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 AM  Vol.12 No.8 , August 2021
A Mathematical Model for the Control of Cholera Epidemic without Natural Recovery
Abstract: In this research work, we present a mathematical model for the control of cholera outbreak without natural recovery. This follows a slight modification as compared to previous cholera models for the Nigerian case. Our model incorporates treatment, water hygiene as well as environmental sanitation. The model employs a system of nonlinear ordinary differential equations, which is analyzed in detail for its stability properties. We compute the basic reproduction ratio R0 for the various control parameters and discover that with proper combination of control measures, the spread of cholera could be minimized. Numerical simulation of the cholera model is done using MathCAD14, and the graphical profiles of the main variables are depicted. We conclude that improvement in treatment, water hygiene and the environmental sanitation is indeed effective in eradicating the cholera epidemic.

Introduction

Cholera is a contagious infectious disease that is characterized by extreme vomiting, profuse watery diarrhoea and leg pain. It has been found that transmission transpires mostly via absorption of contaminated drinking water or food. Worldwide, almost every year, there are estimated 3 - 5 million cholera cases and 100,000 - 130,000 deaths due to cholera a year as of 2010 [1]. It has a very short incubation period which starts from a few hours to five days. The health of an infected person disintegrates rapidly and death may occur if treatment is not promptly given. Cholera was first discovered in the Indian subcontinent in 1817. The disease reaches all the way through Asian continent in the 1960s, getting into Africa in 1970 and Latin America in 1991 [2] [3]. In many parts of Africa and Asia, the disease is still endemic. Cholera is a disastrous water-borne infectious disease that is caused by the bacterium vibrio cholera. It is a very serious problem in many developing countries due to inadequate access to safe drinking water supply, improper treatment of reservoirs and improper sanitation. In 2012, WHO reported 245, 393 cholera cases and 3034 death cases across 48 countries in which 67% cases occurred in African countries [4]. In 2005, Nigeria had 4477 cases and 174 deaths. There were reported cases of cholera in 2008 in Nigeria in which there were 429 deaths out of 6330 cases. Furthermore, 2304 cases were reported in Niger State in which 114 were death cases [5]. Also in 2009, Nigeria reported 13,691 cases and 431 deaths [6] (WHO). Tian and Wang [7] evidenced that recent years have seen a strong trend of cholera outbreak in developing countries, such as in India (2007), Iraq (2008), Congo (2008), Zimbabwe (2008-2009), Haiti (2010), Kenya (2010) and Nigeria (2010).

In Nigeria, outbreaks of the disease have been taking place with ever-increasing occurrence ever since the earliest outbreak in recent times in 1970, [8] [9]. In summary, the United Nation (UN) unit reports: “despite Nigeria’s oil wealth, more than 70% of the country’s 126 million people live below the poverty line and cholera outbreaks are common in poor urban areas which lack proper sanitation and clean drinking water” (UN Office for the Coordination of Humanitarian Affairs Integrated Regional Information Networks (IFIN) 2005). In the last few decades, several researchers [10] [11] [12] [13] [14] have designed mathematical models to explore the transmission dynamics and control of the disease. However, in the literature, not many studies on cholera in Nigeria using bio-mathematical models are prevalent, hence, the need for this research study.

1.1. Model Formulation

A mathematical model for cholera transmission is developed by making a slight modification to the previous model by Sulayman Fatima et al. [15]. Our model is devoid of natural recovery, this is because from the literature we find that cholera is a fatal disease, and a large percentage of those who recover, do so because of treatment. Our model on the other hand incorporates treatment, water hygiene and environmental sanitation. The model contains five variables which are susceptible, infected, recovered, total human population, and the concentration of vibrio cholera in water. The susceptible population is generated either through birth or through immigration. They acquire infection and move to the infected class at the rate:

α ( 1 c w ) B K + B (1)

a = rate of exposure to contaminated water;

cw = rate of compliance with water hygiene;

K = concentration of vibrio cholera in water that yields 50% chance of catching cholera;

B = concentration of vibrio cholera in contaminated water.

