Loading [MathJax]/jax/output/SVG/jax.js
Research article

On stiff, fuzzy IRD-14 day average transmission model of COVID-19 pandemic disease

  • COVID-19, a new pandemic disease is becoming one of the major threats for surviving. Many new models are arrived to study the disease mathematically. Here we are introducing a new model in which instead of studying a day by day changes we are studying the average of 14 day transmission because its life or the patients incubation period is about an average of 14 days. Also, since this is pandemic, and being not aware of susceptible population among the world's population, we considered the model without S-susceptible population. i.e., IRD-Infectious, Recovered, Death-model. In this new model, we are also introducing a new method of calculating new number called N0-average transmission number. This is used to study the average spread of infection instead of basic reproduction number R0. The motto of this paper is not to predict the daily cases but to control the current spread of disease and deaths by identifying the threshold number, exceeding which will increase the spread of infection and number of deaths due to this pandemic. Also if the 14 day average IRD-populations are maintained under this threshold number, will definitely control this pandemic disease globally. Stability analysis and test for stiff system of differential equations are studied. Our main aim is to present the medical world, a threshold population of infected, recovered and death cases for every average of 14 days to quickly overcome this pandemic disease COVID-19.

    Citation: Prasantha Bharathi Dhandapani, Dumitru Baleanu, Jayakumar Thippan, Vinoth Sivakumar. On stiff, fuzzy IRD-14 day average transmission model of COVID-19 pandemic disease[J]. AIMS Bioengineering, 2020, 7(4): 208-223. doi: 10.3934/bioeng.2020018

    Related Papers:

    [1] Roshan Ara, Saeed Ahmad, Zareen A. Khan, Mostafa Zahri . Threshold dynamics of stochastic cholera epidemic model with direct transmission. AIMS Mathematics, 2023, 8(11): 26863-26881. doi: 10.3934/math.20231375
    [2] Yuqin Song, Peijiang Liu, Anwarud Din . Analysis of a stochastic epidemic model for cholera disease based on probability density function with standard incidence rate. AIMS Mathematics, 2023, 8(8): 18251-18277. doi: 10.3934/math.2023928
    [3] Ying He, Bo Bi . Threshold dynamics and density function of a stochastic cholera transmission model. AIMS Mathematics, 2024, 9(8): 21918-21939. doi: 10.3934/math.20241065
    [4] Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi . A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model. AIMS Mathematics, 2023, 8(2): 3484-3522. doi: 10.3934/math.2023178
    [5] Moh. Mashum Mujur Ihsanjaya, Nanang Susyanto . A mathematical model for policy of vaccinating recovered people in controlling the spread of COVID-19 outbreak. AIMS Mathematics, 2023, 8(6): 14508-14521. doi: 10.3934/math.2023741
    [6] Ahmed Alshehri, Saif Ullah . Optimal control analysis of Monkeypox disease with the impact of environmental transmission. AIMS Mathematics, 2023, 8(7): 16926-16960. doi: 10.3934/math.2023865
    [7] Naret Ruttanaprommarin, Zulqurnain Sabir, Rafaél Artidoro Sandoval Núñez, Soheil Salahshour, Juan Luis García Guirao, Wajaree Weera, Thongchai Botmart, Anucha Klamnoi . Artificial neural network procedures for the waterborne spread and control of diseases. AIMS Mathematics, 2023, 8(1): 2435-2452. doi: 10.3934/math.2023126
    [8] Maysaa Al-Qurashi, Sobia Sultana, Shazia Karim, Saima Rashid, Fahd Jarad, Mohammed Shaaf Alharthi . Identification of numerical solutions of a fractal-fractional divorce epidemic model of nonlinear systems via anti-divorce counseling. AIMS Mathematics, 2023, 8(3): 5233-5265. doi: 10.3934/math.2023263
    [9] Oluwaseun F. Egbelowo, Justin B. Munyakazi, Manh Tuan Hoang . Mathematical study of transmission dynamics of SARS-CoV-2 with waning immunity. AIMS Mathematics, 2022, 7(9): 15917-15938. doi: 10.3934/math.2022871
    [10] Abeer Alshareef . Quantitative analysis of a fractional order of the SEIcIηVR epidemic model with vaccination strategy. AIMS Mathematics, 2024, 9(3): 6878-6903. doi: 10.3934/math.2024335
  • COVID-19, a new pandemic disease is becoming one of the major threats for surviving. Many new models are arrived to study the disease mathematically. Here we are introducing a new model in which instead of studying a day by day changes we are studying the average of 14 day transmission because its life or the patients incubation period is about an average of 14 days. Also, since this is pandemic, and being not aware of susceptible population among the world's population, we considered the model without S-susceptible population. i.e., IRD-Infectious, Recovered, Death-model. In this new model, we are also introducing a new method of calculating new number called N0-average transmission number. This is used to study the average spread of infection instead of basic reproduction number R0. The motto of this paper is not to predict the daily cases but to control the current spread of disease and deaths by identifying the threshold number, exceeding which will increase the spread of infection and number of deaths due to this pandemic. Also if the 14 day average IRD-populations are maintained under this threshold number, will definitely control this pandemic disease globally. Stability analysis and test for stiff system of differential equations are studied. Our main aim is to present the medical world, a threshold population of infected, recovered and death cases for every average of 14 days to quickly overcome this pandemic disease COVID-19.


    1. Introduction

    Cholera is an acute diarrhoeal illness caused by infection of the intestine with bacterium vibrio cholerae, which lives in an aquatic organism [16]. The ingestion of contaminated water can cause cholera outbreaks, as John Snow proved in 1854 [24]. This is a possibility of transmission of the disease, but others do exist. For example, susceptible individuals can become infected if they become in contact with contaminated people. When individuals are at an increased risk of infection, then they can transmit the disease to those who live with them, by reflecting food preparation or using water storage containers [24]. An individual can be infected with or without symptoms and some of these can be watery diarrhea, vomiting and leg cramps. If contaminated individuals do not get treatment, then they become dehydrated, suffering acidosis and circulatory collapse. This situation can lead to death within 12 to 24 hours [18,24]. Some studies and experiments suggest that a recovered individual can be immune to the disease during a period of 3 to 10 years. Nevertheless, recent researches conclude that immunity can be lost after a period of weeks to months [19,24].

