------------- 45 # Plotting Total Mortality vs. Proportion of White Oaks # ------------------------------- plt.figure(figsize=(8, 6)) plt.plot(rho_w_values, D_total_values, label=’Total Mortality’, color=’blue’, marker=’o’) plt.xlabel(’Proportion of White Oaks ($\\rho_w$)’, fontsize=12) plt.ylabel(’Total Mortality’, fontsize=12) plt.title(’Total Mortality vs. Proportion of White Oaks’, fontsize=14) plt.grid(True) plt.legend() # Adding parameter values to the plot params_text = ( f’$\\beta_{{rr}}$ = {beta_rr}\n’ f’$\\beta_{{br}}$ = {beta_br}\n’ f’$\\beta_{{bw}}$ = {beta_bw}\n’ f’$\\mu_r$ = {mu_r}\n’ f’$\\mu_w$ = {mu_w}’ ) # Positioning the text box plt.text( 0.8, 0.8, params_text, transform=plt.gca().transAxes, fontsize=10, verticalalignment=’top’, bbox=dict(boxstyle=’round’, facecolor=’white’, alpha=0.7) ) plt.tight_layout() plt.show() References [1] Marc D. 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Acknowledgements Dedication Abstract List of Tables List of Figures Introduction Objectives Significance of the Study Structure of the Thesis Literature Review Oak Wilt Pathology Biological Differences Between Red and White Oaks Transmission Mechanisms Previous Mathematical Models Management Strategies Empirical Evidence Supporting Species Composition Effects Gaps in the Literature Simple Model Model Overview Assumptions Model Equations Detailed Explanation of Equations Justification of Model Structure Non-Dimensionalization Analysis of the Model Interpretation of Parameters Sensitivity Analysis Results for Simple Model Preliminary Definitions and Assumptions Proof that Increasing rho_w Decreases Total Mortality Proof that Increasing rho_w Decreases Infection Rate Extended Model Extended Model with Reproduction and background Mortality Computation of the Basic Reproduction Number R0 Total Mortality vs white oak ratio for Extended Model Sensitivity Analysis of Beetle Transmission Including Climate Change Effects Discussion and Conclusion Empirical and Theoretical Support Insights from the Decline of White Oaks Implications for Forest Management Limitations and Future Research Directions Conclusions Data and Code DATA himelickoakwilt Bifurcation Diagram Code Total Mortality VS White Oak ratio code References