Mathematical modeling of cholera dynamics with intrinsic growth considering constant interventions

A mathematical model that describes the dynamics of bacterium vibrio cholera within a fixed population considering intrinsic bacteria growth, therapeutic treatment, sanitation and vaccination rates is developed. The developed mathematical model is validated against real cholera data. A sensitivity analysis of some of the model parameters is also conducted. The intervention rates are found to be very important parameters in reducing the values of the basic reproduction number. The existence and stability of equilibrium solutions to the mathematical model are also carried out using analytical methods. The effect of some model parameters on the stability of equilibrium solutions, number of infected individuals, number of susceptible individuals and bacteria density is rigorously analyzed. One very important finding of this research work is that keeping the vaccination rate fixed and varying the treatment and sanitation rates provide a rapid decline of infection. The fourth order Runge–Kutta numerical scheme is implemented in MATLAB to generate the numerical solutions.

of bacterial spreading have been created.To more properly depict the pattern of the disease infection spread, a few of these models included two types of infection routes [1][2][3][4][5][6][7][8][9][10][11]14,16,25 . However,a significant drawback of the modelling studies that are currently being conducted on the transmission of cholera is the lack of attention given to the intrinsic dynamics of the bacteria, which results in an inadequate comprehension of the development of the bacteria and how it affects the dynamics of infection.The majority of mathematical models of cholera often make the premise that bacteria cannot survive without human assistance.In [2][3][4][5][6][7][8][9][10][11]17,18,20,25,28,30 , mathematical models that do not consider bacterial intrinsic growth are presented.This is predicated on a cholera ecological early hypothesis documented in 27 .A straightforward depiction of the rate of change for the bacterial density is made possible by the assumption.Regretfully, new ecological studies have provided compelling evidence that the bacteria may reproduce and thrive on their own in a variety of aquatic habitats.These ecological findings require more modelling effort to better understand the internal dynamics of cholera diseases and the connection between environmental persistence and disease outbreaks 16 .In 14,16 , efforts are done to develop and analyze mathematical models considering bacterial intrinsic growth.However, the main gap in the existing mathematical models of cholera are there is no mathematical model that incorporated both bacterial intrinsic growth and intervention strategies.This gap is also available in the recent publications by 14,16 .
Our goal in this work is to develop and analyze a mathematical model that incorporates both bacterial intrinsic growth and intervention strategies with an intention to fill the gap in 14,16 .Such kind of mathematical model will provide valuable advice for efficient preventive and intervention techniques against cholera outbreaks.The current mathematical model examines cholera dynamics using intrinsic bacterial growth rate and control measures that are integrated into the mathematical model of 14,16,25,28 in order to achieve this goal.In addition to the inherent bacteria growth rate, we modify the existing models [1][2][3][4][5][6][7][8][9][10][11]14,16,25 by including three controlling mechanisms, Namely; immunization, therapeutic treatment, and water sanitation. Rigorousmathematical theories are applied to examine the impacts of intrinsic bacteria growth rate, various control measures, and several cholera transmission channels 26 .
This article contains seven sections and is organized as follows.A detailed introduction to cholera modeling is presented in section "Mathematical model formulation".In section "Mathematical model formulation", model assumptions are presented and a corresponding mathematical model is formulated.The positivity of the domain of biological interest is also analyzed in this Section.The mathematical model is validated against a WHO real cholera data in section "Model validation".In section "Mathematical analysis of equilibrium solutions", is rigorously presented.The existence and stability of epidemic and endemic equilibrium solutions are examined in this Section.Section "Numerical test problems for stability of equilibrium solutions" provides information on how model parameters affect the presence and stability of equilibrium solutions.Furthermore, in section "Model parameter sensitivity analysis", sensitivity analysis of a few model parameters is given.Lastly, Section presents the conclusion and subsequent actions.Section "Conclusion and future works".

Mathematical model formulation
We have taken into account the following hypotheses when creating the mathematical model.1.
Because the infection phase is brief, there is a low risk of death and births.This assumption leads us to classify the total human population (N) into susceptible number (S), infective number (I), and recovered number (R) so that where t represents time.2. Individuals are born susceptible.3. The bacteria has an intrinsic rate of growth of r and a weight capacity of κ .Its concentration in the environ- ment is always there, and we represent it as B. 4. The bacterium can spread from person to person, from environment to environment, and from human to environment at rates of α , ξ , and β , respectively.5.The susceptible population is vaccinated at a rate of v, resulting in the removal of vS(t) people from the susceptible class and their addition to the recovered class each time.6.In order to remove aI(t) persons from the affected class and add them to the recovered class, therapeutic treatment is administered to those who have been infected at a rate of a. 7. A recovered individual is assumed to develop immunity.8. Bacteria perish as a result of water sanitation at a rate of w.Based on the previous supposition, we derive the subsequent dynamic system: (1) The rates of birth, infection recovery, natural death, and contribution from infected individuals to the environmental bacterial population are represented by the variables µ , γ , and ξ in Eqs. ( 2)-( 5).Positive values are assumed for each of these parameters.To evaluate the following system of differential equations in our mathematical analysis, we will first utilise R = N − I − S to remove Eq. ( 4).
Lemma 1 For any time t ≥ 0 , all solutions of the dynamical system in Eqs.(2)-( 5) with positive initial conditions are non-negative within the region of biological interest.

An expression for the aforementioned equation is
The particular solution to Eq. ( 12) at t = τ is derived to be   Through the same procedure, it can be easily verified that I(τ ), R(τ ), B(τ ) > 0 .Now, adding Eqs. ( 2)-( 4) , we have that Integrating the above equation with respect to time , we have that where c is a constant real number.Now, let S(0), I(0), R(0) > 0 be initial values in Ŵ , then Thus, Ŵ is positively invariant.

Model validation
The importance of the developed mathematical model is validated against the WHO cholera data for Bangladesh recorded from 1950 to 2000 29 .The model solution of infected individuals is compared with the real data of number of reported cases of cholera, see Fig. 2 and Table 1.The difference between the real data and the model solution is calculated by the relative error (RE) formula given as 31,32 where I is model solution for the infected individuals, Ĩ is the real data of infected individuals and || × || is a vector norm .Thus, it is calculated that RE = 0.5365 which indicates a good fit between the model solution and the real data, see Fig. 2.

Mathematical analysis of equilibrium solutions
In this Section, the general properties of equilibrium solutions to the dynamical system in Eqs. ( 6)-( 8) are presented.

General properties of epidemic cholera dynamics
Equations ( 6)- (8) in the model provide a disease-free state as follows: (13) Real data (see Table 1) fitted with a model solution.
We considered the infective compartments to be I and B so that the infective subsystem of the mathematical model to be The linearized system of the Eqs.( 19)-( 20) about the DFE is given as From Eqs. ( 21)-( 22), the matrix of transmissions and matrix of transsission are, respectively given as The basic reproduction number is the spectral radius of −T −1 .The eigenvalues of −T −1 are Therefore, the basic reproduction number is provided that r − w − γ < βξ α .Moreover, the expression r − w − γ < 0 must hold.Epidemiologically, this relation is meant the bacterium growth rate is lesser than the sum of the natural death rate of the bacterium and sanitation rate.The equation for the basic reproduction number's denominator demonstrates how heavily (19) Table 1.WHO cholera data for Bangladesh, see 29 .Proof Theorem 2 in 33 indicates that when ℜ 0 < 1 , the disease-free equilibrium is locally asymptotically stable.

Year
On the other hand, if the controls are insufficiently robust to ensure that ℜ 0 > 1 , the disease breakout happens and the DFE becomes unstable.
Theorem 2 (Global stability of the equilibrium point devoid of illness) The dynamical system in Eqs. ( 6)-( 8) has a cholera-free equilibrium point E 0 that is globally asymptotically stable if ℜ 0 < 1 , and unstable if ℜ 0 > 1.
Proof To demonstrate that the disease-free equilibrium E 0 is globally asymptotically stable for ℜ 0 < 1 and unsta- ble for ℜ 0 > 1 , we will employ the LaSalle invariance principle 30,35 .Let L(t) := B(t) be a Lyapunov function defined.The following Eqs.are thus valid.
From the above equation, the function dL dt is negative semi-definite for ℜ 0 ≤ 1 provided that N < (µ+δ+a)(µ+v) αµ .From the definition of ℜ 0 , we saw that r − γ − w < 0 .Therefore, the largest compact invariant set in Ŵ such that dL dt = 0 whenever ℜ 0 < 1 is the singleton disease-free equilibrium.Thus, the global asymptotically stability of the disease-free equilibrium in Ŵ is guaranteed by the LaSalle invariance principle 35 whenever ℜ 0 ≤ 1 and globally unstable for ℜ 0 > 1 .

General properties of endemic cholera dynamics
As previously stated, the DFE becomes unsustainable and the sickness will persist if the effects of the constraints are insufficient to bring ℜ 0 below unity.Now let's examine the endemic balance in order to understand the dynamics of cholera over the long run.

Proof
The following nonlinear algebraic system has solutions, which are the endemic equilibrium solutions to the dynamical system in Eqs. ( 6)- (8).
The function I = f (B) , defined as follows, is the answer to the problem above.
( Once more, we create a function B = h(I) from Eq. ( 33) in the following way.
The following derivatives make it simple to verify that the functions f and h are connected.

Lemma 2
The fact that h(I) = B > 0 implies the following relations.
Proof The results in Lemma 2 are direct consequences of the inequality We denote the expression in Eq. ( 32) as Thus, the existence of endemic equilibrium depends on the existence of solutions to the equation Assume that n = min θ, µ+v α .Then, from the results of Lemma 2, we have that 0 < θ − I < n and µ+v α θ n 2 .This leads us to the following inequality.

Considering the above equation and differentiating the functions f and g, we have
The derivatives presented in the above equations are very important in examining the behavior of the function f and g.Evaluating f ′ and g ′ at B = 0 , we have A direct consequence of the above equations is Now, lets consider the following case to analyze the existence of solution to the equation f (B) = g(B) .( 34) www.nature.com/scientificreports/Case I: f (0) > g(0) .In this case, it is clear that g(0) = 0 .Therefore, f (0) > g(0) implies that αµN (µ+v)(µ+δ+a) > 1 which means that I > 0 .Thus, since f is concave downward and g is concave upward, there exists a unique nontrivial equilibrium solution in this case.Case II: f (0) = g(0) .In this case, the relation αµN (µ+v)(µ+δ+a) 1 holds.Thus, the existence depends on the relative slopes of f and g at B = 0 .
(i) Consider the case αµN (µ+v)(µ+δ+a) = 1 .From Eqs. (44)-(45), we have that Therefore, since f is concave downward and g is concave upward, there exists a unique nontrivial equilibrium solution in this case.(ii) Consider the case αµN < 1 .From the expression in Eq. ( 46), The next two cases are the ones that follow.
Case (a): f ′ (0) > g ′ (0) .This is meant that ℜ 0 > 1 .In this case, there exists a unique nontrivial equi- librium solution as f is concave downward and g is concave upward.Case (b): f ′ (0) < g ′ (0) .This is meant that ℜ 0 < 1 .In this case, there doe not exist a nontrivial equi- librium solution as f is concave downward and g is concave upward.
Theorem 4 (Local stability of endemic equilibrium) The dynamical system in Eqs.(2)-( 5) is locally asymptotically stable in its endemic equilibrium.
Proof Assume that EE = (S 1 , I 1 , R 1 , B 1 ) represents the endemic equilibrium of the dynamical system in Eqs.
(2)-( 5).The dynamical system's Jacobian matrix (J) at EE is provided as The Jacobian matrix's characteristic polynomial is provided as where a 1 , a 2 and a 3 are given as in the following.
Theorem 5 (Global stability of endemic equilibrium) If N δ+a−v 2α holds, the dynamical system in Eqs.(2)-( 5) is in an endemic equilibrium that is globally stable.
Vol  36 to prove the global stability of endemic equilibrium EE = (S, I, R, B) .For simplicity, we drop the rate of change of the recovered individuals.Thus, the Jacobian matrix of the dynamical system is given as κ + (γ + w) and defining A(t) = I 3 + tJ , the second additive compound matrix of J is given as is the second compound matrix of A(t) , I 3 is the 3 × 3 identity matrix and t is a scalar.Thus, Define P = diag 1, I B , I B and let f denote the vector field of the dynamical system.Moreover, define P f to be the derivative of P along the direction of f.Then and Now, we define a matrix Q : where there block matrices are given as Let the Lozinski measure with respect to L ∞ vector norm be denoted by m.Then, by a direct calculation, we found that where From the assumption N ≤ δ+a−v 2α we have that 2αS + v − δ − a ≤ 0 which implies that sup(0, 2α + v − δ − a) = 0 .Moreover, from Equation ( 7), we have that βS B I = İ I − + (µ + δ + a) .Thus, we have the following results.which implies that m(Q) ≤ İ I − µ .Since 0 ≤ I(t) ≤ N , there exists T > 0 such that for t > T .The above equation implies that .
Vol:.( 1234567890) www.nature.com/scientificreports/Thus, according to the geometric approach originally proposed by Li and Muldowney, the quantity q2 is given as Therefore, the idea of geometric approach tells us that the endemic equilibrium is globally stable for N ≤ δ+a−v 2α .

Numerical test problems for stability of equilibrium solutions
In all the numerical test problems, we consider N = 250000 and κ = 10 6 .

Numerical test problems for epidemic cholera dynamics
In this sub section, we are examining the results in Theorem 1 and 2 numerically for the epidemic cholera dynamics.
Example 1 We take into account the parameter values in Table 2 in this test problem.
With the parameter values in Table 2, ℜ 0 = 0.000127412 < 1 and the only equilibrium solution to the dynami- cal system in Eqs. ( 6)-( 8) is the disease free equilibrium given as The above disease free equilibrium is, hence, stable by Theorem 1 and 2.

Example 2
In this test problem, we consider the values of parameters in Table 3.
Considering the values of parameters in Table 3, ℜ 0 = 1.6171 > 1 .In this case, we obtained two equilibrium solutions, Namely; the epidemic equilibrium and the endemic equilibrium for the dynamical system in Eqs. ( 6)-( 8).These equilibrium solutions are given as (57) q2 := lim sup  a n d (S(0), I(0), B(0)) = (10, 10, 0.001 × κ) were considered to produce the results in Fig. 4. The results are eventu- ally going away from the disease free equilibrium.Rather, the dynamical system is approaching to the endemic equilibrium.

Numerical test problems for endemic cholera dynamics
In this sub section, we are examining the results in Theorem (3-5) numerically for the endemic cholera dynamics.

Example 3
In this test problem, we consider the values of parameters in Table 4.
As can be seen from Fig. 5, the endemic equilibrium exists and is given as Moreover, the epidemic equilibrium also exists and is given as

Model parameter sensitivity analysis
The sensitivity of a model parameter (p) is meant its effect on the values of the basic reproduction number and is measured by the elasticity index defined as 37,38 From the above equation, if the sign of Y ℜ 0 p is positive, the value of ℜ 0 increases with an increase in the value of the model parameter.Moreover, if the sign of Y ℜ 0 p is negative, the value of ℜ 0 decreases with an increase in the value of the model parameter 37 .The elasticity index is very important to guide an intervention by indicating the most important model parameters to target.Based on Equation (58), the elasticity index of the intervention strategies and intrinsic bacteria growth are derived to be  Based on the values of the parameters in Table 2, the effects of vaccination, therapeutic treatment, and sanitation are shown in Figs. 7, 8 and 9.The impact of vaccination on the dynamical system is shown in Fig. 7.It is evident that a higher vaccination rate lowers the population of vulnerable people, the number of sick people, and the density of germs.As the vaccine is given to susceptible individuals, it drastically reduces the number of susceptible individuals within a short period of time as compared to the number of infected individuals and bacteria density.
Figure 8, the effect of therapeutic treatment is displayed.From the theoretical point of view, therapeutic treatment helps individuals to recover and they can not contribute bacteria to the environment.Because of this, when the rate of therapeutic treatment increases, so does the concentration of germs in the environment and the number of infected persons.The rate of therapeutic treatment and the rate of immunisation, however, are negatively correlated.This important result recommends the development for an optimal control problem.The values of the triplet control parameters (v, a, w) that simultaneously reduce the number of susceptible, infective and bacteria density can be obtained using optimal control problem.
The effect of sanitation on the dynamics of the cholera disease is presented in Fig. 9.It shows that the bacteria concentration in the environment vanishes with an increase in the sanitation rate.However, in this test situation, there have been no appreciable changes in the number of susceptible and infected persons with an increase in sanitation rate.This suggests that in order to see a change, adjustments must be made to the remaining model The effect of bacteria growth rate on the infection dynamics is presented in Fig. 10.It can be observed that an increase in the growth rate results in an increase in the concentration of the bacteria.Moreover, it is displayed in Fig. 10b that the number of susceptible individuals starts to rise after decaying due to the growth rate.

Conclusion and future works
In our assumption, we considered the bacteria intrinsic growth, vaccination, water sanitation and therapeutic treatment rate as very important parameters.The importance of these model parameters is embedded in the mathematical expression of the basic reproduction number.According to the results from sensitivity analysis, an increase in the bacteria intrinsic growth rate contributes positively to the value of the basic reproduction number.Nonetheless, it is determined that higher intervention rates have a detrimental impact on the fundamental reproduction number's value.A significant discovery of this study is that a quick drop in infection may be achieved by maintaining a constant vaccination rate while adjusting treatment and sanitation rates.
The values of model parameters are obtained either from previous research works or are assumed by the researcher.In future works, the values of the model parameters can be estimated from real data using appropriate theories of approximations and covariance of model parameters has to be carried out to examine their relationship.Moreover, from an optimization point of view, intervention strategies should have to vary with time.Therefore, the formulation of an optimal control problem for the mathematical model is recommended in future works.

Figure 1 .
Figure 1.Flow diagram of the mathematical model.

Figure 5 .
Figure 5.Comparison of I = f (B) and I = g(B) for endemic equilibrium.

0 Figure 7 .
Figure 7. Effect of vaccine on the cholera dynamics.

Figure 8 .
Figure 8.Effect of therapeutic treatment on the cholera dynamics.

Figure 9 .Figure 10 .
Figure 9.Effect of sanitation on the cholera dynamics.

Case number Death number CFR Year Case number Death number CFR
values of the control factors affect it.The fundamental reproduction number's weight can be decreased by raising the control parameter settings.Theorem 1 (Local equilibrium point stability in the absence of illness) The dynamical structure in Eqs.(6)-(8) has an unaffected by illness equilibrium point E 0 that is locally asymptotically stable if ℜ 0 < 1 , and unstable if ℜ 0 > 1.

Table 2 .
Parameter values in the case of 1.

Table 3 .
Parameter values for Example 2.

Table 4 .
Parameter values for Example 3.