Chapter 6 Conclusion and Future Work

1. Summary and Future Works for Chapter 3

Single-pesticide control strategies often failed due to the development of resistance in the pest. As a result, combining pesticide application with biological control agents has emerged as an effective approach to reduce pest abundance while supporting sustainable agricultural practices. In Chapter LABEL:chapter2, we presented a stage-structured host–parasitoid model in which both the host and parasitoid populations are categorized into juvenile and adult developmental stages. We first established the conditions for the existence and stability of the two boundary equilibria, and the local stability conditions for the coexistence of both species, using a bifurcation-theorem approach presented in Chapter LABEL:chapter1.5. We then determined the conditions under which both species persist using persistence theory. Next, we incorporated continuous pesticide spraying into the host-parasitoid model and treated parasitoid species as biological control agents to investigate the effect of multiple control strategies. We analyzed the model through numerical simulations by altering the order of events, such as pesticide spraying, parasitism, maturation, and reproduction, within a unit of time. We observed that pesticide spraying at the end of the time unit is more effective at pest control than spraying at the beginning. Moreover, when the pesticide directly affects parasitoids, increasing in pesticide-induced mortality resulted in larger amounts of pests, particularly when the pesticide is less effective.

Our future works related to this chapter are as follows:

• The stage-structured host-parasitoid model assumes two-species interactions, meaning that hosts are attacked mainly by adult parasitoid species. However, many natural systems involve an intricate web of interacting species. We would like to extend the host-parasitoid model with multiple controls by adding two hosts attacked by a common natural enemy to investigate how a shared natural enemy may influence the coexistence of host species competing both intraspecifically and interspecifically for the same resources. Specifically, we would like to start by examining the effects of apparent competition, a phenomenon in which a natural enemy shared between non-competing host species, with multiple control mechanisms, such as pesticide spraying and biological controls.

• The host-parasitoid model assumes compensatory density dependence in the host reproductions, such as the Beverton-Holt nonlinearity. Incorporating an Allee effect into host reproduction can induce bistability in the host dynamics, making eradication of the host (pest) more difficult and potentially detrimental to invading species ([li2025bistability, tobin2011exploiting]). We would like to extend the host-parasitoid model by incorporating an Allee effect in host reproduction. For Allee effect, we define, $f(X)=\frac{\beta (1+aX)}{1+cX+a{X}^2},$ where $a,c\ge 0$. When $a>c$, a low-density host positively impacts the recruitment.

• Our result of the coexistence of host-parasitoid dynamics is limited to local analysis only. Hence, global stability remains an open question. We would like to analyze the global stability of the coexistence equilibrium of the host -parasitoid model.

2. Summary and Future Works for Chapter 4

The formulation of the stage-structured host-parasitoid model presented in Chapter LABEL:chapter2 may be unrealistic in certain situations, particularly when the model time step corresponds to a single day, such as daily pesticide spraying. In this chapter, we extended the stage-structured host-parasitoid model to an impulsive difference system that included periodic pesticide spraying and supplementation with additional parasitoid species. We established conditions for the global stability of the host-eradication cycles and conditions for uniform robust persistence for both host and parasitoid species. Finally, using numerical simulations, we investigated the periodicity of the control strategies in the host-parasitoid dynamics. Similar to the case of continuous pesticide application (e.g., spraying at every discrete time step), if the pesticide directly impacted parasitoids, higher pesticide-induced mortality may resulted in larger pest populations in the impulsive model, particularly when the pesticide is relatively ineffective. In addition, the effectiveness of the control strategies may depended on the initial conditions; consequently, the same control strategy can result in either pest outbreaks or persistence.

Our future research plans for this chapter are:

• While the host-parasitoid impulsive model focuses on the dynamics of the host eradication cycle we are also interested in studying how the control strategies can affect the coexistence of both species when control measures are applied periodically. Specifically, using a bifurcation-theorem approach, we would like to investigate the local stability of the coexistence cycles under a periodic multiple-control mechanism.

• The impulsive host-parasitoid is also a two-species interacting model. We would also like to extend the host-parasitoid impulsive model by incorporating apparent competition or direct competition on the host species.

3. Summary and Future Works for Chapter 5

In disease dynamics, vaccination is widely recognized as one of the most effective control strategies for reducing the spread of infection within a population. By increasing the proportion of immune individuals, vaccination lowers the number of susceptible individuals available for transmission, thereby decreasing the overall infection rate. However, when vaccination is only partially effective or provides limited protection, the epidemiological outcome may become worse. We developed and analyzed two discrete-time SIS models with vaccination, in which vaccination reduces the likelihood of infection but does not fully eliminate it. The two models differ in the order in which vaccination and disease transmission take place within a discrete time unit. We established a condition for global stability of the disease-free equilibrium and the determined conditions for the existence of backward bifurcation when the disease-free equilibrium destabilized at $R_0=1$ by an extension of the Fundamental Bifurcation Theorem. The two models exhibited significant differences in their outcomes. In particular, when vaccination occurs before, we found that disease recovery, individual returning to the susceptible class, and partial vaccine efficacy are necessary for backward bifurcation. In contrast, when disease transmission occurred before the vaccination, we observed backward bifurcation even if the vaccination was completely effective.

Our future projects regarding this chapter lie in the following:

• The discrete-time SIS model with vaccinations assumes partially effective and completely effective vaccination to investigate infection dynamics, but not disease-induced mortality. We would like to extend the model with disease-induced mortality and a limited treatment resource to understand the effect of limited treatment on the infection dynamics, which may produce backward bifurcation in the endemic equilibria (e.g., [wang2009backward, guerrero2018different, wang2006backward]).

• The discrete-time SIS model with vaccination addresses only the local stability around the disease-free equilibrium. However, conditions for the general local stability have not yet been established. Therefore, we aim to analyze the general local stability of the infection dynamics using the Jury condition for the SISV model.

• In the discrete-time SISV model, we focus only on the equilibrium dynamics. However, the compartmental epidemic model can exhibit rich dynamics, such as two- and four-cycle dynamics and a route to chaos. Hence, in the future, we could analyze the interior dynamics of the compartmental discrete-time epidemic models with a demographic two-cycle by the bifurcation theorem approach.

• The compartmental epidemic models are limited to local analysis. We would like to analyze the endemic dynamics using the nullcline approach method to obtain the global result for some specific epidemic models ([ackleh2021nullcline]).

Future research on backward bifurcation will help better understand infection dynamics.