    Since 1979, several mathematical models for the transmission of cholera have been proposed: see, e.g., [2,3,8,9,11,14,16,17,18,19,22,24] and references cited therein. In [19], the authors propose a SIR (Susceptible-Infectious-Recovered) type model and consider two classes of bacterial concentrations (hyperinfectious and less-infectious) and two classes of infectious individuals (asymptomatic and symptomatic). In [24], another SIR-type model is proposed that incorporates hyperinfectivity (where infectivity varies with time since the pathogen was shed) and temporary immunity, using distributed delays. The authors of [18] incorporate in a SIR-type model public health educational campaigns, vaccination, quarantine and treatment, as control strategies in order to curtail the disease.

    Between 2007 and 2018, several cholera outbreaks occurred, namely in Angola, Haiti, Zimbabwe and Yemen [1,24,31]. Mathematical models have been developed and studied in order to understand the transmission dynamics of cholera epidemics, mostly focusing on the epidemic that occurred in Haiti, 2010–2011 [1]. In [16], a SIQR (Susceptible-Infectious-Quarantined-Recovered) type model, which also considers a class for bacterial concentration, is analysed, being shown that it fits well the cholera outbreak in the Department of Artibonite - Haiti, from 1st November 2010 until 1st May 2011. Furthermore, an optimal control problem is formulated and solved, where the optimal control function represents the fraction of infected individuals that are submitted to treatment through quarantine [16]. Many studies have been developed with the purpose to find and evaluate measures to contain the cholera spread. Nevertheless, it was not possible yet to obtain solutions in real time that can stop the cholera epidemics [1].

    Recently, the biggest outbreak of cholera in the history of the world has occurred in Yemen [26]. The epidemic began in October 2016 and in February-March 2017 was in decline. However, on 27th April 2017 the epidemic returned. This happened ten days after Sana'a's sewer system had stopped working. Problems in infrastructures, health, water and sanitation systems in Yemen, allowed the fast spread of the disease [28]. Between 27th April 2017 and 15th April 2018 there were 1 090 280 suspected cases reported and 2 275 deaths due to cholera [32]. In [20], Nishiura et al. study mathematically this outbreak, trying to forecast the cholera epidemic in Yemen by explicitly addressing the reporting delay and ascertainment bias. A vaccine for cholera is currently available, but poor sanitation and the lack of access to vaccines promote the spread of the disease [1].

    Following World Health Organization (WHO) recommendations, the first-ever oral vaccination campaign against cholera had been launched on 6th May 2018 in Yemen, but it was concluded on 15th May 2018 [30], due to the lack of national governmental authorization to do the vaccination [23]. Aid workers say that one reason of the delayed of the campaign vaccination is due to some senior Houthi officials who objected to vaccination. This campaign coincided with the rainy season and some health workers fear that this could spread the disease [23]. This campaign just covered four districts in Aden, which were at a high risk of fast spread of the disease, and just 350 000 individuals (including pregnant women) were vaccinated [23,26]. It is important to note that the vaccinated individuals represent, approximately, 1.21% of the total population, since the total population of Yemen, in 2018, is 28 915 284 [34]. Lorenzo Pizzoli, WHO's cholera expert, said that the campaign hoped to cover at least four million people in areas at risk (corresponding, approximately, to 14% of total population) and Michael Ryan, WHO's Assistant Director-General, revealed that they were negotiating with Yemen health authorities in order to vaccinate people from all high risks zones [23]. The International Coordinating Group on Vaccine Provision had planned one million cholera vaccines for Yemen in July 2017, but WHO and Yemen local authorities decided to postpone it and the doses were diverted to South Sudan. WHO and GAVI, the Vaccine Alliance, affirmed that the largest cholera vaccination is now being carried out in five countries (Kenya, Malawi, South Sudan, Uganda and Zambia). It is expected that this campaign targets more than two million people across Africa [23].

    In this paper, we propose a SITRV (Susceptible-Infectious-Treated-Recovered-Vaccinated) type model, which includes a class of bacterial concentration. In Section 2, we formulate and explain the mathematical model. Then, in Section 3, we show that the model is mathematically well posed and it has biological meaning. We prove the existence and uniqueness of the disease-free and endemic equilibrium points and compute the basic reproduction number. The sensitivity of the basic reproduction number with respect to all parameters of the model is analysed in Subsection 3.3 and the stability analysis of equilibria is carried out in Subsection 3.4. In Section 4, we show that the model fits well the cholera outbreak in Yemen, between 27th April 2017 and 15th April 2018. Through numerical simulations, we illustrate the impact of vaccination of susceptible individuals in Yemen. We end with Section 5, by deriving some conclusions about the importance of vaccination campaigns on the control and eradication of a cholera outbreak.

    We trust that our work is of great significance, because it provides a mathematical model for cholera that is deeply studied and allows to obtain important conclusions about the relevance of vaccination campaigns in cholera outbreaks. We show that, if it had existed a vaccination campaign from the beginning of the outbreak in Yemen, then the epidemic would have been extinguished. Actually, we believe that the absence of this type of prevention measures in Yemen, was one of the responsible for provoking the biggest cholera outbreak in world's history [26], killing 2 275 individuals until 15th April 2018.


