Modelling Rabies Transmission with Vaccination: Incorporating Pharmaceutical and Particle Processing for Pre-Exposure Prophylaxis Optimization ^†

Abstract

Rabies remains a persistent zoonotic threat, particularly in regions where domestic dogs are the main source of human and animal infections. This mathematical model studies the dynamics of rabies transmission between canine populations (dog-to-dog) and from canines to humans (dog-to-human). The model incorporates susceptible, infected, and vaccinated compartments for both species, with pre-exposure vaccination as the key control strategy. Processes such as encapsulation, stability enhancement, and controlled release are modelled as parameters influencing vaccination rates in both dogs and humans. Specifically, the model introduces processing-dependent vaccination functions that reflect improved bioavailability, immunogenicity, and delivery efficiency due to advanced formulation techniques. This interdisciplinary approach bridges mathematical epidemiology and pharmaceutical technology. Earlier rabies models focus on transmission and static vaccination, often ignoring vaccine formulation and delivery. Our current work fills this gap by incorporating pharmaceutical and particle engineering parameters into the vaccination terms of the model, thereby providing a more comprehensive framework for optimizing rabies control strategies in endemic regions. Positivity and boundedness analyses confirm that all model variables remain biologically feasible and bounded over time. Stability analysis identifies thresholds for disease elimination or persistence. Numerical simulations show that enhancing pharmaceutical parameters increases vaccination impact, reducing peak infection prevalence in dogs from 18% to 5% and in humans from 4% to 0.8%, and shortening elimination time from 8 years to 3 years. Formulations with controlled release and improved stability maintain over 90% reduction in transmission for more than 5 years, compared to 60% over 3 years for conventional vaccines. This will ensure that the model’s predictions are validated against realistic conditions and can effectively guide rabies control strategies.

1. Introduction

Rabies remains one of the most fatal yet preventable zoonotic diseases, claiming thousands of human lives annually, predominantly in developing regions where canine rabies is endemic. The rabies virus (RABV) is a single-stranded, neurotropic RNA virus belonging to the Rhabdoviridae family, genus Lyssavirus. It causes acute, progressive encephalitis in humans and animals. Rabies is endemic [1] in more than 150 countries, with the highest burden in Asia and Africa. India alone accounts for approximately 36% of global rabies deaths, and the Health and Family Welfare Department has reported nearly 3.1 lakh dog-bite incidents in Karnataka between January and August 2025. The first human rabies vaccination was successfully administered by Louis Pasteur in 1885 to Joseph Meister, a nine-year-old boy bitten multiple times by a rabid dog [2]. Diagnostic techniques have advanced from the Mouse Inoculation Test to modern assays such as the Direct Fluorescent Antibody Test (dFA), Enzyme-Linked Immunosorbent Assay (ELISA), and Polymerase Chain Reaction (PCR).

Sahusilawane et al. [3] identified a research gap: this model involves pharmaceutical and particle processing in vaccination. The present work addresses this gap by developing a nonlinear differential model that integrates pharmaceutical and particle engineering factors—such as encapsulation efficiency, stability enhancement, and controlled release—into the vaccination dynamics. The model considers susceptible, infected, vaccinated, and quarantined dog populations, and examines dog-to-dog and dog-to-human transmission pathways. Using the basic reproduction number ($R_0$) and Routh–Hurwitz stability criteria [4,5], the existence and stability conditions of disease-free and endemic equilibria are analyzed.

2. Mathematical Model

The nonlinear dynamical system is modeled with a Crowley–Martin type functional response and a vaccination component modified by pharmaceutical processing enhancements,

Dog to dog transmission

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_d}{dt}}& ={\Lambda }_d-{\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}}-{\nu }_d{E}_d{\sigma }_d{R}_d{S}_d-{\mu }_d{S}_d,\hfill \\ \hfill {\displaystyle \frac{d{V}_d}{dt}}& ={\nu }_d{E}_d{\sigma }_d{R}_d{S}_d-{\mu }_d{V}_d,\hfill \\ \hfill {\displaystyle \frac{d{I}_d}{dt}}& ={\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}}-{\alpha }_d{I}_d\hfill \end{array}\right.$$

(1)

Dog to human transmission

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_h}{dt}}& ={\Lambda }_h-{\displaystyle \frac{{\beta }_{dh}{S}_h{I}_d}{1+a_{dh}{S}_d+b_{dh}{I}_d}}-{\nu }_h{E}_h{\sigma }_h{R}_h{S}_h-{\mu }_h{S}_h,\hfill \\ \hfill {\displaystyle \frac{d{V}_h}{dt}}& ={\nu }_h{E}_h{\sigma }_h{R}_h{S}_h-{\mu }_h{V}_h,\hfill \\ \hfill {\displaystyle \frac{d{I}_h}{dt}}& ={\displaystyle \frac{{\beta }_{dh}{S}_h{I}_d}{1+a_{dh}{S}_d+b_{dh}{I}_d}}-{\alpha }_h{I}_h\hfill \end{array}\right.$$

(2)

The above system of equation describes the transmission dynamics of rabies between dogs and humans. The dog–to–dog transmission occurs within the reservoir population, while the dog–to–human transmission occurs through contact with infectious dogs. In both subsystems, $S_x$, $V_x$, and $I_x$ ($x=d,h$) denote the susceptible, vaccinated, and infected populations, respectively. Recruitment takes place at a constant rate ${\Lambda }_x$, natural mortality rate at ${\mu }_x$. The nonlinear incidence terms ${\displaystyle \frac{{\beta }_{xy}{S}_y{I}_d}{1+a_{xy}{S}_d+b_{xy}{I}_d}}$ follow the Crowley–Martin functional response, which shows how the contact rate reduces at a higher population densities due to behavioral interference between individuals.