The number of infected individuals decreases through recovery from the disease at the rate τ, being recovery due to treatment. Bacterial populations in the aquatic environment grow in the water at the rate determined by the environmental factors such as temperature. The variables and parameters of the model are described in Table 1 and Table 2: A compartmental representation of our model is given in Figure 1: The model consists of the following system of nonlinear ordinary differential equations given in (2) to (5):

Table 1. Variables of the cholera model.

Table 2. Parameters of the cholera model.

Figure 1. Compartmental model of controlled cholera transmission model without natural recovery.

d S d t = β N a ( 1 c w ) B K + B S + ω R μ S (2)

d I d t = a ( 1 c w ) B K + B S ( τ + μ + δ ) I (3)

d B d t = e ( 1 c s ) I ρ B (4)

d R d t = τ I ( ω + μ ) R (5)

With

N = S + I + R (6)

1.2. Remark

The variables and parameters of the model are presented in Table 1 and Table 2, while their respective baseline values are given in Table 3 and Table 4.

1.3. Model Analysis

Our Cholera model represented by the system of nonlinear differential Equations (2)-(5) is now analyzed qualitatively to obtain insights into its features, which will give a better understanding of the control features on the transmission dynamics of the cholera virus.

1.3.1. Disease Free Equilibrium (DFE)

The disease free equilibrium of model system (1)-(4) is obtained by setting

Table 3. Baseline values for variables of the cholera model in Nigeria as at 2016.

Table 4. Baseline values for parameters of the cholera model.

d S d t = d I d t = d B d t = d R d t = 0 , and in the absence of disease, I = B = 0 so that:

β N a ( 1 c w ) B K + B S + ω R μ S = 0 (7)

a ( 1 c w ) B K + B S ( τ + μ + δ ) I = 0 (8)

e ( 1 c s ) I ρ B = 0 (9)

τ I ( ω + μ ) R = 0 (10)

Since I = B = 0 , Equation (4) gives R = 0 and Equation (1) gives S = β N μ .

Hence DFE is

( S 0 , I 0 , B 0 , R 0 ) = ( β N μ , 0 , 0 , 0 ) (11)

1.3.2. Positivity of Solutions

Since the model monitors human population, we need to show that all the state variables remain non-negative for all times.

1.3.3. Theorem

Let Ω = { ( S , I , R , B ) + 4 : S ( 0 ) > 0 , I ( 0 ) > 0 , R ( 0 ) > 0 , B ( 0 ) > 0 } then the solutions of { S ( t ) , I ( t ) , R ( t ) , B ( t ) } of the system Equation (2)-(5) are positive for all t 0 .

Proof: Consider the first equation of the system;

d S d t = β N a ( 1 c w ) B K + B S + ω R μ S

d S d t β N μ S

We consider the solution of

d S d t + μ S = β N

Employing the integrating factor method, we obtain

S ( t ) = β N μ + C e μ t (C = constant of integration)

With the initial condition S ( 0 ) = S 0 C = S 0 β N μ ,

S ( t ) = β N μ + ( S 0 β N μ ) e μ t 0

We now consider the second equation;

d I d t = a ( 1 c w ) B K + B S ( τ + μ + δ ) I

d I d t ( τ + μ + δ ) I

Integration yields

I ( t ) I 0 e ( τ + μ + δ ) t 0

We consider also the third equation

d B d t = e ( 1 c s ) I ρ B

d B d t ρ B

B ( t ) B 0 e ρ B t

We observe that as t , B ( t ) B 0 .

Finally from the fourth equation

d R d t = τ I ( ω + μ ) R

d R d t ( ω + μ ) R

R ( t ) R 0 e ( ω + μ ) t 0

1.3.4. Invariant Region

The system Equations (2)-(5) have solutions which are contained in the feasible region:

Ω = { ( S , I , R , B ) + 4 : 0 S ( 0 ) , 0 I ( 0 ) , 0 R ( 0 ) , 0 B ( 0 ) , S + I + R β N μ }

where the model makes biological sense, and can be shown to be positively invariant and globally attracting in + 4 with respect to the system of ODE describing our model.

1.4. Model Analysis

The model system (2)-(5) is analyzed qualitatively to get insights into its dynamical features which give better understanding of the impact control strategies on the transmission dynamics of the cholera disease. First, we study the stability properties of the cholera model.