    2. Model formulation

    We modify the model studied in [16], adding a vaccination class and considering different kinds of cholera's treatment. The model is a SITRV (Susceptible-Infectious-Treated-Recovered-Vaccinated) type model and considers a class of bacterial concentration for the dynamics of cholera. The total human population N(t) is divided into five classes: susceptible S(t), infectious with symptoms I(t), in treatment T(t), recovered R(t) and vaccinated V(t) at time t, for t0. Furthermore, we consider a class B(t) that reflects the bacterial concentration at time t. We assume that there is a positive recruitment rate Λ into the susceptible class S(t) and a positive natural death rate μ, for all time t under study. Susceptible individuals can be vaccinated at rate φ0 and become infected with cholera at rate βB(t)κ+B(t)0, which is dependent on time t. Note that β>0 is the ingestion rate of the bacterium through contaminated sources, κ>0 is the half saturation constant of the bacterial population, and B(t)κ+B(t) measures the possibility of an infected individual to have the disease with symptoms, given a contact with contaminated sources [18]. Any recovered and vaccinated individual can lose the immunity at rate ω10 and ω20, respectively, becoming susceptible again. Different types of treatment for cholera infected individuals are considered based on [29]. The infected individuals can get a proper treatment, at rate δ0, and the individuals in treatment can recover at rate ε0. The disease-related death rates associated with the individuals that are infected and in treatment are α10 and α20, respectively. Each infected individual contributes to the increase of the bacterial concentration at rate η>0. On the other hand, the bacterial concentration can decrease at mortality rate d>0. These assumptions are translated into the following mathematical model:

    {S(t)=ΛβB(t)κ+B(t)S(t)+ω1R(t)+ω2V(t)(φ+μ)S(t),I(t)=βB(t)κ+B(t)S(t)(δ+α1+μ)I(t),T(t)=δI(t)(ε+α2+μ)T(t),R(t)=εT(t)(ω1+μ)R(t),V(t)=φS(t)(ω2+μ)V(t),B(t)=ηI(t)dB(t). (2.1)

    3. Model analysis

    Throughout the paper, we assume that the initial conditions of system (2.1) are non-negative:

    S(0)=S00, I(0)=I00, T(0)=T00, R(0)=R00, V(0)=V00, B(0)=B00. (3.1)

    3.1. Positivity and boundedness of solutions

    Our first lemma shows that the considered model (2.1)-(3.1) is biologically meaningful.

    Lemma 1. The solutions (S(t),I(t),T(t),R(t),V(t),B(t)) of system (2.1) are non-negative for all t0 with non-negative initial conditions (3.1) in (R+0)6.

    Proof. We have

    {dS(t)dt|ξ(S)=Λ+ω1R(t)+ω2V(t)>0,dI(t)dt|ξ(I)=βB(t)κ+B(t)S(t)0,dT(t)dt|ξ(T)=δI(t)0,dR(t)dt|ξ(R)=εT(t)0,dV(t)dt|ξ(V)=φS(t)0,dB(t)dt|ξ(B)=ηI(t)0,

    where ξ(υ)={υ(t)=0 and S,I,T,R,V,BC(R+0,R+0)} and υ{S,I,T,R,V,B}. Therefore, due to Lemma 2 in [33], any solution of system (2.1) is such that (S(t),I(t),T(t),R(t),V(t),B(t))(R+0)6 for all t0.

    Lemma 2 shows that it is enough to consider the dynamics of the flow generated by (2.1)-(3.1) in a certain region ΩV.

    Lemma 2. Let

    ΩH={(S,I,T,R,V)(R+0)5|0S(t)+I(t)+T(t)+R(t)+V(t)Λμ} (3.2)

    and

    ΩB={BR+0|0B(t)Λημd}. (3.3)

    Define

    ΩV=ΩH×ΩB. (3.4)

    If N(0)Λμ and B(0)Λημd, then the region ΩV is positively invariant for model (2.1) with non-negative initial conditions (3.1) in (R+0)6.

    Proof. Let us split system (2.1) into two parts: the human population, i.e., S(t), I(t), T(t), R(t) and V(t), and the pathogen population, i.e., B(t). Adding the first five equations of system (2.1) gives

    N(t)=S(t)+I(t)+T(t)+R(t)+V(t)=ΛμN(t)α1I(t)α2T(t)ΛμN(t).

    Assuming that N(0)Λμ, we conclude that N(t)Λμ. For this reason, (3.2) defines the biologically feasible region for the human population. As it is proved in [16], the region (3.3) defines the biologically feasible region for the pathogen population. From (3.2) and (3.3), we know that N(t) and B(t) are bounded for all t0. Therefore, every solution of system (2.1) with initial conditions in ΩV remains in ΩV for all t0. In other words, in region ΩV defined by (3.4), our model is epidemiologically and mathematically well posed in the sense of [10].


    3.2. Equilibrium points and basic reproduction number

    From now on, let us consider that a0=φ+μ, a1=δ+α1+μ, a2=ε+α2+μ, a3=ω1+μ and a4=ω2+μ. The disease-free equilibrium (DFE) of model (2.1) is given by

    E0=(S0,I0,T0,R0,V0,B0)=(Λa4a0a4φω2,0,0,0,Λφa0a4φω2,0). (3.5)

    Remark 1. Note that, because μ>0, one has a0a4φω2=(φ+μ)(ω2+μ)φω2>0.

    Next, following the approach of [18,27], we compute the basic reproduction number R0.

    Proposition 3(Basic reproduction number of (2.1)). The basic reproduction number of model (2.1) is given by

    R0=βΛηa4(a0a4φω2)κda1. (3.6)

    Proof. Consider that Fi(t) is the rate of appearance of new infections in the compartment associated with index i, V+i(t) is the rate of transfer of "individuals" into the compartment associated with index i by all other means and Vi(t) is the rate of transfer of "individuals" out of compartment associated with index i. In this way, the matrices F(t), V+(t) and V(t), associated with model (2.1), are given by

    F(t)=[0βB(t)S(t)κ+B(t)0000],V+(t)=[Λ+ω1R(t)+ω2V(t)0δI(t)εT(t)φS(t)ηI(t)] and V(t)=[βB(t)S(t)κ+B(t)+a0S(t)a1I(t)a2T(t)a3R(t)a4V(t)dB(t)].