The vaccination terms ${\nu }_x{E}_x{\sigma }_x{R}_x{S}_x$ describe the rate at which susceptible individuals gain immunity through pharmaceutical-enhanced vaccination. Here, $E_x$ represents the vaccine’s effective bioavailability or immunogenicity, while ${\sigma }_x$ denotes the stability factor of the formulation that helps maintain its potency over time. The parameter $R_x$ accounts for the controlled-release or encapsulation effect, capturing advanced delivery mechanisms that extend antigen exposure and sustain immune protection. Together, these parameters modify the baseline vaccination rate ${\nu }_x$, reflecting how modern pharmaceutical processes—such as nanoparticle encapsulation, adjuvant formulation, and stabilization technologies—improve both the strength and duration of vaccine-induced immunity [6]. The parameters used in the rabies model are listed in

Table 1. Parameter descriptions with dimensions.

Parameter	Biological Meaning	Dimension
${\Lambda }_d$	Recruitment coefficient of dogs	time−1
${\Lambda }_h$	Recruitment coefficient of humans	time−1
${\beta }_{dd}$	Dog-to-dog transmission rate coefficient	population−1 time−1
${\beta }_{dh}$	Dog-to-human transmission rate coefficient	population−1 time−1
$a_{dd},b_{dd}$	Saturation in dog-to-dog transmission	population−1
$a_{dh},b_{dh}$	Saturation in dog-to-human transmission	population−1
$\mu$	Natural mortality coefficient	time−1
$\alpha$	Rabies-induced death coefficient	time−1
${\nu }_d,{\nu }_h$	Vaccination rate coefficients	time−1
$E_d,E_h$	Encapsulation effectiveness	dimensionless
${\sigma }_d$, ${\sigma }_h$	Vaccine stability factor	dimensionless
$R_d,R_h$	Controlled release factor	dimensionless

.

3. Well-Posedness of the Solution

Theorem 1.

Consider the nonlinear system of differential equations

$${\displaystyle \frac{dx}{dt}}=F\left(x\left(t\right)\right),x\left(0\right)=x_0\in {\mathbb{R}}_{+}^n,$$

(3)

where

$$x\left(t\right)={\left[\begin{array}{c}{S}_d\left(t\right),I_d\left(t\right),V_d\left(t\right),S_h\left(t\right),I_h\left(t\right),V_h\left(t\right)\end{array}\right]}^T.$$

The system admits a unique and continuously differentiable solution

$$x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right))$$

on a finite time interval $t\in [0,T)$ for some $T>0$.

Proof. 

The local existence and uniqueness of the solution are verified using the Picard–Lindelöf theorem.

(i) The nonlinear function $F\left(x\right)$ includes infection terms of the form

$${\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}},{\displaystyle \frac{{\beta }_{dh}{S}_h{I}_d}{1+a_{dh}{S}_d+b_{dh}{I}_d}},$$

which follow the Crowley–Martin functional response [7]. These expressions are continuous for all $S_d,I_d,S_h,I_h\ge 0$, ensuring that $F\left(x\right)$ is continuous within the biologically feasible region ${\mathbb{R}}_{+}^n$.

(ii) The model also enhances pharmaceutical parameters such as $E_x$, ${\sigma }_x$, and $R_x$, denoting effective bioavailability, formulation stability, and controlled-release characteristics of vaccine particles. These quantities appear in the vaccination rate functions and this remains as smooth, bounded functions of the state variables. This preserves the differentiability of $F\left(x\right)$. In a mathematically consistent manner, the system represents the behavior of real formulations.

(iii) Because each component of $F\left(x\right)$ is a rational function with differentiable numerator and denominator, the partial derivatives ${\displaystyle \frac{\partial f_i}{\partial x_j}}$ exist and remain bounded on any compact subset $D\subset {\mathbb{R}}_{+}^n$. $F\left(x\right)$ satisfies the local Lipschitz condition on D. □

Remark 1.

The pharmaceutical and particle-processing parameters, $E_x$, $S_x$, and $R_x$, can be quantitatively associated with real formulation data. For instance, $E_x$ corresponds to the experimentally measured vaccine bioavailability or immunogenicity; ${\sigma }_x$ is linked to formulation stability derived from encapsulation kinetics; and $R_x$ represents controlled-release rates obtained from in vitro particle-release.

4. Positivity and Boundedness

Theorem 2.

The system $\left(1\right)$ and $\left(2\right)$ has only positive solutions in ${\mathbb{R}}_{+}^6$ for all $t\ge 0$ (i.e.) The solution $\left[S_d\left(t\right)I_d\left(t\right)V_d\left(t\right)S_h\left(t\right)I_h\left(t\right)V_h\left(t\right)\right]$ lies in the positive orthant of ${R_{+}}^6$.

Proof. 

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_d}{dt}}& ={\Lambda }_d\ge 0\left(Sinceallthetermsunder{S}_{d}vanish\right);\hfill \\ \hfill {\displaystyle \frac{d{V}_d}{dt}}& =0\left(Sinceallthetermsunder{V}_{d}vanish\right);\hfill \\ \hfill {\displaystyle \frac{d{I}_d}{dt}}& ={\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}}\ge 0(Since{S}_d,I_d\ge 0);\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_h}{dt}}& ={\Lambda }_h\ge 0\left(Sinceallthetermsunder{S}_{h}vanish\right);\hfill \\ \hfill {\displaystyle \frac{d{V}_h}{dt}}& =0\left(Sinceallthetermsunder{V}_{h}vanish\right);\hfill \\ \hfill {\displaystyle \frac{d{I}_h}{dt}}& ={\displaystyle \frac{{\beta }_{dh}{S}_h{I}_d}{1+a_{dh}{S}_d+b_{dh}{I}_d}}\ge 0(Since{S}_h,I_h\ge 0);\hfill \end{array}$$

(5)

The system remains within the feasible region ${\mathbb{R}}_{+}^6$. Hence the state variables are positive for all $t\ge 0$. □

Theorem 3.

The system $\left(1\right)$ and $\left(2\right)$ admits solution that are bounded within the region ${R_{+}}^6$.