1.4.1. Stability Analysis of the Cholera Model

Here we consider the Jacobian matrix associated with the system of differential equations for our Cholera model

J ( S , I , B , R ) = ( f 1 S f 1 I f 1 B f 1 R f 2 S f 2 I f 2 B f 2 R f 3 S f 3 I f 3 B f 3 R f 4 S f 4 I f 4 B f 4 R ) (12)

where

f 1 ( S , I , B , R ) = β N a ( 1 c w ) B K + B S + ω R μ S

f 2 ( S , I , B , R ) = a ( 1 c w ) B K + B S ( τ + μ + δ ) I

f 3 ( S , I , B , R ) = e ( 1 c s ) I ρ B

f 4 ( S , I , B , R ) = τ I ( ω + μ ) R

Computing the respective partial derivatives, we obtain;

J ( S , I , B , R ) = ( a ( 1 c w ) B K + B μ 0 a K ( 1 c w ) ( K + B ) 2 ω a ( 1 c w ) B K + B ( τ + μ + δ ) a K ( 1 c w ) ( K + B ) 2 0 0 e ( 1 c s ) ρ 0 0 τ 0 ( ω + μ ) ) (13)

Evaluating the Jacobian at the disease free equilibrium state

( S 0 , I 0 , B 0 , R 0 ) = ( β N μ , 0 , 0 , 0 ) gives

J * ( S 0 , I 0 , B 0 , R 0 ) = ( μ 0 a ( 1 c w ) K ω 0 ( τ + μ + δ ) a ( 1 c w ) K 0 0 e ( 1 c s ) ρ 0 0 τ 0 ( ω + μ ) ) (14)

The eigenvalues of J * are given by

| J * λ I | = 0

Giving the characteristic equation;

λ 4 + A λ 3 + B λ 2 + C λ + D = 0 (15)

where

A = 3 μ + ρ + τ + ω + δ

B = 3 K μ 2 a e + a e c s + a e c w + 3 K μ ρ + 2 K μ τ + K ρ τ + 2 K μ δ + 2 K μ ω + K ρ δ + K ρ ω + K τ ω + K ω δ a e c s c w K

C = K μ 3 + 3 K μ 2 ρ + K μ 2 τ + K μ 2 ω + K μ 2 δ 2 a e μ a e ω + 2 a e μ c s + 2 a e μ c w + a e ω c s + a e ω c w + 2 K μ ρ τ + 2 K μ ρ ω + 2 K μ ρ δ + K μ τ ω + K ρ τ ω + K μ ω δ + K ρ ω δ 2 a e μ c s c w a e ω c s c w K

D = K μ 3 ρ a μ 2 e a μ ω e + a μ 2 c s e + a μ 2 c w e + K μ 2 ρ τ + K μ 2 ρ ω + K μ 2 ρ δ + a μ ω c s e + K μ ρ τ ω + K μ ρ ω δ a μ 2 c s c w e a μ ω c s c w e K

1.4.2. Remark

We have asymptotic stability of the disease free equilibrium provide all the roots of the quartic Equation (15) have negative real parts.

1.5. The Basic Reproduction Ratio, R0

The basic reproduction ratio (or number) denoted by R0 is the average number of secondary infections caused by an infectious individual during his or her entire period of infectiousness (Diekmann et al. [16] ). The basic reproduction ratio is an important non-dimensional quantity in epidemiology because it sets the threshold in the study of any given disease, both for predicting its outbreak and for evaluating its control strategies.

1.5.1. The Next Generation Matrix Approach

In generalR0 is found through the study and computation of the eigenvalues of the Jacobianmatrix at the disease- or infectious-free equilibrium. Diekmann et al. [16] follow a different approach which is the next generation matrix method. This procedure converts a system of ordinary differential equations of a model of infectious disease dynamics to an operator (or matrix) that translate from one generation of infectious individuals to the next. The basic reproductive number is then defined as the spectral radius (dominant eigenvalue) of this operator.