    Therefore, by considering V(t)=V(t)V+(t), we have that

    [S(t)I(t)T(t)R(t)V(t)B(t)]T=F(t)V(t).

    The Jacobian matrices of F(t) and of V(t) are, respectively, given by

    F=[000000βB(t)κ+B(t)0000βκS(t)(κ+B(t))2000000000000000000000000] and V=[βB(t)κ+B(t)+a000ω1ω2βκS(t)(κ+B(t))20a100000δa200000εa300φ000a400η000d].

    In the disease-free equilibrium E0 defined by (3.5), we obtain the matrices F0 and V0 given by

    F0=[00000000000βΛa4(a0a4φω2)κ000000000000000000000000] and V0=[a000ω1ω2βΛa4(a0a4φω2)κ0a100000δa200000εa300φ000a400η000d].

    The basic reproduction number of model (2.1) is then given by

    R0=ρ(F0V10)=βΛηa4(a0a4φω2)κda1,

    found with the help of the computer algebra system Maple. This concludes the proof.

    Now we prove the existence of an endemic equilibrium when R0 given by (3.6) is greater than one.

    Proposition 4 (Endemic equilibrium). If the basic reproduction number (3.6) is such that R0>1, then the model (2.1) has an endemic equilibrium given by

    E=(S,I,T,R,V,B), (3.7)

    where

    {S=a1a4{κd(a1a2a3δεω1)+Ληa2a3}η˜D,I=a2a3{βΛηa4(a0a4φω2)κda1}η˜D,T=a3δ{βΛηa4(a0a4φω2)κda1}η˜D,R=δε{βΛηa4(a0a4φω2)κda1}η˜D,V=a1φ{κd(a1a2a3δεω1)+Ληa2a3}η˜D,B=a2a3{βΛηa4(a0a4φω2)κda1}d˜D

    and ˜D=a1a2a3(a0a4φω2)+βa4(a1a2a3δεω1).

    Proof. We note that

    1. a1=δ+α1+μ>0, because δ, α10 and μ>0;

    2. a2=ε+α2+μ>0, because ε, α20 and μ>0;

    3. a3=ω1+μ>0, because ω10 and μ>0;

    4. β, κ, d>0 and δ,ε,φ0;

    5. a0a4φω2>0 (see Remark 1);

    6. a1a2a3δεω1=(δ+α1+μ)(ε+α2+μ)(ω1+μ)δεω1>0, because α1, α20 and μ>0;

    7. Ληa2a3>0, because Λ, η, a2, a3>0.

    With the above inequalities, we conclude that ˜D>0 and, consequently, that S>0 and V0. The basic reproduction number is given by βΛηa4(a0a4φω2)κda1. Thus, it follows that

     βΛηa4=R0(a0a4φω2)κda1 βΛηa4(a0a4φω2)κda1=R0(a0a4φω2)κda1(a0a4φω2)κda1 βΛηa4(a0a4φω2)κda1=(a0a4φω2)κda1(R01).

    Therefore, we have that

    {I=a1a2a3κd(a0a4φω2)(R01)η˜D,T=a1a3κdδ(a0a4φω2)(R01)η˜D,R=a1κdδε(a0a4φω2)(R01)η˜D,B=a1a2a3κ(a0a4φω2)(R01)˜D.

    In order to obtain an endemic equilibrium, we have to ensure that I, B>0. Thus, we obtain I, B>0 if and only if R01>0R0>1. In this case (R0>1), we also have that T, R0.


    3.3. Sensitivity of the basic reproduction number

    In this section, we are going to study the sensitivity of R0 with respect to all parameters p of model (2.1), computing the respective normalized forward sensitive indexes ΥR0p given in Definition 1. They are presented in Table 1.

    Table 1. The normalized forward sensitivity indexes ΥR0p with respect to all parameters of model (2.1).
    Parameter pΥR0p
    Λ1
    μ μ(1a41a1φ+ω2+2μa0a4φω2)
    β1
    κ-1
    ω10
    ω2 φω2a4(φ+ω2+μ)
    φ φφ+ω2+μ
    δ δa1
    ε0
    α1 α1a1
    α20
    η1
    d-1
     | Show Table
    DownLoad: CSV

    Definition 1 (See [6,15,25]). The normalized forward sensitivity index of a variable z that depends differentiably on a parameter p is defined by

    Υzp=zp×p|z|.

    Remark 2. When a parameter p is one of the most sensitive parameters with respect to a variable z, then we have Υzp=±1. If Υzp=1, then an increase (decrease) of p by γ% provokes an increase (decrease) of z by γ%. On the other hand, if Υzp=1, then an increase (decrease) of p by γ% provokes a decrease (increase) of z by γ% (see [25]).


    3.4. Stability analysis

    Now we prove the local stability of the disease-free equilibrium E0.

    Theorem 5 (Stability of the DFE (3.5)). The disease-free equilibrium E0 of model (2.1) is

    1. locally asymptotic stable, if R0<1;

    2. unstable, if R0>1.

    Moreover, if R0=1, then a critical case occurs.

    Proof. The characteristic polynomial associated with the linearised system of model (2.1) is given by

    pV(χ)=det(F0V0χI6).

    In order to compute the roots of the polynomial pV, we have that

    |a0χ00ω1ω2βΛa4(a0a4φω2)κ0a1χ000βΛa4(a0a4φω2)κ0δa2χ00000εa3χ00φ000a4χ00η000dχ|=0,

    that is,

    χ2+(a0+a4)χ+(a0a4φω2)=0χ2+(a1+d)χ+a1dβΛηa4(a0a4φω2)κ=0χ=a2χ=a3.

    As the coefficients of polynomial χ2+(a0+a4)χ+(a0a4φω2) have the same sign (see Remark 1), then it follows from Routh's criterion that their roots have negative real part (see, e.g., pp.~55-56 of [21]). Furthermore, using similar arguments, the roots of the polynomial

    χ2+(a1+d)χ+a1dβΛηa4(a0a4φω2)κ

    have negative real part if and only if

    a1dβΛηa4(a0a4φω2)κ>0R0<1.