The normalized Canine population is $N_d\left(t\right)=S_d+V_d+I_d$

The normalized population of humans is $N_h\left(t\right)=S_h+V_h+I_h$

$$N_d\left(t\right)\le N_d^{max},N_h\left(t\right)\le N_h^{max},\forall t\ge 0,$$

where $N_d^{max}={\displaystyle \frac{{\Lambda }_d}{{\mu }_d}}$ and $N_h^{max}={\displaystyle \frac{{\Lambda }_h}{{\mu }_h}}$.

Proof. 

Dog population yields,

$${\displaystyle \frac{d{N}_d}{dt}}={\Lambda }_d-{\mu }_d{N}_d\left(t\right)$$

$$N_d\left(t\right)\le {\displaystyle \frac{{\Lambda }_d}{{\mu }_d}}=N_d^{max}$$

Human population yields,

$${\displaystyle \frac{d{N}_h}{dt}}={\Lambda }_h-{\mu }_h{N}_h\left(t\right)$$

$$N_h\left(t\right)\le {\displaystyle \frac{{\Lambda }_h}{{\mu }_h}}=N_h^{max}$$

Therefore, the system is bounded in the region ${R_{+}}^6$. □

Pharmaceutical and Particle-Processing Perspective: The system remains bounded even when pharmaceutical-enhancement terms such as ${\nu }_x{E}_x{\sigma }_x{R}_x{S}_x$$(x=d,h)$ are introduced. The parameters such as bioavailability $\left(E_x\right)$, formulation stability ${\sigma }_x$, and release-rate constant $\left(R_x\right)$ are biologically constrained. The particle-processing factors (e.g., encapsulation efficiency, polymer degradation rate, or thermal stability index) ensures that $E_x$, ${\sigma }_x$, and $R_x$ remain positive and bounded. Hence, the overall system remains biologically feasible and mathematically bounded in ${\mathbb{R}}_{+}^6$ for all $t\ge 0$.

5. Equilibrium Points

5.1. Trivial Equilibrium

The trivial equilibrium $E_1=(0,0,0,0,0,0)$ exists only when ${\Lambda }_d=0$ and ${\Lambda }_h=0$, indicating the absence of both canine and human populations.

5.2. Disease-Free Equilibrium

When infection does not persist, we have $I_d=0$ and $I_h=0$.

Dog subsystem:

$$S_d^{*}={\displaystyle \frac{{\Lambda }_d}{K_d}},V_d^{*}={\displaystyle \frac{{\kappa }_d}{{\mu }_d}}{S}_d^{*}={\displaystyle \frac{{\kappa }_d{\Lambda }_d}{{\mu }_d{K}_d}}.$$

Since

$${\kappa }_d={\nu }_d{E}_d{\sigma }_d{R}_d,K_d={\kappa }_d+{\mu }_d,$$

the parameter ${\kappa }_d$ represents the overall vaccination rate enhanced by pharmaceutical and particle-processing effects, where $E_d$ quantifies the vaccine’s effective bioavailability or immunogenicity, ${\sigma }_d$ denotes the stability of the formulation (for example, through improved excipient compatibility or nanoparticle encapsulation), and $R_d$ represents the sustained-release efficiency arising from the controlled particle matrix design.

Human subsystem:

$$S_h^{*}={\displaystyle \frac{{\Lambda }_h}{K_h}},V_h^{*}={\displaystyle \frac{{\kappa }_h}{{\mu }_h}}{S}_h^{*}={\displaystyle \frac{{\kappa }_h{\Lambda }_h}{{\mu }_h{K}_h}}.$$

Since

$${\kappa }_h={\nu }_h{E}_h{\sigma }_h{R}_h,K_h={\kappa }_h+{\mu }_h,$$

the parameters ${\kappa }_h$, $E_h$, and ${\sigma }_h$ similarly reflect the impact of formulation stability and pharmaceutical carrier properties in enhancing vaccination among humans.

Thus, the disease-free equilibrium (DFE) is

$$E_1=\left(S_d^{*},V_d^{*},I_d^{*},S_h^{*},V_h^{*},I_h^{*}\right)=\left({\displaystyle \frac{{\Lambda }_d}{K_d}},{\displaystyle \frac{{\kappa }_d{\Lambda }_d}{{\mu }_d{K}_d}},0,{\displaystyle \frac{{\Lambda }_h}{K_h}},{\displaystyle \frac{{\kappa }_h{\Lambda }_h}{{\mu }_h{K}_h}},0\right).$$

At this equilibrium, the disease is eradicated in both species, and the steady-state immunity is sustained through efficient vaccine delivery and particle-based pharmaceutical enhancements.

5.3. Endemic Equilibrium

From the infected dog equation, we obtain

$$S_d^{*}={\displaystyle \frac{{\alpha }_d(1+b_{dd}{I}_d^{*})}{D_d}},(\mathrm{where}{D}_d={\beta }_{dd}-{\alpha }_d{a}_{dd}),$$

which holds only when $D_d\ne 0$. From the susceptible dog equation,

$$I_d^{*}={\displaystyle \frac{{\Lambda }_d{D}_d-{\alpha }_d{K}_d}{{\alpha }_d\left(D_d+K_d{b}_{dd}\right)}},(\mathrm{since}{\Lambda }_d={\alpha }_d{I}_d^{*}+K_d{S}_d^{*}),$$

valid only when $D_d+K_d{b}_{dd}\ne 0$. From the vaccinated dog equation,

$$V_d^{*}={\displaystyle \frac{{\kappa }_d}{{\mu }_d}}{S}_d^{*},$$

where ${\kappa }_d={\nu }_d{E}_d{\sigma }_d{R}_d$ captures the pharmaceutical-enhanced vaccination rate, combining vaccine bioavailability ($E_d$), formulation stability (${\sigma }_d$), and sustained antigen release ($R_d$) from particle-processing or encapsulation effects. These factors determine how effectively vaccination reduces canine infection persistence at equilibrium. From $I_d^{*}$ and $S_d^{*}$, the human subsystem gives

$${\Lambda }_h={\displaystyle \frac{{\beta }_{dh}{S}_h^{*}{I}_d^{*}}{1+a_{dh}{S}_d^{*}+b_{dh}{I}_d^{*}}}+K_h{S}_h^{*},$$

which leads to a quadratic in $S_h^{*}$:

$$A_h{\left(S_h^{*}\right)}^2+B_h{S}_h^{*}+C_h=0,$$

where

$$A_h=K_h{a}_{dh},B_h={\beta }_{dh}{I}_d^{*}+K_h(1+b_{dh}{I}_d^{*})-{\Lambda }_h{a}_{dh},C_h=-{\Lambda }_h(1+b_{dh}{I}_d^{*}).$$

