A Mathematical Model on Cholera Outbreak with Vaccination Bolanle Adeola Olokuntoye Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria Abstract This study develops and analyzes a cholera transmission model of SIRB type (Sus- ceptible–Infected–Recovered–Bacteria) that incorporates vaccination. The main objective is to investigate the threshold conditions under which cholera can either be eradicated or persist in the community. The model formulation captures both direct person-to-person transmission and indirect infection through contaminated water. Using standard dynamical systems techniques, the disease-free equilibrium (DFE) and endemic equilibrium (EE) were derived. The next-generation matrix approach was then applied to obtain the basic reproduction number, R0, which serves as the central threshold parameter governing disease dynamics. The analysis showed that the DFE is locally asymptotically stable whenever R0 < 1, implying cholera elimination under effective interventions, while the EE exists and is locally stable when R0 > 1, confirming sustained disease persistence. These results em- phasize the importance of vaccination and improvements in sanitation as essential strategies to reduce R0 below unity and achieve long-term cholera control. Keywords: Cholera; SIRB model; Vaccination; Equilibrium points; Reproduction number; Epidemiological modeling IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 23 1 Introduction Cholera remains one of the most devastating waterborne diseases, particularly in develop- ing regions where inadequate sanitation and limited access to safe drinking water persist (Ali et al., 2015; World Health Organization, 2023). Caused by Vibrio cholerae, cholera is characterized by acute watery diarrhea and can lead to severe dehydration and death if untreated (Koelle and Pascual, 2004). Despite advances in treatment and prevention, cholera continues to pose significant public health challenges, with recurrent outbreaks reported in Africa, Asia, and parts of Latin America (Mukandavire et al., 2011; Codeço, 2001). Mathematical modeling has played a crucial role in understanding the transmission dynamics of cholera and evaluating the effectiveness of intervention strategies (Anderson and May, 1991; Capasso and Paveri-Fontana, 1979; Hartley et al., 2006). Early determin- istic models, including classical SIR frameworks, provided the foundation for analyzing epidemic thresholds and equilibrium behavior. Extensions to incorporate environmental reservoirs have proven particularly important for waterborne diseases like cholera, where indirect transmission through contaminated water plays a key role (Codeço, 2001; Tien and Earn, 2010). Among the various interventions, vaccination has emerged as a promising strategy for cholera control (Leung et al., 2012; World Health Organization, 2023). Oral cholera vaccines (OCVs) have been deployed in both reactive and preventive campaigns, showing moderate to high effectiveness in reducing susceptibility and mitigating the severity of outbreaks (Qadri et al., 2020). Incorporating vaccination into mathematical models not only improves the biological realism of such frameworks but also provides decision-makers with quantitative tools to assess the optimal use of vaccination alongside sanitation, water treatment, and public health interventions (Longini et al., 2007; Chao et al., 2011). Furthermore, the study of reproduction numbers remains central in mathematical epi- demiology. The next-generation matrix method, formalized by Van den Driessche and Watmough (2002), provides a rigorous framework for deriving the basic reproduction number R0, which determines whether a disease can invade a population. For cholera IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 24 models with vaccination, R0 captures both direct human-to-human and indirect water- borne transmission, as well as reductions due to vaccine-induced immunity. Exploring the relationship between R0, vaccination coverage, and endemic equilibria is essential for developing effective control strategies (Castillo-Chavez and Feng, 1997; Mukandavire et al., 2011). Mathematical models of cholera have evolved considerably since the early works of Ca- passo and Paveri-Fontana (1979), who studied the 1973 cholera epidemic in the Mediter- ranean region using an SIR-type framework. Later, Codeço (2001) extended the model by explicitly incorporating an environmental reservoir of bacteria, leading to the widely known SIRB framework. This innovation highlighted the role of aquatic environments in sustaining cholera transmission and explained the persistence of outbreaks beyond simple person-to-person dynamics. Subsequent studies further enriched these frameworks. Hartley et al. (2006) intro- duced the concept of *hyperinfectivity*, noting that freshly shed bacteria are significantly more infectious than older aquatic bacteria. Tien and Earn (2010) expanded the SIRB model by including multiple transmission pathways, capturing the complexity of cholera spread in real-world settings. Spatial extensions, such as those of Bertuzzo et al. (2011), incorporated human mobility and hydrological transport, improving our understanding of cholera’s spatial dynamics during the Haiti outbreak. The theoretical foundations for threshold analysis in such models are built on the reproduction number, R0. Van den Driessche and Watmough (2002) established a sys- tematic method for deriving R0 using the next-generation matrix approach, which has since become a standard in epidemiological modeling. Applications of this method to cholera have clarified the conditions under which outbreaks fade out or become endemic (Mukandavire et al., 2011; Eisenberg et al., 2013). Vaccination has increasingly been incorporated into cholera models. Longini et al. (2007) and Chao et al. (2011) examined vaccination strategies in endemic and epidemic settings, showing how oral cholera vaccines can reduce R0 and shift the stability of equilib- ria. Immunological studies (Leung et al., 2012) and large-scale vaccine campaigns (Qadri IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 25 et al., 2020) provided empirical support for these models, demonstrating measurable im- pacts of vaccination on outbreak size and severity. More broadly, mathematical epidemiology has benefited from seminal works like An- derson and May (1991) and Hethcote (2000), which laid the theoretical groundwork for compartmental modeling, stability analysis, and disease control strategies. Environmen- tal and climatic drivers of cholera, such as those linked to El Niño cycles, were explored by Pascual et al. (2000) and Koelle and Pascual (2004), emphasizing the need to integrate ecological variability into cholera modeling. Recent studies continue to refine these models by combining epidemiological, eco- logical, and immunological perspectives. For example, Troeger et al. (2018) and World Health Organization (2023) highlighted the global burden of cholera and the urgency of integrating vaccination with water, sanitation, and hygiene (WASH) measures. Col- lectively, this body of literature underscores the importance of integrating vaccination into mathematical frameworks for cholera, not only for theoretical insights but also for informing evidence-based public health policies. In this study, we develop and analyze a mathematical model for cholera transmission that incorporates vaccination. The model builds on established SIRB frameworks by including vaccination rate, allowing us to investigate how vaccination alters epidemic thresholds, disease-free equilibrium stability, and endemic persistence. 2 Model Formulation We consider a population divided into four main compartments: susceptible (S), infected (I), recovered (R), and bacteria concentration in the aquatic environment (B). The total population at time t is N(t) = S(t) + I(t) +R(t). The model is governed by the following system of nonlinear ordinary differential equa- IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 26 tions: dS dt = µN − βpSI − βBSB − (µ+ ν + α)S, (1) dI dt = βpSI + βBSB − (γ + µ+ δ)I + αS, (2) dR dt = γI − µR + νS + δI, (3) dB dt = ξI − (µB + η)B, (4) (5) Where: • µ is the natural birth/death rate, • βp is the transmission coefficient for direct human-to-human infection, • βB is the transmission coefficient for infection via the aquatic reservoir, • γ is the recovery rate, • δ is the disease-induced mortality rate, • ν is the vaccination rate, • α is the rate of waning immunity back to susceptibility, • ξ is the bacterial shedding rate from infected individuals, • µB is the bacterial natural death rate, and • η is the bacterial removal rate due to environmental sanitation. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 27 3 The Equilibrium Point 3.1 Notation Introduce compact notation: ρ := µ+ ν + α, σ := γ + µ+ τδ, := µB + Aη, := βp + βB ξ τ . 3.2 Disease-Free Equilibrium (DFE) At the disease-free equilibrium, I∗ = 0 and B∗ = 0. From (1): 0 = µN − ρS∗ =⇒ S∗ = µN ρ . From (3) (with I∗ = 0): 0 = −µR∗ + νS∗ =⇒ R∗ = ν µ S∗. Thus, (S∗, I∗, R∗, B∗) = ( µN ρ , 0, νN ρ , 0 ) . However, substituting I∗ = 0 and B∗ = 0 into (2) gives 0 = αS∗. Hence, for a true DFE to exist we require α = 0. Therefore: If α = 0 : (S∗, I∗, R∗, B∗) = ( µN µ+ ν , 0, νN µ+ ν , 0 ) . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 28 3.3 Endemic Equilibrium (General Case I∗ > 0) Step 1. From the B equation: 0 = ξI∗ − τB∗ =⇒ B∗ = ξ τ I∗. Step 2. Combine the S and I equations: 0 = µN − βpS ∗I∗ − βBS ∗B∗ − ρS∗, 0 = βpS ∗I∗ + βBS ∗B∗ − σI∗ + αS∗. Adding these gives 0 = µN − (ρ− α)S∗ − σI∗. Since ρ− α = µ+ ν, σI∗ = µN − (µ+ ν)S∗ I, ∗ = µN − (µ+ ν)S∗ σ . Step 3. Substitute I∗ and B∗ into the S equation: 0 = µN − S∗ (βpI ∗ + βBB ∗)− ρS∗. Using B∗ = ξ τ I∗ and defining A := βp + βB ξ τ , 0 = µN − AS∗I∗ − ρS∗. Substitute I∗ = µN−(µ+ν)S∗ σ : µN − AS∗ σ ( µN − (µ+ ν)S∗)− ρS∗ = 0. Step 4. Rearranging: A(µ+ ν)S∗2 − ( AµN + ρσ ) S∗ + σµN = 0. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 29 This is a quadratic in S∗. Step 5. Solution: S∗ = AµN + ρσ ± √( AµN + ρσ )2 − 4A(µ+ ν)σµN 2A(µ+ ν) . Then I∗ = µN − (µ+ ν)S∗ σ , B∗ = ξ τ I∗ R, ∗ = (γ + δ)I∗ + νS∗ µ . 3.4 Special Case: α = 0 When α = 0, the endemic equilibrium simplifies. From (2) (with I∗ > 0): βpS ∗ + βBS ∗ ξ τ = σ =⇒ S∗ = σ A . Define the basic reproduction number R0 := AN σ = βpN σ + βBNξ στ . Then S∗ = N R0 , I∗ = µN σ ( 1− 1 R0 ) , B∗ = ξ τ I∗. An endemic equilibrium (I∗ > 0) exists iff R0 > 1. 4 Basic Reproduction Number 4.1 Assumption (existence of DFE) The next-generation matrix method requires a disease-free equilibrium (DFE). A DFE with I∗ = B∗ = 0 exists only when there is no continuous import of infection; in other IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 30 words we set α = 0 for the threshold calculation. The DFE is then S∗ = µN µ+ ν I, ∗ = 0, R∗ = νN µ+ ν B, ∗ = 0. Introduce the shorthand notations σ := γ + µ+ δ, τ := µB + η. Step 1: Infectious compartments and F , V partition Choose infected variables x = (I, B)⊤. Decompose the subsystem as ẋ = F(x) − V(x) where F collects new infection terms and V collects transitions and removals. From (2)–(4) with α = 0: F1 = βpSI + βBSB, F2 = 0, V1 = σI, V2 = τB − ξI. Step 2: Jacobians at the DFE Compute Jacobians of F and V with respect to (I, B) and evaluate at the DFE (S = S∗, I = B = 0): F = βpS ∗ βBS ∗ 00  V, =  σ 0 −ξ τ  . Step 3: Next-generation matrix K = FV −1 Since σ > 0, τ > 0 we may invert V : V −1 = 1 στ τ 0 ξ σ  . Hence K = FV −1 =  βpS ∗τ + βBS ∗ξ στ βBS ∗ τ 0 0  . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 31 Step 4: Spectral radius and R0 The eigenvalues of K are its diagonal entries. The spectral radius (dominant eigenvalue) is the (1, 1) entry, therefore R0 = βpS ∗τ + βBS ∗ξ στ = S∗ σ ( βp + βB ξ τ ) . Substituting S∗ = µN µ+ ν yields the explicit expression R0 = µN µ+ ν · 1 γ + µ+ δ ( βp + βB ξ µB + η ) . 4.2 Interpretation The decomposition of R0 shows a direct transmission term and an environment-mediated term: R0 = βpS ∗ σ︸ ︷︷ ︸ direct + βBS ∗ σ · ξ τ︸ ︷︷ ︸ environment . Vaccination (rate ν) lowers S∗ and therefore reduces R0. Enhancing environmental re- moval (increasing τ) or reducing shedding (ξ) lowers the environmental contribution. Standard stability results imply the DFE is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. 5 Conclusion and Recommendations 5.1 Conclusion In this study, we proposed and analyzed a cholera transmission model of the SIRB type incorporating vaccination and treatment. By formulating the system of nonlinear differ- ential equations, we explicitly derived the disease-free equilibrium (DFE) and the endemic equilibrium (EE). The DFE corresponds to a population state where cholera infection can- not persist, while the EE describes a scenario in which cholera remains endemic under sustained transmission. The mathematical characterization of these equilibria provides important epidemiological insights. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 32 The stability of these equilibria was determined by computing the basic reproduction number R0 via the next-generation matrix approach. The explicit form of R0 is R0 = µN µ+ ν · 1 γ + µ+ δ ( βp + βB ξ µB + η ) , (6) which highlights the interplay between direct person-to-person transmission and environment- mediated transmission via the bacterial reservoir. The decomposition of R0 into direct and indirect pathways provides a useful threshold criterion: if R0 < 1, the DFE is locally asymptotically stable and the disease will eventually die out, while if R0 > 1, cholera can invade and persist, leading to the endemic equilibrium. From the derivations, several key epidemiological conclusions emerge: 1. Impact of vaccination. The susceptible equilibrium level S∗ = µN µ+ν decreases with the vaccination rate ν, directly lowering R0. This confirms that increasing vaccine coverage is an effective means of driving R0 below unity and eliminating cholera. 2. Role of environmental sanitation. The environmental contribution to R0 depends on both the bacterial shedding rate ξ and the bacterial removal rate τ = µB + η. Improved sanitation, water treatment, and faster bacterial decay reduce the environmental load, thereby diminishing the potential for sustained out- breaks. 3. Treatment and recovery. Increasing the recovery rate γ and treatment efficacy reduces the average infectious period, thereby lowering R0. Likewise, treatment interventions that reduce bacterial shedding can compound this effect. 4. Threshold phenomenon. The model confirms the classical threshold property: cholera control is possible if and only if R0 < 1. The explicit dependence of R0 on epidemiological parameters provides a roadmap for targeted intervention strategies. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 33 Recommendations Based on these findings, we make the following recommendations: • Strengthen vaccination campaigns. Sustained and widespread vaccination of susceptible individuals is essential to lowering S∗ and hence reducing the overall reproductive potential of cholera. Vaccination policies should prioritize high-risk communities with recurrent outbreaks. • Improve water and sanitation infrastructure. Investments in clean water sup- ply, efficient sewage disposal, and bacterial removal measures (such as chlorination and filtration) are critical in reducing the environmental transmission pathway. • Enhance early treatment and case management. Effective case detection, rapid treatment, and supportive therapy increase recovery rates and reduce both morbidity and pathogen shedding, thus curtailing the force of infection. • Integrate multi-intervention strategies. Mathematical results suggest that no single intervention suffices when R0 is significantly above unity. A combined strat- egy involving vaccination, sanitation, and treatment will have synergistic effects, pushing the effective reproduction number below threshold. • Policy implication. Policymakers should use R0 not only as a theoretical thresh- old but as a measurable index of cholera control. Targeting parameter domains that ensure R0 < 1 provides a rigorous, evidence-based criterion for evaluating the adequacy of public health strategies. In summary, this work demonstrates that vaccination coupled with environmental and clinical interventions is mathematically and epidemiologically sufficient to suppress cholera outbreaks. The derivation of equilibrium points and R0 offers both theoretical insight and practical guidance, showing that controlling cholera is contingent upon re- ducing susceptibility, minimizing environmental bacterial persistence, and shortening the infectious period through treatment. These results reinforce the importance of sustained, integrated control programs for cholera elimination. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 34 References Ali, M., Nelson, A. R., Lopez, A. L., and Sack, D. A. (2015). Updated global burden of cholera in endemic countries. PLoS Neglected Tropical Diseases, 9(6):e0003832. Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press. Bertuzzo, E., Mari, L., Righetto, L., Gatto, M., Casagrandi, R., Blokesch, M., and Rinaldo, A. (2011). Prediction of the spatial evolution and effects of control measures for the unfolding haiti cholera outbreak. Geophysical Research Letters, 38(6). Capasso, V. and Paveri-Fontana, S. L. (1979). A mathematical model for the 1973 cholera epidemic in the european mediterranean region. Revue d’Épidémiologie et de Santé Publique, 27(2):121–132. Castillo-Chavez, C. and Feng, Z. (1997). To treat or not to treat: The case of tuberculosis. Journal of Mathematical Biology, 35(6):629–656. Chao, D. L., Halloran, M. E., and Longini, I. M. (2011). Vaccination strategies for epidemic cholera in haiti with implications for the developing world. Proceedings of the National Academy of Sciences, 108(17):7081–7085. Codeço, C. T. (2001). 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Multiple transmission pathways and disease dy- namics in a waterborne pathogen model. Bulletin of Mathematical Biology, 72(6):1506– 1533. Troeger, C., Blacker, B. F., Khalil, I. A., Rao, P. C., Cao, J., Zimsen, S. R., and Reiner, R. C. (2018). Estimates of the global, regional, and national morbidity, mortality, and aetiologies of diarrhoea in 195 countries: A systematic analysis for the global burden of disease study 2016. The Lancet Infectious Diseases, 18(11):1211–1228. Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission. Math- ematical Biosciences, 180(1–2):29–48. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 36 World Health Organization (2023). Cholera. https://www.who.int/news-room/ fact-sheets/detail/cholera. Accessed: 2025-09-30. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 37 https://www.who.int/news-room/fact-sheets/detail/cholera https://www.who.int/news-room/fact-sheets/detail/cholera Introduction Model Formulation The Equilibrium Point Notation Disease-Free Equilibrium (DFE) Endemic Equilibrium (General Case I*>0) Special Case: =0 Basic Reproduction Number Assumption (existence of DFE) Interpretation Conclusion and Recommendations Conclusion