    Therefore, the DFE E0 is locally asymptotic stable if R0<1; unstable if R0>1; critical if R0=1.

    We end this section by proving the local stability of the endemic equilibrium E. Our proof is based on the Center Manifold Theory [4], as described in Theorem 4.1 of [5].

    Theorem 6 (Local asymptotic stability of the endemic equilibrium (3.7)). The endemic equilibrium E of model (2.1) (see Proposition 4) is locally asymptotic stable for R0>1.

    Proof. In order to apply the method described in Theorem 4.1 of [5], we are going to do the following change of variables. Let us consider that

    X=(x1,x2,x3,x4,x5,x6)=(S,I,T,R,V,B).

    Consequently, we have that the total number of individuals is given by N=5i=1xi. Thus, the model (2.1) can be written as follows:

    {x1(t)=f1=Λβx6(t)κ+x6(t)x1(t)+ω1x4(t)+ω2x5(t)a0x1(t),x2(t)=f2=βx6(t)κ+x6(t)x1(t)a1x2(t),x3(t)=f3=δx2(t)a2x3(t),x4(t)=f4=εx3(t)a3x4(t),x5(t)=f5=φx1(t)a4x5(t),x6(t)=f6=ηx2(t)dx6(t). (3.8)

    Choosing β as bifurcation parameter and solving for β, from R0=1 we have that

    β=(a0a4φω2)κda1Ληa4.

    Considering β=β, the Jacobian of the system (3.8) evaluated at E0 is given by

    J0=[a000ω1ω2a1dη0a1000a1dη0δa200000εa300φ000a400η000d].

    The eigenvalues of J0 are obtained solving the equation det(J0χI6)=0. Thus, we have that

     det(J0χI6)=0 χ=0  χ=a1d  χ=a2  χ=a3  χ=12(a0+a4±(a0a4)2+4φω2).

    Note that the eigenvalue χ=12(a0+a4(a0a4)2+4φω2) is a negative real number, because

    12(a0+a4(a0a4)2+4φω2)=12(φ+μ+ω2+μ(φ+μω2μ)2+4φω2)=12(φ+ω2+2μφ22φω2+ω22+4φω2)=12(φ+ω2+2μ(φ+ω2)2)=φ+ω2012(φ+ω2+2μ(φ+ω2))=μ<0.

    Therefore, we can conclude that a simple eigenvalue of J0 is zero, while all other eigenvalues of J0 have negative real part. So, the Center Manifold Theory [4] can be applied to study the dynamics of (3.8) near β=β. Theorem 4.1 in [5] is used to show the local asymptotic stability of the endemic equilibrium point of (3.8), for β near β. The Jacobian J0 has, respectively, a right eigenvector and a left eigenvector (associated with the zero eigenvalue),

    w=[w1  w2  w3  w4  w5  w6]T and v=[v1  v2  v3  v4  v5  v6],

    given by

    w=[a4(a1a2a3δεω1)a2a3(a0a4φω2)  1  δa2  δεa2a3   φa4  ηd]Tw2

    and

    v=[0  1  0  0  0   a1η]v2.

    Remember that fl represents the right-hand side of the lth equation of the system (3.8) and xl is the state variable whose derivative is given by the lth equation, l=1,,6. The local stability near the bifurcation point β=β is determined by the signs of two associated constants a and b defined by

    a=6i,j,k=1wiwjvk[2fkxixj(E0)]β=β and b=6i,k=1wivk[2fkxiϕ(E0)]β=β

    with ϕ=ββ. As v1=v3=v4=v5=0, we only have to consider the following non-zero partial derivatives at the disease free equilibrium E0:

    [2f2x1x6(E0)]β=β=[2f2x6x1(E0)]β=β=βκ  and  [2f2x26(E0)]β=β=2βΛa4a0a4φω2.

    Therefore, the constant a is

    a=2βηa4d(a0a4φω2)(a1a2a3δεω1a2a3κ+Ληd)v2w22<0.

    Furthermore, we have that

    b=v2w6[2fx6ϕ(E0)]β=β=Ληa4κd(a0a4φω2)v2w2>0.

    Thus, as

    {a<0b>0ϕ=ββ=a1κd(a0a4φω2)Ληa4(R01)>0{a<0b>0R0>1,

    we conclude from Theorem 4.1 in [5] that the endemic equilibrium E of (2.1) is locally asymptotic stable for a value of the basic reproduction number such that R0>1.


    4. Numerical simulations

    In this section, we simulate the worst cholera outbreak that ever occurred in human history. It occurred in Yemen, from 27th April 2017 to 15th April 2018 [31]. As the first-ever oral cholera vaccination campaign had been launched only on 6th May 2018 and was concluded on 15th May 2018 [30], to describe such reality of Yemen, a numerical simulation of our model is carried out with φ=ω2=V(0)=0, that is, in absence of vaccination, and with all the other values as in Table 2. We also simulate an hypothetical situation that includes vaccination from the beginning of the outbreak, considering in that case all values of Table 2. Let us denote the numerical simulation without and with vaccination by (NS) and (NSV), respectively. The curves of infected individuals for (NS) and (NSV) can be observed in Figure 1, respectively in blue solid line and in blue dashed line. Our results allow us to state that if a vaccination campaign had been considered earlier in time, the number of infected individuals would have been significantly lower. Furthermore, the basic reproduction number of the simulation without vaccination is R06.132305>1 and the one with vaccination is R00.753969<1. This means that if vaccination had been considered from the beginning of the outbreak, then the spread of cholera would have been extinguished. Consequently, there would not have been so many deaths. Note that the decrease of R0 with the introduction of a vaccination campaign is expected, because