Hence,

$$S_h^{*}={\displaystyle \frac{-B_h\pm \sqrt{B_h^2-4A_h{C}_h}}{2A_h}},I_h^{*}={\displaystyle \frac{{\beta }_{dh}{S}_h^{*}{I}_d^{*}}{{\alpha }_h(1+a_{dh}{S}_d^{*}+b_{dh}{I}_d^{*})}},V_h^{*}={\displaystyle \frac{{\kappa }_h}{{\mu }_h}}{S}_h^{*},$$

subject to ${\Delta }_h=B_h^2-4A_h{C}_h\ge 0$ and $S_h^{*}>0$. When $a_{dh}=0$, $A_h=0$, and the quadratic becomes linear:

$$S_h^{*}={\displaystyle \frac{{\Lambda }_{h}D}{{\beta }_{dh}{I}_d^{*}+K_{h}D}},I_h^{*}={\displaystyle \frac{{\beta }_{dh}{S}_h^{*}{I}_d^{*}}{{\alpha }_{h}D}},V_h^{*}={\displaystyle \frac{{\kappa }_h}{{\mu }_h}}{S}_h^{*}.$$

5.4. Dog Endemic and Human Disease-Free Equilibrium

$$E_d^{*}=\left(S_d^{*},V_d^{*},I_d^{*},S_h^{*}={\displaystyle \frac{{\Lambda }_h}{K_h}},V_h^{*}={\displaystyle \frac{{\kappa }_h}{{\mu }_h}}·{\displaystyle \frac{{\Lambda }_h}{K_h}},I_h^{*}=0\right).$$

This equilibrium represents the persistence of infection in dogs while humans remain disease-free. The equilibrium reflects the balance between epidemiological transmission and vaccine-mediated immunity, where ${\kappa }_d$ and ${\kappa }_h$ integrate the effects of pharmaceutical formulation stability, carrier particle design, and controlled release kinetics, ensuring that enhanced vaccine processing reduces infection persistence and stabilizes immunity over time.

6. Local Stability Analysis

To investigate the local stability of the endemic equilibrium of the coupled dog–human rabies model, we linearize the nonlinear system around the equilibrium point. The Jacobian matrix, $J\left(x\right)$, evaluated at the equilibrium, captures the rates of change in each compartment with respect to small perturbations in all state variables.

Here, the variables are ordered as $(S_d,V_d,I_d,S_h,V_h,I_h)$, and the terms $F_{dd}$ and $F_{dh}$ represent the dog-to-dog and dog-to-human infection functions, respectively. The eigenvalues of this Jacobian determine the local stability of the equilibrium: if all eigenvalues have negative real parts, the equilibrium is locally asymptotically stable.

$$J(x)=\left[\begin{array}{cccccc}-(1+\frac{K_d}{{\kappa }_d})\frac{\mathrm{\partial }{F}_{dd}}{\mathrm{\partial }{S}_d}-{\mu }_d& -\frac{\mathrm{\partial }{F}_{dd}}{\mathrm{\partial }{V}_d}& \frac{\mathrm{\partial }{F}_{dd}}{\mathrm{\partial }{I}_d}& 0& 0& 0\\ 0& -{\mu }_d& 0& 0& 0& 0\\ -\frac{\mathrm{\partial }{F}_{dd}}{\mathrm{\partial }{S}_d}& 0& \frac{\mathrm{\partial }{F}_{dd}}{\mathrm{\partial }{I}_d}-{\alpha }_d& -\frac{\mathrm{\partial }{F}_{dh}}{\mathrm{\partial }{I}_d}& 0& 0\\ 0& 0& -\frac{\mathrm{\partial }{F}_{dh}}{\mathrm{\partial }{I}_d}& -(1+\frac{K_h}{{\kappa }_h})\frac{\mathrm{\partial }{F}_{dh}}{\mathrm{\partial }{S}_h}-{\mu }_h& 0& 0\\ 0& 0& 0& 0& -{\mu }_h& 0\\ 0& 0& \frac{\mathrm{\partial }{F}_{dh}}{\mathrm{\partial }{I}_d}& -\frac{\mathrm{\partial }{F}_{dh}}{\mathrm{\partial }{S}_h}& 0& -{\alpha }_h\end{array}\right].$$

$$F_{dd}=\frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{I}_d+b_{dd}{S}_d},F_{dh}=\frac{{\beta }_{dh}{S}_h{I}_d}{1+a_{dh}{I}_d+b_{dh}{S}_h}.$$