1.5.2. Remark

We deduce that whether a disease becomes persistent or dies out in a community is totally dependent on the value of the basic reproduction ratio, R0. Furthermore, stability of equilibria can be analyzed usingR0. If R 0 < 1 , it means that every infectious individual will cause less than one secondary infection and hence the disease will die out and when R 0 > 1 , every infectious individual will cause more than one secondary infection and hence the disease will invade the population.

A large value of R0 may indicate the possibility of a major epidemic. For the case of a model with a single infected class, R0 is simply the product of the infection rate and the mean duration of the infection.

1.5.3. The Basic Reproduction Ratio for the Cholera Model

The basic reproduction ratio is obtained by taking the dominant eigenvalue of

F V 1 = [ F i ( E 0 ) x j ] [ V i ( E 0 ) x j ] 1 (16)

where F i is the rate of appearance of new infection in compartment i, V i is the transfer of infections from one compartment i to another and E 0 is the disease-free equilibrium. From the system of differential equations we identify the equations with infectious classes as

d I d t = a ( 1 c w ) B K + B S ( τ + μ + δ ) I (17)

and

d B d t = e ( 1 c s ) I ρ B (18)

We thus have

F i = [ a ( 1 c w ) B S K + B 0 ] (19)

Hence

F = [ F 1 I F 1 B F 2 I F 2 B ] = [ 0 a ( 1 c w ) S 0 K 0 0 ] (20)

Similarly

V i = [ ( τ + μ + δ ) I e ( 1 c s ) I + ρ B ] (21)

Hence

V = [ V 1 I V 1 B V 2 I V 2 B ] = [ ( τ + μ + δ ) 0 e ( 1 c s ) ρ ] (22)

V 1 = [ ( τ + μ + δ ) 0 e ( 1 c s ) ρ ] 1 = [ 1 ( τ + μ + δ ) 0 e ( 1 c s ) ρ ( τ + μ + δ ) 1 ρ ] (23)

F V 1 = [ 0 a ( 1 c w ) S 0 K 0 0 ] [ 1 ( τ + μ + δ ) 0 e ( 1 c s ) ρ ( τ + μ + δ ) 1 ρ ] = [ a e ( 1 c w ) ( 1 c s ) S 0 K ( τ + μ + δ ) a ( 1 c w ) S 0 K ρ 0 0 ]

where S 0 = β N μ . The eigenvalues of F V 1 are given by

| F V 1 λ I | = 0 (24)

or

| [ a e ( 1 c w ) ( 1 c s ) S 0 K ( τ + μ + δ ) a ( 1 c w ) S 0 K ρ 0 0 ] λ [ 1 0 0 1 ] | = 0 (25)

Giving the characteristic equation;

K ρ ( τ + μ + δ ) λ 2 S 0 a e ( 1 c w ) ( 1 c s ) λ K ρ ( τ + μ + δ ) = 0

i.e.,

K ρ ( τ + μ + δ ) λ 2 S 0 a e ( 1 c w ) ( 1 c s ) λ = 0 (26)

λ [ K ρ ( τ + μ + δ ) λ S 0 a e ( 1 c w ) ( 1 c s ) ] = 0

λ = 0 , λ = a e ( 1 c w ) ( 1 c s ) S 0 K ρ ( τ + μ + δ ) (27)

Hence

R 0 = a e ( 1 c w ) ( 1 c s ) S 0 K ρ ( τ + μ + δ ) (28)

Substituting S 0 = β N μ .

R 0 = β N a e ( 1 c w ) ( 1 c s ) K ρ μ ( τ + μ + δ ) (29)

1.6. Control Measures with Various Strategies

As shown in Equations (2) to (5), our model incorporates three new control measures namely treatment, water hygiene and environmental sanitation. In this study, it is of main interest to see the effect of control measures at different levels of control strategies. Using the baseline values for variables and parameters as in Table 3 and Table 4 we compute the effect of control measures Cw, Cs, and τ at seven different control strategies as explained below. The seven control strategies are:

1) No control: This is a situation where there is no control ( τ = C w = C s = 0 ).

In this case

R 0 = β N a e K ρ μ ( μ + δ ) (30)

2) Low without treatment: Low efficiency level of the rate of compliance with water hygiene and rate of compliance with environmental sanitation at 25% ( τ = 0 , C w = C s = 0.25 ).

R 0 = 9 β N a e 16 K ρ μ ( μ + δ ) (31)

3) Moderate without treatment: Moderate effectiveness level of rate of compliance with water hygiene and rate of compliance with environmental sanitation at 50% ( τ = 0 , C w = C s = 0.50 ).

R 0 = β N a e 4 K ρ μ ( μ + δ ) (32)

4) High without treatment: High effectiveness level of rate of compliance with water hygiene and rate of compliance with environmental sanitation at 75% ( τ = 0 , C w = C s = 0.75 ).

R 0 = β N a e 16 K ρ μ ( μ + δ ) (33)

5) Low universal: Low level of all the three control strategies at 25% ( τ = C w = C s = 0.25 ).

R 0 = 9 β N a e 4 K ρ μ ( 1 + 4 μ + 4 δ ) (34)

6) Moderate universal: Moderate level of all the three control strategies at 50% ( τ = C w = C s = 0.50 ).

R 0 = β N a e 2 K ρ μ ( 1 + 2 μ + 2 δ ) (35)

7) High universal: High level of all the three control strategies at 75% ( τ = C w = C s = 0.75 ).

R 0 = β N a e 4 K ρ μ ( 3 + 4 μ + 4 δ ) (36)

1.7. MathCAD Simulation

β : = 0.000107 , a : = 0.00000289 , K : = 10 6 , τ : = 0.2 , ρ : = 0.33 , ω : = 0.003 , μ : = 0.0000526 , δ : = 0.2 , Ω : = 740 , c w : = 0.25 , c s : = 0.25 , e : = 10 .

Define a function that determines a vector of derivative values at any solution point (t, Y):

D ( t , Y ) : = [ β Ω a ( 1 c w ) Y 2 Y 0 K + Y 2 + ω Y 3 μ Y 0 a ( 1 c w ) Y 2 K + Y 2 Y 0 ( τ + μ + δ ) Y 1 e ( 1 c s ) Y 1 ρ Y 2 τ Y 1 ( ω + μ ) Y 3 ]

Define additional arguments for the ODE solver:

t0: = 0 : Initial value of independent variable.

Y0: = ( 640 100 10 6 0 ) : Vector of initial function values.

t1: = 1 0 : Final value of independent variable.

num: = 1 × 10 3 : Number of solution values on [t0, t1].

t: = S 1 0 : Independent variable values.

S: = S 1 1 : First solution function values.

I: = S 1 2 : Second solution function values.

B: = S 1 3 : Third solution function values.

R: = S 1 4 : Fourth solution function values.

The table of values for the matrix S is given in Figure 2.

1.8. Results and Discussion

The graphical profiles in Figures 3(a)-(d) depict respectively the variation of the

Figure 2. Cross section of solution matrix.

Figure 3. (a) Variation of susceptible population with time. (b) Variation of infective population with time. (c) Variation of concentration of vibriocholera population with time. (d) Variation of recovered human population with time.

intrinsic variables of the our model, namely the susceptible class S, the infected class I, the concentration of vibrio cholera population B, and the removed class R. The variation of the susceptible population shows a sharp rise from the initial value over time. However, the infective population deceases exponentially over the same time period, while the concentration of vibrio cholera decreases exponentially to zero due, to intervention measures. This in turn enables an increase in the recovered population, which reaches a saturation level over time.

1.9. Conclusion

An SIR-B model was modified and analyzed. The model incorporates treatment, water hygiene as well as environmental sanitation. The model employs a system of nonlinear ordinary differential equations, which is studied in detail for its stability properties. The basic reproduction ratio R0 was computed for the various control parameters and it is discovered that with proper combination of control measures, the spread of cholera could be minimized. Numerical simulation of the cholera model was performed using the numerical software MathCAD14, and the graphical profiles of the intrinsic variables are depicted. We conclude that improvement in treatment, water hygiene and the environmental sanitation is indeed effective in eradicating the cholera epidemic.

Cite this paper: Maliki, O. and Chibuike, O. (2021) A Mathematical Model for the Control of Cholera Epidemic without Natural Recovery. Applied Mathematics, 12, 655-668. doi: 10.4236/am.2021.128046.
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