    Table 2. Parameter values and initial conditions for the SITRVB model (2.1).
    ParameterDescriptionValueReference
    ΛRecruitment rate28.4N(0)/365 000 (day1) [12]
    μNatural death rate1.6×105 (day1) [13]
    βIngestion rate0.01694 (day1)Assumed
    κHalf saturation constant 107 (cell/ml)Assumed
    ω1Immunity waning rate0.4/365 (day1) [19]
    ω2Efficacy vaccination waning rate1/1 460 (day1) [7]
    φVaccination rate5/1 000 (day1)Assumed
    δTreatment rate1.15 (day1)Assumed
    εRecovery rate0.2 (day1) [18]
    α1Death rate (infected)6×106 (day1) [13,31]
    α2Death rate (in treatment)3×106 (day1)Assumed
    ηShedding rate (infected)10 (cell/ml day1 person1) [3]
    dBacteria death rate0.33 (day1) [3]
    S(0)Susceptible individuals at t=028 249 670 (person) [34]
    I(0)Infected individuals at t=0750 (person) [31]
    T(0)Treated individuals at t=00 (person)Assumed
    R(0)Recovered individuals at t=00 (person)Assumed
    V(0)Vaccinated individuals at t=00 (person) [30]
    B(0)Bacterial concentration at t=0 275×103 (cell/ml)Assumed
     | Show Table
    DownLoad: CSV
    Figure 1. Numerical solution of model (2.1)-(3.1) with the parameters of Table 2 without (blue solid line) and with (blue dashed line) vaccination and real data (black solid line) of infected individuals I(t) in Yemen from 27th April 2017 to 15th April 2018.
    ΥR0φ=φφ+ω2+μ0.877050<0.

    Furthermore, for (NS), we obtain an endemic equilibrium point given by

    (S,I,T,R,V,B)=(2.943350×107, 1.035599×105, 5.954131×105, 1.070992×108, 0, 3.138180×106)

    and for (NSV) we have a disease-free equilibrium point given by

    (S0,I0,T0,R0,V0,B0)=(1.689119×107, 0, 0, 0, 1.204910×108, 0).

    Note that the previous figures correspond to the equilibrium points for the parameter values of Table 2, which can be obtained numerically for a final time of approximately 1 370 years. We also call attention to the fact that the recruitment rate Λ of Yemen is big and this leads to a huge growth of the population.


    5. Conclusion

    In this paper, we proposed and analysed, analytically and numerically, a SITRVB model for cholera transmission dynamics. In order to fit the biggest cholera outbreak worldwide, which has occurred very recently in Yemen, we simulated the outbreak of Yemen without vaccination. Indeed, vaccination did not exist in Yemen from 27th April 2017 to 15th April 2018. Simulations of our mathematical model, with and without vaccination, show that the introduction of vaccination from the beginning of the epidemic could have changed the situation in Yemen substantially, to the case R0<1, where the disease extinguishes naturally. Therefore, our research motivates and fortify the importance of vaccination in cholera epidemics.


    Acknowledgments

    This research was supported by the Portuguese Foundation for Science and Technology (FCT) within projects UID/MAT/04106/2013 (CIDMA), and PTDC/EEI-AUT/2933/2014 (TOCCATA), funded by Project 3599 - Promover a Produção Científica e Desenvolvimento Tecnológico e a Constituição de Redes Temáticas, and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionalização (POCI). Lemos-Paião is also supported by the FCT Ph.D. fellowship PD/BD/114184/2016, Silva by the postdoctoral grant SFRH/BPD/72061/2010.

    The authors are very grateful to an anonymous referee for reading their paper carefully and for several constructive remarks, questions and suggestions.


    Conflict of interest

    The authors declare that there is no conflicts of interest in this paper.



    Acknowledgments



    This research paper was not supported by any funds or grants from any government or non-government sectors. The authors would like to thank the anonymous reviewers for their useful suggestions in making the paper a better one.

    Conflict of interest



    All authors declare no conflict of interest.