Using the explicit derivatives gives,

$$\left[\begin{array}{cccccc}-\left(1+{\displaystyle \frac{K_d}{{\kappa }_d}}\right){\beta }_{dd}{I}_d{\displaystyle \frac{1+a_{dd}{I}_d}{D_{dd}^2}}-{\mu }_d& 0& -{\beta }_{dd}{S}_d{\displaystyle \frac{1+b_{dd}{S}_d}{D_{dd}^2}}& 0& 0& 0\\ 0& -{\mu }_d& 0& 0& 0& 0\\ {\beta }_{dd}{I}_d{\displaystyle \frac{1+a_{dd}{I}_d}{D_{dd}^2}}& 0& {\beta }_{dd}{S}_d{\displaystyle \frac{1+b_{dd}{S}_d}{D_{dd}^2}}-{\alpha }_d& -{\beta }_{dh}{S}_h{\displaystyle \frac{1+b_{dh}{S}_h}{D_{dh}^2}}& 0& 0\\ 0& 0& -{\beta }_{dh}{S}_h{\displaystyle \frac{1+b_{dh}{S}_h}{D_{dh}^2}}& -\left(1+{\displaystyle \frac{K_h}{{\kappa }_h}}\right){\beta }_{dh}{I}_d{\displaystyle \frac{1+a_{dh}{I}_d}{D_{dh}^2}}-{\mu }_h& 0& 0\\ 0& 0& 0& 0& -{\mu }_h& 0\\ 0& 0& {\beta }_{dh}{S}_h{\displaystyle \frac{1+b_{dh}{S}_h}{D_{dh}^2}}& -{\beta }_{dh}{I}_d{\displaystyle \frac{1+a_{dh}{I}_d}{D_{dh}^2}}& 0& -{\alpha }_h\end{array}\right]$$

$$D_{dd}=1+a_{dd}{I}_d+b_{dd}{S}_d,D_{dh}=1+a_{dh}{I}_d+b_{dh}{S}_h.$$

Pharmaceutical and particle-processing factors influence these dynamics indirectly by altering the vaccination terms, ${\kappa }_d={\nu }_d{E}_d{\sigma }_d{R}_d$ and ${\kappa }_h={\nu }_h{E}_h{\sigma }_h{R}_h$. These parameters capture the effects of vaccine bioavailability ($E_x$), formulation stability (${\sigma }_x$), and controlled-release rate ($R_x$), which together determine the overall efficacy of the immunization process.

6.1. Stability at the Trivial Equilibrium $E_0$

At the trivial equilibrium $E_0=(0,0,0,0,0,0)$, the Jacobian reduces to

$$J\left(E_0\right)=diag(-K_d,-{\mu }_d,-{\alpha }_d,-K_h,-{\mu }_h,-{\alpha }_h).$$

Since all diagonal entries are strictly negative, each eigenvalue has a negative real part. Hence, $E_0$ is locally asymptotically stable. This equilibrium corresponds to the extinction of all populations, a biologically unrealistic but mathematically stable state in the absence of recruitment or vaccination input (${\Lambda }_d={\Lambda }_h=0$) [8].

6.2. Stability at the Disease-Free Equilibrium

At the disease-free equilibrium, infection terms vanish ($I_d=I_h=0$), and the vaccinated and susceptible populations depend primarily on the vaccination and mortality parameters:

$$S_d^0={\displaystyle \frac{{\Lambda }_d}{K_d}},V_d^0={\displaystyle \frac{{\kappa }_d}{{\mu }_d}}{S}_d^0,S_h^0={\displaystyle \frac{{\Lambda }_h}{K_h}},V_h^0={\displaystyle \frac{{\kappa }_h}{{\mu }_h}}{S}_h^0.$$

Applying the next-generation matrix method, the basic reproduction number ${\mathcal{R}}_0$ is obtained as

$$\begin{array}{|c|}\hline {\mathcal{R}}_0={\displaystyle \frac{{\beta }_{dd}{S}_d^0}{{\alpha }_d(1+a_{dd}{S}_d^0)}}={\displaystyle \frac{{\beta }_{dd}}{{\alpha }_d}}·{\displaystyle \frac{{\Lambda }_d/K_d}{1+a_{dd}{\Lambda }_d/K_d}}.\\ \hline\end{array}$$

If ${\mathcal{R}}_0<1$, all eigenvalues of the Jacobian have negative real parts, and the disease-free equilibrium is locally asymptotically stable. If ${\mathcal{R}}_0>1$, instability arises and infection invades the population. Pharmaceutical enhancement (increasing $E_d$ or ${\sigma }_d$) effectively increases ${\kappa }_d$, thereby enlarging $K_d$ and reducing $S_d^0$—ultimately lowering ${\mathcal{R}}_0$ and enhancing stability.

6.3. Stability of the Dog-Endemic, Human Disease-Free Equilibrium

At the dog-endemic equilibrium, the Jacobian assumes block structure

$$J\left(E^{*}\right)=\left[\begin{array}{cc}A& 0\\ B& C\end{array}\right],$$

where A represents the dog subsystem and C the human subsystem. The Routh–Hurwitz conditions applied to A yield the stability of the endemic dog population, while C governs the transmission spillover to humans.

6.4. Eigen Value Conditions

Dog subsystem: Let the characteristic equation of A be

$${\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3=0,$$

with coefficients

$$a_1=-(J_{11}+J_{22}+J_{33})>0,a_2=J_{11}{J}_{22}+J_{11}{J}_{33}+J_{22}{J}_{33}-J_{13}{J}_{31}-J_{12}{J}_{21}>0,a_3=det\left(A\right)>0.$$

According to the Routh–Hurwitz criterion [9], the equilibrium is locally asymptotically stable if $a_1{a}_2>a_3.$

Enhanced vaccine formulations, characterized by improved stability and release control, increase ${\kappa }_d$, effectively reducing the susceptible pool and ensuring that the Routh–Hurwitz conditions hold [10], thus stabilizing endemic prevalence.

Human subsystem: For the human block C, the eigenvalues are

$${\lambda }_1=-{\mu }_h<0,{\lambda }_2=-{\alpha }_h<0,{\lambda }_3={\displaystyle \frac{{\beta }_{dh}{S}_h^{*}{I}_d^{*}}{1+a_{dh}{S}_d^{*}+b_{dh}{I}_d^{*}}}-K_h.$$

Hence, the human subsystem remains stable if

$${\displaystyle \frac{{\beta }_{dh}{S}_h^{*}{I}_d^{*}}{1+a_{dh}{S}_d^{*}+b_{dh}{I}_d^{*}}}<K_h.$$

Pharmaceutical and particle-processing improvements that increase ${\kappa }_h$ (via higher immunogenicity or sustained release) elevate $K_h$, thereby suppressing ${\lambda }_3$ and maintaining human-disease-free stability.

6.5. Overall Interpretation

The Jacobian matrix derived in this section clarifies how each compartment of the rabies model responds to small perturbations around the equilibrium state. By examining the signs and structure of the partial derivatives, we can identify which parameters play the most influential roles in determining whether infection in dogs or humans will grow or decline. The negative diagonal entries associated with natural mortality, disease-induced removal, and quarantine show that these processes consistently push the system back toward equilibrium. In contrast, the off-diagonal transmission terms—especially those involving ${\beta }_{dd}$ and ${\beta }_{dh}$—represent the main channels through which disturbances can propagate.

The Jacobian analysis confirms the biological expectation: the stability of both the dog-only and the combined dog-human equilibrium is controlled by the same mechanisms that define the basic reproduction numbers. High transmission strengthens disturbances, while natural removal and quarantine work in the opposite direction. Importantly, the pharmaceutical-enhanced vaccination terms contribute additional stabilizing effects by reducing the size of the effective susceptible pools and by lowering the infection pressure in both species. These vaccination terms do not create nonlinear feedback in the Jacobian; instead, they appear only in damping components that push the system back toward equilibrium. Because of this structure, stronger pharmaceutical vaccination-through improved stability, release rate, or immunogenic response-moves the dominant eigenvalues farther into the negative region, supporting the overall stabilizing trend of the model. Thus, the mathematical structure of the Jacobian aligns with the numerical and epidemiological findings and verifies that pharmaceutical-enhanced vaccination plays a direct role in strengthening stability and suppressing potential outbreaks as shown in Figure 1.

7. Sensitivity Analysis

A local sensitivity analysis was performed to examine how variations in the model parameters influence the basic reproduction number, ${\mathcal{R}}_0$. Each parameter was perturbed slightly from its baseline value, and the normalized sensitivity index [$S_p=\frac{p}{{\mathcal{R}}_0}\frac{\partial {\mathcal{R}}_0}{\partial p}$] was estimated numerically. The findings indicate that ${\mathcal{R}}_0^{*}$ responds most strongly to changes in the dog-to-dog transmission rate ${\beta }_{dd}^{*}$, implying that even small increases in this parameter can markedly intensify rabies transmission within dog populations. Parameters such as the recruitment rate ${\Lambda }_d$ and the vaccination rate ${\kappa }_d$ also exert a noticeable positive influence on ${\mathcal{R}}_0$, underscoring the need for sustained vaccination coverage and effective control of host population dynamics. Conversely, increases in the disease-induced mortality rate ${\alpha }_d$ and the natural death rate ${\mu }_d$ tend to lower ${\mathcal{R}}_0^{*}$, as faster removal of infected individuals reduces potential transmission. The saturation constant $a_{dd}^{*}$ plays a stabilizing role, moderating the spread when infection levels increase [11]. A notable observation emerges from the role of the pharmaceutical-enhanced vaccination rate ${\kappa }_d$, expressed as [${\kappa }_d={\nu }_d{E}_d{\sigma }_d{R}_d,$] where ${\nu }_d$ denotes the base vaccination rate, $E_d$ represents bioavailability or immunogenicity, ${\sigma }_d$ characterizes formulation stability, and $R_d$ captures the controlled-release rate as shown in

Table 2. Parametric values.

Parameter	Description	Value/Range	Description Units
$S_d\left(0\right)$	Initial susceptible dogs	1000	individuals
$I_d\left(0\right)$	Initial infectious dogs	1–5	individuals
$S_h\left(0\right)$	Initial susceptible humans	10,000	individuals
$I_h\left(0\right)$	Initial infectious humans	0	individuals
${\mu }_d$	Baseline mortality coefficient (dogs)	0.0009	day−1
${\mu }_h$	Baseline mortality coefficient (humans)	$3.9\times {10}^{-5}$	day−1
${\alpha }_d$	Disease-induced mortality coefficient (dogs)	0.14	day−1
${\alpha }_h$	Disease-induced mortality coefficient (humans)	0.14	day−1
${\kappa }_d$	Vaccination rate coefficient (dogs)	0.001	day−1
${\kappa }_h$	Vaccination rate coefficient (humans)	0.0001	day−1
$K_d={\kappa }_d+{\mu }_d$	Recovery rate coefficient in dogs	0.00191	day−1
$K_h={\kappa }_h+{\mu }_h$	Recovery rate coefficient in humans	0.000139	day−1
${\Lambda }_d=K_d{S}_d^{*}$	Dog recruitment coefficient	1.91	individuals/day
${\Lambda }_h=K_h{S}_h^{*}$	Human recruitment coefficient	1.39	individuals/day
$R_0$	Basic reproduction number	1.5	dimensionless
${\beta }_{dd}$	Dog-to-dog transmission rate coefficient	0.00021	(individuals)−1 day−1
${\beta }_{dh}$	Dog-to-human transmission rate coefficient	$2\times {10}^{-5}$	(individuals)−1 day−1
$a_{dd},b_{dd}$	Saturation coefficients (dog-to-dog)	${10}^{-4}$–${10}^{-3}$	(individuals)−1
$a_{dh},b_{dh}$	Saturation coefficients (dog-to-human)	${10}^{-4}$–${10}^{-3}$	(individuals)−1

. Enhancements in any of these pharmaceutical parameters effectively strengthen ${\kappa }_d$, resulting in a lower ${\mathcal{R}}_0$. This emphasizes that improved vaccine formulations—through higher stability, sustained release, and optimized immunogenic response-can substantially contribute to long-term rabies control in dog populations. As shown in Figure 2, the sensitivity of $R_0$ to model parameters is illustrated.

8. Bifurcation Analysis

In this section we perform a bifurcation analysis for the rabies model with pharmaceutical-enhanced vaccination. We concentrate on the dog subsystem

$$\begin{array}{cc}\hfill {\dot{S}}_d& ={\Lambda }_d-{\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}}-{\nu }_d{E}_d{\sigma }_d{R}_d{S}_d-{\mu }_d{S}_d,\hfill \\ \hfill {\dot{V}}_d& ={\nu }_d{E}_d{\sigma }_d{R}_d{S}_d-{\mu }_d{V}_d,\hfill \\ \hfill {\dot{I}}_d& ={\displaystyle \frac{{\beta }_{dd}{S}_d{I}_d}{1+a_{dd}{S}_d+b_{dd}{I}_d}}-{\alpha }_d{I}_d,\hfill \end{array}$$

and show that the system undergoes a transcritical (forward) bifurcation at the threshold $R_0=1$. The analysis follows the standard next-generation approach to obtain $R_0$, then the Castillo–Chavez–Song center manifold calculation to determine the bifurcation direction [6].

8.1. Disease-Free Equilibrium and Basic Reproduction Number

Let

$${\kappa }_d:={\nu }_d{E}_d{\sigma }_d{R}_d,K_d:={\mu }_d+{\kappa }_d.$$

The disease-free equilibrium (DFE) for the dog subsystem is

$$E_0=\left(S_d^0,V_d^0,I_d^0\right)=\left({\displaystyle \frac{{\Lambda }_d}{K_d}},{\displaystyle \frac{{\kappa }_d}{{\mu }_d}}{\displaystyle \frac{{\Lambda }_d}{K_d}},0\right).$$

Applying the next-generation method to the infected equation yields the basic reproduction number

$$R_0={\displaystyle \frac{{\beta }_{dd}{S}_d^0}{{\alpha }_d\left(1+a_{dd}{S}_d^0\right)}}={\displaystyle \frac{{\beta }_{dd}}{{\alpha }_d}}·{\displaystyle \frac{{\Lambda }_d/({\mu }_d+{\kappa }_d)}{1+a_{dd}{\Lambda }_d/({\mu }_d+{\kappa }_d)}}.$$

(6)

The critical value of the transmission parameter ${\beta }_{dd}$ at which $R_0=1$ is therefore

$${\beta }_c={\alpha }_d·{\displaystyle \frac{{\mu }_d+{\kappa }_d+a_{dd}{\Lambda }_d}{{\Lambda }_d}}={\alpha }_d{\displaystyle \frac{{\kappa }_d+{\Lambda }_d{a}_{dd}+{\mu }_d}{{\Lambda }_d}}.$$

(7)

8.2. Linearization at the DFE and Simple Zero Eigenvalue

Linearizing the dog subsystem about $E_0$ gives the Jacobian matrix $J\left(E_0\right)$. Evaluating the Jacobian at ${\beta }_{dd}={\beta }_c$ (equivalently $R_0=1$) one finds the block-triangular structure

$$J\left(E_0\right){|}_{{\beta }_{dd}={\beta }_c}=\left(\begin{array}{ccc}-({\kappa }_d+{\mu }_d)& 0& -{\alpha }_d\\ {\kappa }_d& -{\mu }_d& 0\\ 0& 0& 0\end{array}\right),$$

where ${\kappa }_d={\nu }_d{E}_d{\sigma }_d{R}_d$. The spectrum contains two negative eigenvalues and a simple zero eigenvalue, so the Castillo–Chavez center manifold reduction applies.

8.3. Center-Manifold Coefficients

We follow the notation of Castillo–Chavez and Song. Let $x=(S_d,V_d,I_d)$. Denote by $f(x,{\beta }_{dd})$ the right-hand side of the dog subsystem and let $f_k$ be its k-th component. Let w be the right eigenvector and v the left eigenvector of $J\left(E_0\right)$ associated with the zero eigenvalue, normalized so that $v^{\top }w=1$. At ${\beta }_{dd}={\beta }_c$ one convenient choice (up to scaling) is

$$w=\left(\begin{array}{c}{w}_1\\ w_2\\ w_3\end{array}\right)=\left(\begin{array}{c}-{\displaystyle \frac{{\alpha }_d}{{\kappa }_d+{\mu }_d}}\\ -{\displaystyle \frac{{\kappa }_d{\alpha }_d}{{\mu }_d({\kappa }_d+{\mu }_d)}}\\ 1\end{array}\right),v=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),$$

which satisfy $J\left(E_0\right)w=0$ and $v^{\top }J\left(E_0\right)=0$ and $v^{\top }w=1$. Define the bifurcation coefficients

$$\begin{array}{cc}\hfill a& =\sum _{k,i,j}{v}_k{w}_i{w}_j{\left.{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}}\right|}_{(E_0,{\beta }_c)},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill b& =\sum _{k,i}{v}_k{w}_i{\left.{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial {\beta }_{dd}}}\right|}_{(E_0,{\beta }_c)}.\hfill \end{array}$$

(9)

Because $v={(0,0,1)}^{\top }$ only the third component $f_3$ (the infected equation) contributes; hence

$$a=\sum _{i,j}{w}_i{w}_j{\left.{\displaystyle \frac{{\partial }^2{f}_3}{\partial x_i\partial x_j}}\right|}_{(E_0,{\beta }_c)},b=\sum _i{w}_i{\left.{\displaystyle \frac{{\partial }^2{f}_3}{\partial x_i\partial {\beta }_{dd}}}\right|}_{(E_0,{\beta }_c)}.$$

A direct symbolic computation (evaluation of the second derivatives of the incidence term and substitution at the DFE and ${\beta }_{dd}={\beta }_c$) yields closed-form expressions for a and b:

$$\begin{array}{cc}\hfill a& ={\displaystyle \frac{2{\alpha }_d(-{\Lambda }_d{b}_{dd}-{\alpha }_d)({\kappa }_d+{\mu }_d)}{{\Lambda }_d\left({\kappa }_d+{\Lambda }_d{a}_{dd}+{\mu }_d\right)}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill b& ={\displaystyle \frac{{\Lambda }_d}{{\kappa }_d+{\Lambda }_d{a}_{dd}+{\mu }_d}}.\hfill \end{array}$$

(11)

8.4. Sign of the Coefficients and Bifurcation Type

All biological parameters are positive, and $b_{dd},{\Lambda }_d,{\alpha }_d>0$; therefore $-{\Lambda }_d{b}_{dd}-{\alpha }_d<0$. From (10) we see that $a<0$. From (11) we have $b>0$. According to the Castillo–Chavez–Song criterion:

$$a<0,b>0⟹\mathrm{a}\mathrm{forward}\left(\mathrm{transcritical}\right)\mathrm{bifurcation}\mathrm{at}{R}_0=1.$$

Thus the disease-free equilibrium is locally asymptotically stable for $R_0<1$ and loses stability at $R_0=1$, where an endemic equilibrium branches off continuously for $R_0>1$. No backward bifurcation arises from this leading-order center-manifold calculation under the present model assumptions (i.e., with the given form of incidence and vaccination terms).

8.5. Pharmaceutical Parameters as Bifurcation Drivers

Equation (6) explicitly shows how the pharmaceutical parameters enter the threshold via ${\kappa }_d={\nu }_d{E}_d{S}_d^{\mathrm{stab}}{R}_d$. Writing $R_0$ as a function of the effective vaccination contribution, as shown in Figure 3.

$$R_0\left({\kappa }_d\right)={\displaystyle \frac{{\beta }_{dd}}{{\alpha }_d}}·{\displaystyle \frac{{\Lambda }_d/({\mu }_d+{\kappa }_d)}{1+a_{dd}{\Lambda }_d/({\mu }_d+{\kappa }_d)}},$$

makes transparent the policy-relevant interpretation: increasing encapsulation efficiency $E_d$, stability ${\sigma }_d$ or controlled-release factor $R_d$ (which raise ${\kappa }_d$) reduces $R_0$ and thus shifts the bifurcation threshold. In particular, the critical vaccination effectiveness ${\kappa }_d^c$ needed to force $R_0<1$ for fixed ${\beta }_{dd}$ can be found by solving $R_0\left({\kappa }_d^c\right)=1$, yielding

$${\kappa }_d^c={\displaystyle \frac{{\beta }_{dd}{\Lambda }_d-{\alpha }_d({\mu }_d+{\Lambda }_d{a}_{dd})}{{\alpha }_d+a_{dd}{\beta }_{dd}}}(\mathrm{rearranged}\mathrm{form}).$$

9. Numerical Simulations

To evaluate the epidemiological consequences of pharmaceutical enhancement in rabies control, numerical simulations of the coupled dog–human system were conducted. The model equations were integrated numerically using the classical fourth-order Runge–Kutta scheme with a daily time step, under two distinct immunization regimes:

(i)

Baseline immunization plan: standard vaccination rates ${\kappa }_d$ and ${\kappa }_h$ without additional formulation effects ($E_d=E_h=1$, ${\sigma }_d={\sigma }_h=1$, $R_d=R_h=1$);

(ii)

Enhanced immunization plan: inclusion of pharmaceutical processing parameters, where encapsulation efficiency, stability, and controlled release were incorporated as $E_x>1$, ${\sigma }_x>1$, and $R_x>1$ ($x=d,h$). As shown in Figure 4 these parameters represent advanced vaccine formulations with extended antigen exposure and improved immunogenicity [12].

9.1. Results and Interpretation

Under the baseline scenario, the system exhibits a high and persistent endemic level in both species, consistent with $R_0=1.5>1$. Incorporating pharmaceutical processing effects effectively increases ${\kappa }_d$ and ${\kappa }_h$ through enhanced factors $(E_x,{\sigma }_x,R_x)$, thereby reducing the effective reproduction number $R_0^{\mathrm{eff}}<1$ as predicted by the bifurcation analysis in Section 8.

The simulation outcomes show marked improvements in epidemiological indicators:

• The peak infection in dogs decreases from approximately 18% under conventional vaccination to 5% under enhanced vaccination.
• The human infection peak drops from 4% to 0.8%.
• The elimination time reduces from about 8 years to 3 years.
• Long-term prevalence remains suppressed: a 90% reduction in cases is sustained beyond five years, compared to only 60% for traditional vaccines.

These results confirm the theoretical prediction of a forward (transcritical) bifurcation. As vaccination parameters $(E_d,{\sigma }_d,R_d)$ increase, the disease-free equilibrium becomes globally stable, indicating that pharmaceutical enhancement shifts the bifurcation threshold leftward (i.e., lower $R_0$ values suffice for eradication).

9.2. Discussion

The simulations quantitatively validate the analytical results. The improvement in encapsulation, stability, and controlled release acts synergistically to decrease the effective transmission potential of the pathogen. From a control perspective, these parameters can be interpreted as “technological multipliers’’—each unit increase in stability or release control equivalently gives a substantial rise in the coverage of conventional vaccination campaigns. Therefore, the integration of pharmaceutical engineering principles into vaccination strategies is not merely theoretical but demonstrates measurable epidemiological benefits.

10. Conclusions

The simulation results show that improving vaccine formulation through pharmaceutical techniques greatly reduces rabies infections in both dogs and humans. The nonlinear incidence in the model suggests that when infection levels are high, the spread of the disease naturally slows down, supporting the idea that transmission is self-limiting. This, together with the effects of enhanced vaccine stability, controlled release, and better absorption, helps the disease decline faster and disappear sooner. Overall, these findings highlight that advanced vaccine design not only improves the effectiveness of immunization but also makes rabies control more reliable and sustainable in the long run.