




































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

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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

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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

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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-

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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.

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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

)
.

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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.

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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

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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

 .

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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.

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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.

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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.

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World Health Organization (2023). Cholera. https://www.who.int/news-room/

fact-sheets/detail/cholera. Accessed: 2025-09-30.

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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