    [1] Zadeh LA (1965) Fuzzy sets. Inform Contr 8: 338-353. doi: 10.1016/S0019-9958(65)90241-X
    [2] Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. doi: 10.1109/TSMC.1972.5408553
    [3] Buckley JJ, Feuring T (2000) Fuzzy differential equations. Fuzzy Set Syst 110: 43-54. doi: 10.1016/S0165-0114(98)00141-9
    [4] Dubois D, Prade H (1982) Towards fuzzy differential calculus, Part 3: Differentiation. Fuzzy Set Syst 8: 225-233. doi: 10.1016/S0165-0114(82)80001-8
    [5] Lupulescu V (2009) On a class of fuzzy functional differential equations. Fuzzy set Syst 160: 1547-1562. doi: 10.1016/j.fss.2008.07.005
    [6] Kaleva O (1987) Fuzzy differential equations. Fuzzy Set Syst 24: 301-317. doi: 10.1016/0165-0114(87)90029-7
    [7] Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Set Syst 35: 389-386. doi: 10.1016/0165-0114(90)90010-4
    [8] Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Set Syst 105: 133-138. doi: 10.1016/S0165-0114(97)00233-9
    [9] Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Set Syst 24: 319-330. doi: 10.1016/0165-0114(87)90030-3
    [10] Diamond P, Kloeden P (1984)  Metric Spaces of Fuzzy Sets: Theory and Applications Singapore: World Scientific.
    [11] Chalco-Cano Y, Roman-Flores H (2008) On new solutions of fuzzy differential equation. Chaos Soliton Fract 38: 112-119. doi: 10.1016/j.chaos.2006.10.043
    [12] Abbasbandy S, Viranloo TA (2002) Numerical solution of fuzzy differential equation by Taylor method. Comput Meth Appl mat 2: 113-124. doi: 10.2478/cmam-2002-0006
    [13] Abbasbandy S, Viranloo TA (2004) Numerical solution of fuzzy differential equation by Runge-Kutta method. Nonlinear Stud 11: 117-129.
    [14] Curtiss CF, Hirschfelder JO (1952) Integration of stiff equations. Proc Natl Acad Sci USA 38: 235-243. doi: 10.1073/pnas.38.3.235
    [15] Söderlind G, Jay L, Calvo M (2015) Stiffness 1952–2012: Sixty years in search of a definition. BIT Numer Math 55: 531-558. doi: 10.1007/s10543-014-0503-3
    [16] Shampine LF (1981) Evaluation of a test set for stiff ODE solvers. ACM Trans Math Softw 7: 409-420. doi: 10.1145/355972.355973
    [17] Higham DJ, Trefethen LN (1993) Stiffness of ODEs. BIT 33: 285-303. doi: 10.1007/BF01989751
    [18] Spijker MN (1996) Stiffness in numerical initial-value problems. J Comp Appl Math 72: 393-406. doi: 10.1016/0377-0427(96)00009-X
    [19] Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc Roy Soc Lond A 115: 700-721. doi: 10.1098/rspa.1927.0118
    [20] Palese P, Young JF (1982) Variation of influenza A, B, and C viruses. Science 215: 1468-1474. doi: 10.1126/science.7038875
    [21] Allen LJS (2007)  An Introduction to Mathematical Biology NJ: Prentice Hall.
    [22] He S, Tang S, Rong L (2020) A discrete stochastic model of the COVID-19 outbreak: Forecast and control. MBE 17: 2792-2804. doi: 10.3934/mbe.2020153
    [23] Zhou W, Wang A, Xia F, et al. (2020) Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak. MBE 17: 2693-2707. doi: 10.3934/mbe.2020147
    [24] Yin F, Lv J, Zhang X, et al. (2020) COVID-19 information propagation dynamics in the chinese sina-microblog. MBE 17: 2676-2692. doi: 10.3934/mbe.2020146
    [25] Dai C, Yang J, Wang K (2020) Evaluation of prevention and control interventions and its impact on the epidemic of coronavirus disease 2019 in Chongqing and Guizhou Provinces. MBE 17: 2781-2791. doi: 10.3934/mbe.2020152
    [26] Rong X, Yang L, Chu H, et al. (2020) Effect of delay in diagnosis on transmission of COVID-19. MBE 17: 2725-2740. doi: 10.3934/mbe.2020149
    [27] Tian J, Wu J, Bao Y, et al. (2020) Modeling analysis of COVID-19 based on morbidity data in Anhui, China. MBE 17: 2842-2852. doi: 10.3934/mbe.2020158
    [28] Li C, Xu J, Liu J, et al. (2020) The within-host viral kinetics of SARS-CoV-2. MBE 17: 2853-2861. doi: 10.3934/mbe.2020159
    [29] Volpert V, Banerjee M, Petrovskii S (2020) On a quarantine model of coronavirus infection and data analysis. Math Model Nat Phenom 15: 24. doi: 10.1051/mmnp/2020006
    [30] Tang B, Bragazzi NL, Li Q, et al. (2020) An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov). Infect Dis Model 5: 248-225.
    [31] Yang Z, Zeng Z, Wang K, et al. (2020) Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions. J Thorac Dis 12: 165-174. doi: 10.21037/jtd.2020.02.64
    [32] Tuli S, Tuli S, Tuli R, et al. (2020) Predicting the growth and trend of COVID-19 pandemic using machine learning and Cloud computing. Internet Thing 11: 100222. doi: 10.1016/j.iot.2020.100222
    [33] Pai C, Bhaskar A, Rawoot V (2020) Investigating the dynamics of COVID-19 pandemic in India under lockdown. Chaos Soliton Fract 138: 109988. doi: 10.1016/j.chaos.2020.109988
    [34] Ribeiro MHDM, Da Silva RG, Mariani VC, et al. (2020) Short-term forecasting COVID-19 cumulative confirmed cases: Perspectives for Brazil. Chaos Soliton Fract 135: 109853. doi: 10.1016/j.chaos.2020.109853
    [35] Khan MA, Atangana A (2020) Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. doi: 10.1016/j.aej.2020.02.033
    [36] Gong X, Fatmawati, Khan MA (2020) A numerical solution of the competition model among bank data in Caputo-Fabrizio derivative. doi: 10.1016/j.aej.2020.02.008
    [37] Jan R, Khan MA, Gómez-Aguilar JF (2020) Asymptomatic carriers in transmission dynamics of dengue with control interventions. Optim Control Appl Meth 41: 430-447. doi: 10.1002/oca.2551
    [38] Khan MA, Ullah S, Ullah S, et al. (2020) Fractional order SEIR model with generalized incidence rate. AIMS Mathematics 5: 2843-2857. doi: 10.3934/math.2020182
    [39] Windarto, Khan MA, Fatmawati (2020) Parameter estimation and fractional derivatives of dengue transmission model. AIMS Mathematics 5: 2758-2779. doi: 10.3934/math.2020178
    [40] Dhandapani PB, Baleanu D, Thippan J, et al. (2019) Fuzzy type RK4 solutions to fuzzy hybrid retarded delay differential equations. Front Phys 7: 168. doi: 10.3389/fphy.2019.00168
    [41] Dhandapani PB, Thippan J, Sivakumar V (2019) Numerical solution of fuzzy multiple hybrid single retarded delay differential equations. Int J Recent Technol Eng 8: 1946-1949.
    [42] Dhandapani PB, Thippan J, Sivakumar V (2019) Numerical solutions of fuzzy multiple hybrid single neutral delay differential equations. Int J Sci Technol Res 8: 520-523.
    [43] Daily updates of coronavirus COVID-19 pandemic disease.Available from: https://www.worldometers.info/coronavirus/.
  • This article has been cited by:

    1. Ibrahim M. Hezam, Abdelaziz Foul, Adel Alrasheedi, A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen, 2021, 2021, 1687-1847, 10.1186/s13662-021-03271-6
    2. Maria Francesca Carfora, Isabella Torcicollo, Identification of epidemiological models: the case study of Yemen cholera outbreak, 2020, 0003-6811, 1, 10.1080/00036811.2020.1738402
    3. Lilis Suryani, W. M. Kusumawinahyu, N. Shofianah, Dynamical Analysis on A Model of Cholera Epidemic with Quarantine, Vaccination, and Two Path of Transmissions, 2021, 1776, 1742-6588, 012052, 10.1088/1742-6596/1776/1/012052
    4. Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres, Ezio Venturino, Optimal Control of Aquatic Diseases: A Case Study of Yemen’s Cholera Outbreak, 2020, 185, 0022-3239, 1008, 10.1007/s10957-020-01668-z
    5. Olena Kostylenko, Helena Sofia Rodrigues, Delfim F. M. Torres, The spread of a financial virus through Europe and beyond, 2019, 4, 2473-6988, 86, 10.3934/Math.2019.1.86
    6. Mohamed A. Zaitri, Cristiana J. Silva, Delfim F. M. Torres, Stability Analysis of Delayed COVID-19 Models, 2022, 11, 2075-1680, 400, 10.3390/axioms11080400
    7. Ana P. Lemos-Paião, Helmut Maurer, Cristiana J. Silva, Delfim F. M. Torres, Vitaly Volpert, A SIQRB delayed model for cholera and optimal control treatment, 2022, 17, 0973-5348, 25, 10.1051/mmnp/2022027
    8. Kumama Regassa Cheneke, Koya Purnachandra Rao, Gereme Kenassa Edesssa, Harvinder S. Sidhu, A New Generalized Fractional-Order Derivative and Bifurcation Analysis of Cholera and Human Immunodeficiency Co-Infection Dynamic Transmission, 2022, 2022, 1687-0425, 1, 10.1155/2022/7965145
    9. Wahyudin Nur, Agus Suryanto, Wuryansari Muharini Kusumawinahyu, Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent, 2021, 9, 2227-7390, 1858, 10.3390/math9161858
    10. Olukayode Adebimpe, Isaac Adesina, Babatunde Gbadamosi, Olajumoke Oludoun, Oluwakemi Abiodun, Olukayode Adeyemo, 2022, Chapter 53, 978-981-16-2101-7, 591, 10.1007/978-981-16-2102-4_53
    11. Kumama Regassa Cheneke, Koya Purnachandra Rao, Geremew Kenassa Edessa, Vladimir Mityushev, Application of a New Generalized Fractional Derivative and Rank of Control Measures on Cholera Transmission Dynamics, 2021, 2021, 1687-0425, 1, 10.1155/2021/2104051
    12. Kumama Regassa Cheneke, Koya Purnachandra Rao, Geremew Kenassa Edessa, Fahad Al Basir, Modeling and Analysis of HIV and Cholera Direct Transmission with Optimal Control, 2022, 2022, 1607-887X, 1, 10.1155/2022/5460337
    13. Isa Abdullahi Baba, Usa Wannasingha Humphries, Fathalla A. Rihan, A Well-Posed Fractional Order Cholera Model with Saturated Incidence Rate, 2023, 25, 1099-4300, 360, 10.3390/e25020360
    14. Martins O. Onuorah, F. A. Atiku, H. Juuko, Yuriy Rogovchenko, Mathematical model for prevention and control of cholera transmission in a variable population, 2022, 9, 2768-4830, 10.1080/27658449.2021.2018779
    15. Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi, A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model, 2023, 8, 2473-6988, 3484, 10.3934/math.2023178
    16. Dumitru Baleanu, Fahimeh Akhavan Ghassabzade, Juan J. Nieto, Amin Jajarmi, On a new and generalized fractional model for a real cholera outbreak, 2022, 61, 11100168, 9175, 10.1016/j.aej.2022.02.054
    17. Silvério Rosa, Delfim F. M. Torres, Fractional-Order Modelling and Optimal Control of Cholera Transmission, 2021, 5, 2504-3110, 261, 10.3390/fractalfract5040261
    18. Salma Siddiqua, Anita Chaturvedi, Naba Kumar Goswami, 2022, 2516, 0094-243X, 130002, 10.1063/5.0108808
    19. Moiz Usmani, Kyle D. Brumfield, Yusuf Jamal, Anwar Huq, Rita R. Colwell, Antarpreet Jutla, A Review of the Environmental Trigger and Transmission Components for Prediction of Cholera, 2021, 6, 2414-6366, 147, 10.3390/tropicalmed6030147
    20. Wahyudin Nur, Agus Suryanto, Wuryansari Muharini Kusumawinahyu, Schistosomiasis model with treatment, habitat modification and biological control, 2022, 19, 1551-0018, 13799, 10.3934/mbe.2022643
    21. B. Gbadamosi, O. Adebimpe, Mayowa M. Ojo, O. Oludoun, O. Abiodun, I. Adesina, Modeling the impact of optimal control measures on the dynamics of cholera, 2023, 9, 2363-6203, 1387, 10.1007/s40808-022-01570-9
    22. Kusum Lata, A. K. Misra, Y. Takeuchi, Modeling the effectiveness of TV and social media advertisements on the dynamics of water-borne diseases, 2022, 15, 1793-5245, 10.1142/S1793524521500698
    23. Zafer Bekiryazici, Sensitivity analysis and random dynamics for a mathematical model of tuberculosis transmission, 2023, 0361-0918, 1, 10.1080/03610918.2023.2212884
    24. Phoebe AMADİ, George LAWI, Job BONYO, A Metapopulation Model for Cholera with Variable Media Efficacy and Imperfect Vaccine, 2023, 2636-8692, 20, 10.33187/jmsm.1289684
    25. Shenxing Li, Wenhe Li, Dynamical Behaviors of a Stochastic Susceptible-Infected-Treated-Recovered-Susceptible Cholera Model with Ornstein-Uhlenbeck Process, 2024, 12, 2227-7390, 2163, 10.3390/math12142163
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5607) PDF downloads(295) Cited by(11)

Article outline

Figures and Tables

Figures(7)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog