Rift Valley Fever (RVF) is a viral disease affecting animals and humans, causing symptoms such as fever, liver damage, and bleeding, particularly prevalent in Africa. This study focuses on numerical solutions for a non-linear delayed dynamic epidemiological model of RVF. It extends a control problem incorporating the susceptible, infected, treated, recovered vector to analyze the impact of measures such as mosquito repellent and treatment. The goal is to examine how time delays in implementing control measures affect the dynamics of an epidemic. The model considers delay factors such as mosquito replication, hospitalization, travel restrictions, and isolation due to the lack of proper vaccination. The study explores the model’s aspects, including the reproduction number, equilibrium points, and stability. Local and global implications are examined using techniques such as the Lyapunov function and the Brauer-F lemma. Numerical analysis employs the non-standard finite difference method, establishing the local stability of the equilibrium through the effective reproduction number Rrvf and sensitivity analysis. The research highlights the importance of treatment and delay strategies in reducing RVF transmission, emphasizing the critical need for immunization and preventive measures.

Topics
Geographic location, Multivariable calculus, Finite difference methods, Mathematical modeling, Series expansion, Insects, Diseases and conditions, Organs, Epidemiology, Immunization

A deadly illness that caused lamb deaths in Kenya in 19121 was characterized as having potential adverse economic effects. Two decades later, the same illness was cited as the cause of a schism between a scientist and his co-workers. The mosquito-transmitted phlebovirus has been around for almost a century in many parts of the world. One of the five genera that make up the family Bunyaviridae is the Phlebovirus genus.2 This genus contains viruses that were formerly referred to as uukuviruses and phlebotomus fever viruses. The genus’s prototype, Rift Valley fever virus (RVFV), is one of the most significant viral zoonoses in Africa. Humans are believed to get the virus by direct contact with infected animals, aerosolized blood from infected vertebrates, or infected arthropod vectors. Rift Valley Fever (RVF) has an incubation period of 2–6 days in humans. Symptoms include fever, muscle pain, headache, and sensitivity to light. RVF can be transmitted vertically from the mother to the fetus and horizontally through contact with infected animals or their tissues. Mosquitoes play a crucial role in horizontal transmission by biting infected animals and then spreading the virus to humans.3 It still has the potential to harm people and livestock. A pandemic might have a severe adverse effect on low-income communities. Due to its economic impact and unexpected re-emergence, RVF is considered a threat to animals and humans. Understanding the various eco-epidemiologies of the virus can help develop effective interventions and surveillance strategies. In 1931, a shepherd investigating a property in Kenya’s Rift Valley discovered the RVF virus.4 Pregnant animals miscarry because newborn animals, mainly sheep and goats, have a high death rate because of contact with infected animals and their body fluids, as well as their corpses and organs. These things, including internal organs, during animal handling dissection and stinging can lead to human infection with the Rift Valley fever virus. Only one stranded RNA virus, the West Nile virus, belongs to the Flaviviridae family, which also includes many other significant human infections, including the dengue, eastern encephalitis, and yellow fever viruses.5 

The first case of West Nile virus was identified in 19375 in a feverish patient in Uganda. Consequently, research observed that WNV transmission was endemic6 and major in tropical regions of Africa, South Asia, and northwestern Australia and is also episodic in more temperate regions of Europe. Similar to vector-borne illnesses, the extended transmission seasons made possible by the tropics’ higher temperatures occasionally led to more profound transmission due to quick mosquito and virus growth and increased biting rates. Eighty percent of people over the age of 15 in some African cultures have antibodies to WNV, even though the virus was once thought to be practically asymptomatic and was even tried as a treatment for most malignancies in the 1950s. In Africa, RVF was regarded as an animal disease until 1975. Rare human instances have little symptoms of illness.

Hemorrhagic fever and demise epidemics on a significant scale had been documented in South Africa in the middle of 1975, Egypt in 1977, and Mauritania in 1987. One of the most severe epidemics ever recorded occurred in East Africa in December 1997, with inexplicable fatalities, reported in southern Somalia and the northern regions of southern Kenya.7 This pandemic grew to be regarded as the most dangerous in the area. RVF was first discovered outside of the African continent in Saudi Arabia and Yemen in September 2000, resulting in significant farm animal losses and catastrophic fatalities. The epidemic was first reported in Kenya from 2006 to 2007. Later, the effects spread to Tanzania and Somalia. In 2007 and 2008, Madagascar and South Africa were destroyed. However, 20 outbreaks have been reported in Europe.

On April 17, 2008,8 the Madagascar Ministry of Health announced that RVF was an epidemic. Almost 476 probable cases of RVF, including 19 fatalities, were reported from four provinces between January and June 2008. The Madagascar Ministry of Health recorded 236 suspected cases, including seven fatalities, between December 2008 and May 2009. The South African government recorded 237 verified cases of RVF in people between February and July 2010, including 26 fatal cases from nine regions. On October 4, 2012, the Mauritania Ministry of Health reported an RVF epidemic. The index case first appeared on September 16, 2012, and by November 13, 2012. A total of 36 cases, including 18 deaths, had been recorded from six areas. The rate of animal illness and death was rapidly rising as of June 2022.9 Between August 30, 2022, and October 17, 2022, around 47 cases, of which the majority were animal breeders, were recorded in Mauritania. There were also 23 documented deaths during that time.10 This comprehensive review aimed to consolidate knowledge of RVFV epidemiology from 1937 to 2022 and highlighted knowledge gaps pertinent to plans for human vaccination trials by using different delay factors.

Mosquitoes are particularly susceptible due to the disease incidence in livestock. Advising that isolating people from one another is a workable preventive measure during an outbreak, we can also reduce sickness if people apply insect repellent. To further investigate theoretically the long-term circulation of the disease at the population level, numerical simulations are employed to validate the logical results. A vector–host epidemic model with vertical transmission was examined using the model described in Ref. 11. We expanded12 the model by varying the interaction rates within the host population, as well as vertical transmission between the host and vector populations. As we cannot control the vector but can lower the risk of disease exposure in people, we expand their model by including delay tactics in the host populations.

One significant type of mathematical modeling is delay modeling,20–22 which proves highly beneficial in solving such problems. An example of a serious disease where delay modeling is essential is Rift Valley fever, a terrible illness that leads to numerous deaths. Therefore, delay modeling is vital in combating this disease effectively. In the mathematical modeling of Rift Valley Fever (RVF), incorporating delays into the differential equations mirrors real-world scenarios. The reproduction number is key in understanding the disease’s dynamics: a number below one signifies control, while that above one indicates ongoing transmission. Integrating delays, such as those from quarantine or hospitalization, is vital. Since controlling infection rates is complex, employing tactics such as mosquito control and isolation is crucial for effective pandemic management. To protect themselves from Rift Valley fever, people should steer clear of infected animals or their body fluids, apply insect repellent to fend off mosquito bites, and maintain good hygiene by washing their hands regularly. It is evident that controlling epidemic diseases is crucial. Given that most vector-borne diseases are zoonoses, implementing control measures is vital. Understanding the dynamic nature of these diseases is essential for effective control measures and ensuring that people are treated to prevent further infections.12 

Various mathematical models are used to study and control Rift Valley Fever (RVF). These models primarily focus on different geographical regions, scale levels, and host vector dynamics. Notably, the models include both deterministic and stochastic approaches to assess control strategies, risk mapping, and scenario comparisons. For instance, the EFSA AHAW Panel’s models are stochastic and applied at sub-national and national levels in Mayotte and the Netherlands, respectively, focusing on livestock and vector control through compartmental models (SEIRV and SIR).13 Gachohi et al. utilized a deterministic model for Kenya at a local scale, incorporating multiple vertebrate hosts (cattle and small ruminants) and vectors (Aedes and Culex) within an SEIR framework.14 Similarly, Gaff et al. and Adongo et al. presented theoretical deterministic models for livestock control, emphasizing scenario comparisons through compartmental methods (SEIRV and SEIR).15,16 Chamchod and Yang’s studies also offered deterministic models targeting livestock control with variations in infection states and vector taxa.17,18 Overall, it highlights the diversity in modeling techniques and control strategies across different regions and scales, emphasizing the importance of both deterministic and stochastic methods in understanding and managing RVF outbreaks.

To date, numerous mathematical models have been explored to comprehend the mechanisms underlying real-world phenomena. Researchers have investigated various methods to solve these models both analytically and numerically.28–41 A significant body of the literature has been dedicated to developing models for infectious diseases.42 The model that we used in our research, which was first proposed by Ross and later modified by Macdonald, has significantly influenced both the modeling and the application of control strategies for vector-borne diseases. Lashari et al. presented an analysis of a simple vector-host epidemic model with horizontal transmission. Abdullah et al. extended this model to include vertical transmission in both vector and host populations and introduced a treatment class in the host population with different interaction rates. We have analyzed delay control dynamics of the model developed by Abdullah et al.12 

Given the absence of vaccines or complete treatments for Rift Valley fever globally, delay tactics such as social awareness, travel restrictions, hospitalization, control of the vector, protection against their bites, and isolation have been instrumental in controlling the disease. Risk of animal-to-human transmission of infection can be reduced by avoiding unsafe animal husbandry and slaughtering practices by maintaining good hygiene and by avoiding unsafe consumption of fresh blood, raw milk, or animal tissue in affected regions.43–45 The primary objective of this research is to validate the newly developed model through numerical simulations and to assess its effectiveness in tracking diseases such as Rift Valley fever. For this purpose, we have incorporated the τ rate factor that can reduce disease spread, and a negative sign with delta indicates that our problem is an initial value problem, tending toward initial values when the delay term is applied. Delay mathematical modeling is crucial in epidemiology, particularly for addressing real-world issues such as epidemic diseases.

This is the paper’s approach: A review of the literature on illnesses with a brief history about and illnesses similar to RVF is included in Sec. I. The delayed model’s formulation and mathematical analysis are covered in Sec. II; Sec. III addresses reproductive number analysis. Section IV presents the model’s stability locally and globally, as well as parameter sensitivity. In Sec. V, we present the results of our numerical simulations and discuss the convergence of our NSFD method. Numerical experiments at different points are presented in Sec. VI. The work’s final observations or conclusions are included in Sec. VII.

Rift Valley fever is a very harmful disease in different countries, but in South Africa, it causes deaths in large amounts, so delay modeling is an important work in dealing with this type of dangerous disease. We understand that RVF disease can be reduced if we apply delay strategies on human beings because we cannot reduce mosquitoes. It will take a lot of time, but we can follow some rules to protect ourselves from mosquitoes. Hence, we adopted a model from Ref. 12 and applied intervention measures between susceptible and infected humans. The model formulation can easily be understood as all required information is written in the below-mentioned compartmental information.12 Here, we use different compartments as mentioned:

  • S(t) describes the susceptible human population.

  • I(t) demonstrates the infectious human population.

  • T(t) represents the human population under treatment.

  • R(t) reveals the recovered human population.

For vector population, we have the following:

  • V(t) shows the susceptible population of the vector.

  • W(t) represents here the infected class of vector.

N1(t) and N2(t) are the total number of population sizes for human hosts and vectors, respectively,12 at time t. The susceptible human population of size S(t), the infected individual population of size I(t), the human population receiving treatment of size T(t), and the recovered Homo sapiens population size R(t) make up the host population size N1(t), which is
(1)
The susceptible and infected vector classes are represented by the subclasses V(t) and W(t), respectively, in the vector population N2(t), that is,
(2)
Figure 1 displays the flow diagram for the model. The SITR-VW model, its corresponding parameters, and their physical interpretation are defined as follows:
FIG. 1.
RVF model with a delay strategy.
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RVF model with a delay strategy.

FIG. 1.
RVF model with a delay strategy.
View largeDownload slide

RVF model with a delay strategy.

Close modal
The rate at which new individuals enter the susceptible category depends on factors such as birth and the fraction of newborns infected by their parents’ infection, represented by b1. Disease transmission occurs from the susceptible compartment to the infected compartment due to direct contact with infectious individuals at a rate of β1 and through vector-mediated transmission at a rate of β2. The natural death rate of the susceptible class is indicated by μ1. Therefore, the overall rate of change of susceptible individuals is explained by the following differential equation:
dSdt=(1−ϵ1I(t))b1−β1S(t−τ)I(t−τ)e−δτ−β2S(t)W(t)−μ1S(t).
In the infectious class, b1 represents the birth rate of infected individuals due to contaminated newborns, while e−δτ accounts for delayed direct transmission of the disease. β2 represents transmission via vectors. The terms α, η, δ1, and μ1 represent the treatment, natural recovery, disease-induced mortality, and natural mortality rates, respectively, affecting the infected population. The rate of change of the infected human population I(t) is governed by the following differential equation:
dIdt=ϵ1b1I(t)+β1S(t−τ)I(t−τ)e−δτ+β2S(t)W(t)−αI(t)−ηI(t)−δ1I(t)−μ1I(t).
The rate at which individuals enter the treated compartment T(t), due to infection, is denoted by α. The recovery of the treated sub-population at rate γ, deaths due to illness at rate δ1, and natural mortality at rate μ1 are represented as −γT(t) − δ1T(t) − μ1T(t). Thus, the model differential equation for the rate of change of treated individuals T(t) is
dTdt=αI(t)−γT(t)−δ1T(t)−μ1T(t).
The rate of change of the recovered population R(t) over time is governed by the recovery of infected individuals at rate η, the recovery of treated individuals at rate γ, and the natural mortality rate μ1 affecting the recovered population. Thus, the differential equation describing this process in the model for the recovered population is
dRdt=ηI(t)+γT(t)−μ1R(t).
The rate at which new individuals enter the vector population depends on factors such as recruitment and the fraction of newborn vectors infected by their parents’ infection, represented as (1 − ϵ2W(t))b2. Disease transmission from the vector population to humans due to contact with infectious individuals is denoted by β3V(t)I(t). The term μ2 indicates the natural mortality rate of the vector population. Thus, the differential equation governing the rate of change of the vector population in the model V(t) represented as
dVdt=(1−ϵ2W(t))b2−β3V(t)I(t)−μ2V(t).
The rate at which new vectors enter the host population by the continuous recruitment rate and the fraction of newborn vectors infected by their host parents is given by ϵ2. Disease transmission from vectors to humans due to contact with infectious individuals is denoted by β3. The terms δ2 and μ2 indicate the mortality rates due to disease and natural mortality of vectors, respectively. Therefore, the differential equation governing the rate of change of the vector population in the model for W(t) is represented as
dWdt=ϵ2b2W(t)+β3V(t)I(t)−δ2W(t)−μ2W(t).
Therefore, the system of delayed differential equations that can be used to illustrate the given mathematical model is
(3a)
(3b)
(3c)
(3d)
(3e)
(3f)
with the initial conditions
(3g)

Here, t ≥ 0, τ ≤ t.We cannot control mosquito reproduction, but we can overcome this disease by following some rules and applying a time delay and precautions. The flowchart displayed in the image describes the entire process of the model’s construction.

In this section, we conduct a theoretical analysis of the dynamics exhibited by Rift Valley fever disease. Model equilibrium states, model properties such as the boundedness and positivity of solution, and the reproductive number of our delay model are analyzed.

Equations (3a)–(3f) will be illustrated in this part with three different forms of equilibrium, indicated by E0, E1, and E2, which stand for trivial, disease-free, and endemic equilibrium points:
E0=S0,I0,T0,R0,V0,W0=0,0,0,0,0,0,E1=S1,I1,T1,R1,V1,W1=b1μ1,0,0,0,b2μ2,0,E2=S2,I2,T2,R2,V2,W2,
with coordinates
S2=b1−ϵ1b1I2β1I2e−δτ+β2W2+μ1,V2=mW2β3I2,T2=αI2l,R2=I2(lη+γα)μ1l,W2=μ2β3I2N2μ2m+β3(δ2+μ2)I2,I2=b1−μ1S2α+η+δ1+μ1.

Theorem 1.

For given τ ≤ t, t ≥ 0, and initial conditions, the system holds the positivity of the solution at the system of Eqs. (3a)–(3f).

Proof.
It is clear from the system of Eqs. (3a)–(3f) that
dSdtS=0=(1−ϵ1I)b1≥0,dIdtI=0=β2SW≥0,dTdtT=0=αI≥0,dRdtR=0=ηI+γT≥0,dVdtV=0=(1−ϵ2W)b2≥0,dWdtW=0=β3VI≥0,
which shows that positivity exists in the proposed system.□

Theorem 2.

For any finite time t ≥ 0, there is a unique and bounded solution to the system of equations in a positively invariant set.

Proof.

Every equation in the system has a continuous right side in the convex domain Y = t, S(t), I(t), T(t), R(t), V(t), and W(t)) of (6 + 1)-dimensional space with the partial derivatives that are continuous. Thus, for a given limited period t ≥ 0 and with initial conditions, the problem has a unique solution.

As the total population is divided into two sectors N1(t) and N2(t), from the model equations, we get
(4)
Then
dN1dt≤b1−μ1N1 and dN2dt≤b2−μ2Nv⁠. ⇒N1≤N1(0)e−μ1t+b1μ11−e−μ1t and N2≤N2(0)e−μ2t+b2μ21−e−μ2t⁠, which show that
(5)
The specified starting conditions ensure that N1 ≥ 0 and N2 ≥ 0. As a result, the system’s feasible region is
Ω=(S,I,T,R,V,W)∈R+6,N1≤b1μ1,N2≤b2μ2.
Therefore, the overall population and each sub-population class remain bounded for all finite time t ≥ 0.□

The basic reproductive number is used to analyze the dynamics of the system. Rrvf can defined as the expected number of the secondary cases that an ordinary affected person could cause in an entirely susceptible group throughout the infectious period. In our case, we represent the reproductive number with Rrvf.

Mathematically it is defined as Rrvf ∝ T.C.D, where T=infectioncontact⁠, C=contacttime⁠, and D=timeinfection⁠.

As a threshold, this amount is used to measure whether a disease will spread or disappear from the community. It may be determined with the approach of the next generation matrix. In this system, the infected population is indicated by I, T, and W, whereas those without infection compartments are marked by S, R, and V. The two matrices F and V describe the new rate of infection generation and the rate of transition between stages, respectively.

We are now using the next-generation matrix approach to the system. In this part, to calculate the reproduction number by computation, the following are the transmission and transition matrices. For the next generation matrix method dxdt=f(x,y)−v(x,y)⁠,
F=ϵ1b1I+β1SIe−δτ+β2SW00,V=(α+η+δ1+μ1)I−αI+(γ+δ1+μ1)T−ϵ2b2W−β3VI+(δ2+μ2)W.
Thus, at the disease-free equilibrium (DFE) points, the transmission matrices F and V are
F̄=ϵ1b1+β1Se−δτ0β2S000000,V̄=α+η+δ1+μ100−αγ+δ1+μ10−β3V0δ2+μ2−ϵ2b2.
F̄ and V̄ represent the changes in new infections and transitions approaching the equilibrium, respectively. Here, we have calculated V̄ inverse and F̄ inverse, which correspondingly display the duration of stay in each compartment and the total number of new infections created during the span of the illness, respectively,
F̄V̄−1=ϵ1b1+β1N1e−δτα+η+δ1+μ1+(β2N1)(β3N2)(α+η+δ1+μ1)(−ϵ2b2+δ2+μ2)0β2N1−ϵ2b2+δ2+μ2000000.
Hence, the fundamental reproductive number has the highest spectral radius
(6)
It is worth noting that the F̄V̄−1 spectral radius is referred to as the threshold number and is represented as Rrvf in our system. It is critical for disease control since it switches the illness from one equilibrium point to the next.
Hence,
(7)
Here, k = α + δ1 + μ1 + η and m = δ2 + μ2 − ϵ2b2.

When there is no vertical transmission, ϵ1 = ϵ2 = 0 and the basic reproductive number for a model that only has side-to-side transmission is Rrvf.

The local and global stability analysis at disease-free and endemic equilibrium points using Brauer, F lemma, and Castillo–Chavez’s methods and the Lyapunov function with specific properties is discussed in this section. In addition, we found the sensitivity analysis of the parameters.

Theorem 3.

Trivial steady state E0 is locally asymptotically stable if the reproductive number Rrvf < 1.

Proof.
This can be demonstrated by linearizing a system around E0, which obtains the Jacobian matrix as follows:
J=−μ1−ϵ1b100000A00000α−γ−δ1−μ10000ηγ−μ1000000−μ2−ϵ2b200000ϵ2b2−δ2−μ2,
where A = −ϵ1b1 − α − η − δ1 − μ1.

Notice that all eigenvalues of our system are λ1 = −μ1 < 0, λ2 = −(ϵ1b1 + α + η + δ1) < 0, λ3 = −(γ + δ1 + μ1) < 0, λ4 = −μ1 < 0, λ5 = −μ2 < 0, and λ6 = −m < 0.

Hence, all eigenvalues are negative, so E0 clearly shows that it is locally asymptotically stable.□

Theorem 4.

E1 points that are the disease-free equilibrium points are locally asymptotically stable if Rrvf is less than one; otherwise, if the reproductive number is greater than one, then it will be unstable.

Proof.
This can be illustrated by linearizing E1, which creates the Jacobian matrix that is shown below:
J(E1)=AB000−β2S1−β1I2+β1W1D000β2S10α−γ−δ1−μ10000ηγ−μ1000−β3V200−β3I2−μ2J0β3V100β3I1E,|J(E1)−λI|=A−λB000−β2S1−β1I2+β1W1D−λ000β2S10α−γ−δ1−μ1−λ0000ηγ−μ1−λ000−β3V200−β3I2−μ2−λJ0β3V100β3I1E−λ,
where A = −β1I2e−δτ − β1W1 − μ1, B = −ϵ1b1 − β1S1e−δτ, D = −ϵ1b1 + β1S1e−δτ − α − η − δ1 − μ1, E = ϵ2b2 − δ2 − μ2, and J = −ϵ2b2.

Now the four eigenvalues are −μ1, −μ1, −μ2, and −l. The other eigenvalues are found by the matrix, which is
|J(E1)−λI|=−ϵ1b1+β1b1μ1e−δτ−k−λβ2b1μ1β3b2μ2−m−λ.

Lemma

(Brauer-F).

For the quadratic equation λ2 − cλ + d = 0, where |λi| < 1 i = 1, 2, both roots will be satisfied if and only if the following conditions are satisfied:

  1. 1 − c + d > 0;

  2. 1 + c + d > 0;

  3. d < 1;

Here, c = Trace(j) = ϵ1b1 + β1N1e−δτ − k − m and d = Det(j) = −mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2;

  • 1 − c + d > 0

  • 1 − (ϵ1b1 + β1N1e−δτ − k − m) + (−mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2),

  • 1 − ϵ1b1 − β1N1e−δτ + k + m − mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2.

  • According to the given condition,

  • 1 − ϵ1b1 − β1N1e−δτ + k + m − mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2 > 0.

  • 1 + c + d > 0

  • 1 + (ϵ1b1 + β1N1e−δτ − k − m) + (−mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2),

  • 1 + ϵ1b1 + β1N1e−δτ − k − m − mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2.

  • According to the given condition,

  • 1 + ϵ1b1 + β1N1e−δτ − k − m − mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2 > 0.

  • d < 1

  • −mϵ1b1 − mβ1N1e−δτ + km − β2β3N1N2 < 1.

All these three conditions satisfied means that all eigenvalues are negative and by Brauer-F conditions, the given equilibrium E1 is locally asymptotically stable.□

Theorem 5.

The endemic equilibrium point E2 is locally asymptotically stable if Rrvf is >1⁠.

Proof.
This can be verified by a system of linearization with E2, which gives the following Jacobian matrix:
J=−Y−Z000−β2S2Y−μ1Z−k000β2S20α−l0000ηγ−μ1000−β3V200−β3I2−μ2ϵ2b20β3V100β3I1−m,
whereY = −β1I2e−δτ − β1W2 − μ1 and Z = −ϵ1b1β1S2e−δτ,
|J(E2)−λI|=−Y−λ−Z000−β2S2Y−μ1Z−k−λ000β2S20α−l−λ0000ηγ−μ1−λ000−β3V200−β3I2−μ2−λϵ2b20β3V100β3I1−m−λ.
Two eigenvalues are −μ1 and −l, which we get from the above-mentioned matrix. The remaining eigenvalues can be determined by
|J(E2)−λI|=−Y−λ−Z0β2S2Y−μ1Z−k−λ0β2S20β3V2−β3I2−μ2−λ−ϵ2b20β3V2β3I2−m−λ.
After we found the other eigenvalues, we applied an elementary row operation on the Jacobian matrix. Then we obtain the following matrix:
J2=−Y−Z0β2S20μ1ZY−k0μ1Y−β2S200−μ2−ϵ2b2−m000G,
where
G=m+U+β3I2μ2(m+ϵ2b2)andU=m+μ1β2β3S2V2μ1Z−kY.
Since it is a lower triangle matrix, our eigenvalues will be the elements of the main diagonal. Three of them have negative real parts. The second eigenvalue μ1ZY−k has a negative real part iff μ1ZY−k<0⁠. Using the values of Y and Z, we can check that all the coefficients of the equation will be negative if Rrvf > 1. Hence all eigenvalues have negative real parts, showing that the endemic or disease existing point E2 is locally asymptotically stable if and only if Rrvf > 1.□

To demonstrate global stability at DFE, we use Castillo–Chavez’s method.19 In order to use this strategy, the model must be written in the format specified below:
dPdt=k(P,B),dBdt=M(P,B).
The previously mentioned form of P represents the population of uninfected humans and vectors. Alternatively, B denotes the infected population (both). If the next two requirements are met, DFE is globally asymptotically stable, according to Castillo–Chavez’s method:
C1:dPdt=k(P,O)=0,P̄is globally asymptotically stable,C2:M(P,B)=BB−M(P̄,B),(P̄,B)≥0∀(P,B)∈Ω.
Here, B=DBM(P̄,O) is a matrix with non-negative off-diagonal entries, and Ω is a feasible region.

Theorem 6.

The Rift Valley fever-free equilibrium is globally asymptotically stable if both conditions satisfied C1,C2, when Rrvf < 1.

Proof.
Let P=S,V be the uninfected population from host and vectors and B the infected population (I, T, W) so that
dPdt=k(P,B)=(1−ϵ1I)b1−β1SIe−δτ−β2SW−μ1S(1−ϵ2W)b2−β3VI−μ2V.
At disease-free points, the first condition is satisfied when we put disease-free points in it, and we get a null matrix; hence, the first condition is satisfied,
ddtITW=ϵ1b1+β1Se−δτ−k0β2S1αγ−δ1−μ10β3V10−m×ITW−β1Ie−δτ(S1−S)α(I1−I)β3I(V1−V).

Since S1 ≤ S, I1 ≤ I, and V1 ≤ V, we have M̄(P,B)≥0⁠. It also clearly represents that matrix B has non-negative non-diagonal entries, so B is an M matrix. Therefore, our conditions are satisfied, and we can say that disease-free equilibrium points are globally asymptotically stable.□

Theorem 7.

The Rift Valley fever existing equilibrium E2 is globally asymptotically stable, if Rrvf > 1.

Proof.
The Volterra–Lyapunov function F:Ω→R is defined as20–22 
F=S−S2−S2⁡logSS2+S−I2−I2⁡logII2+T−T2−T2⁡logTT2+R−R2−R2⁡logRR2+V−V2−V2⁡logVV2+W−W2−W2⁡logWW2,
dFdt=ddSS−S2−S2⁡logSS2dSdt+ddII−I2−I2⁡logII2dIdt+ddTT−T2−T2⁡logTT2dTdt+ddRR−R2−R2⁡logRR2dRdt+ddVV−V2−V2⁡logVV2dVdt+ddWW−W2−W2⁡logWW2dWdt,
dFdt=S−S2S(b1−ϵ1b1I−β1SIe−δτ−β2SW−μ1S)+I−I2I(ϵ1b1I+β1SIe−δτ+β2SW−αI−ηI−δ1I−μ1I)+T−T2T(αI−γT−δ1T−μ1T)+R−R2R(ηI+γT−μ1R)+V−V2V(b2−ϵ2Wb2−β3VI−μ2V)+W−W2W(ϵ2b2W+β3VI−δ2W−μ2W).
The above-mentioned condition can be put in the form
dFdt=γ1−γ2,
where
γ1=b1+(ϵ1b1I)S2S+(S2)2S(β1Ie−δτ+β2W+μ1)+β2SW+(I−I2)2I(ϵ1b1+β1Se−δτ)+I2(ϵ1b1+β1Se−δτ)+(I2)2I(α+η+δ1+μ1)+αI+(T2)2T(γ+δ1+μ1)+ηI+γT+(R2)2R(μ1+b2)+ϵ2b2WV2V+(V2)2V(β3I+μ2)+β3VI+(W2)2W(ϵ2b2+δ2+μ2)
and
γ2=b1S2S+ϵ1b1I+(S−S2)2S(β1Ie−δτ+β2W+μ1)+S2(β1Ie−δτ+β2W+μ1)+β2SWI2I+(I−I2)2I(α+η+δ1+μ1)+(I2)2I(ϵ1b1+β1Se−δτ)+I2(α+η+δ1+μ1)+αIT2T+(T−T2)2T(γ+δ1+μ1)+T2(γ+δ1+μ1)+ηIR2R+(R−R2)2Rμ+μ1R2+b2V2v+b2ϵ2W+(V−V2)2V(β3I+μ2)+V2(β3I+μ2)+β3VIW2W+(W−W2)2W(ϵ2b2+δ2+μ2)+W2(ϵ2b2+δ2+μ2).

We know that the parametric values are non-negative, so we have dFdt≤0⁠, when γ1 ≤ γ2, and dFdt=0⟺γ1=γ2⁠.

From the calculation, it can be observed that dFdt=0⟺ if conditions S = S2, I = I2, T = T2, R = R2, V = V2, and W = W2 are satisfied.

Hence, E2 globally asymptotically stable.□

Sensitivity analysis in epidemic models involves studying how changes in specific parameters affect the model’s behavior and outcomes. By varying these parameters systematically, researchers can see how sensitive the model’s predictions are to each change. This helps us identify which parameters have the most significant impact on the epidemic’s spread and control. The sensitivity analysis of the basic reproductive number Rrvf regarding every parameter is as follows:

Positive effect on Rrvf

1. ϵ1: ∂Rrvf∂ϵ1=b1k (positive effect). 2. β1: ∂Rrvf∂β1=N1e−δτk (positive effect) 3. β3: ∂Rrvf∂β3=β2N1N2mk (positive effect). 4. b1: ∂Rrvf∂b1=ϵ1k (positive effect). 5. b2: ∂Rrvf∂b2=β2β3N1N2mk (positive effect). 6. β2: ∂Rrvf∂β2=β3N1N2mk (positive effect).

Negative effect on Rrvf

1. α: ∂Rrvf∂α=−ϵ1b1k2 (negative effect). 2. η: ∂Rrvf∂η=−ϵ1b1k2 (negative effect). 3. δ1: ∂Rrvf∂δ1=−ϵ1b1k2 (negative effect). 4. μ1: ∂Rrvf∂μ1=−ϵ1b1k2 (negative effect). 5. ϵ2: ∂Rrvf∂ϵ2=−β2β3N1N2mk2 (negative effect). 6. δ2: ∂Rrvf∂δ2=−β2β3N1N2mk2 (negative effect). 7. μ2: ∂Rrvf∂μ2=β2β3N1N2mk2 (negative effect). 8. γ: ∂Rrvf∂γ=−1k2 (negative effect).

The analysis reveals that some parameters exhibit positive sensitivity indices, while others have negative sensitivity indices, as shown in Fig. 2, similar to others. This indicates that certain parameters are directly related to Rrvf, while others have an inverse relationship with the reproduction number, that is, if the cure rate of mosquitoes increase, the disease will decrease. In addition, when the disease transmission rate of vector β2 increases, the disease will spread fast. These results indicate that parameters have significant effects on the basic reproductive number of the model.

FIG. 2.
(a) Behavior of Rrvf that decreases with an increasing cure rate α. (b) Rrvf value increases as vertical transmission β2 increases.
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(a) Behavior of Rrvf that decreases with an increasing cure rate α. (b) Rrvf value increases as vertical transmission β2 increases.

FIG. 2.
(a) Behavior of Rrvf that decreases with an increasing cure rate α. (b) Rrvf value increases as vertical transmission β2 increases.
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(a) Behavior of Rrvf that decreases with an increasing cure rate α. (b) Rrvf value increases as vertical transmission β2 increases.

Close modal

In this section, for numerical investigation we prefer the NSFD method, and this method positively preserves the numerical scheme. Further justification proved its convergence and consistency analysis.

For the numerical solution, we chose the NSFD method. Based on Micken’s non-standard finite difference modeling theory,27 we show an unconditionally convergent non-standard finite difference scheme in this section. Because NSFD systems maintain the characteristics of the continuous model, they have been effectively employed.20–25 We also offer the convergence study of our suggested scheme here. This stability made the NSFD method a clear choice for our study. For a continuous dynamical system, the NSFD model is provided as
(8a)
(8b)
(8c)
(8d)
(8e)
After taking the partial derivative of NSFD equations and putting the disease-free point, we got
ES=11+hμ1,EI=(1+hμ1)(−hb1ϵ1)−(S+hb1)(hβ1)(1+hμ1)2,ET=0,EV=0,EW=(−hβ2)(S+hb1)(1+hμ1)2,FS=0,FI=(1+hϵ1b1+hβ1N1e−δτ)1+hα+hη+hδ1+hμ1,FT=0,FV=0,FW=(hβ2N1)1+hα+hη+hδ1+hμ1,GS=0,GI=hα1+hγ+hδ1+hμ1,GT=11+hγ+hδ1+hμ1,GV=0,GW=0,HS=0,HI=−(hβ2)(V+hb2)(1+hμ2)2,HT=0,HV=11+hμ2,HW=−hb2ϵ21+hμ2,KS=0,KI=hβ3N21+hδ2+hμ2,KT=0,KV=0,KW=1+hϵ2b21+hδ2+hμ2.
Now J will be
(9)
(10)
The quadratic equation λ2 − cλ + d = 0 is obtained by the matrix from Eq. (10), where |λi| < 1 i = 1, 2, and the remaining roots will be satisfied if and only if the following conditions are satisfied. From the matrix two by two, the remaining two eigenvalues can be found by Brauer-F lemma,
c=Trace(J)=1+hϵ1b1+hβ1N1e−δτ1+hk+1+hϵ2b21+hδ2+hμ2,d=det(J)=(1+hϵ1b1+hβ1N1e−δτ)(1+hϵ2b2)−hβ2β3N1N2(1+hk)(1+hδ2+hμ2).

All conditions of the lemma (Brauer-F) are satisfied, so we can say that our scheme is stable.

In this section, the consistency analysis of our numerical scheme is performed by using Taylor’s series expansion.26 First, we take Eq. (3a) of the numerical integration scheme and apply Taylor’s series expansion of Sn+1,
(11)
In the following expression,
(12)
We obtain as follows:
(13)
By applying h → 0, we obtain as follows:
(14)
This result implies that our discretized equation is consistent with Eq. (3a) of the model. Similarly, we take Eq. (3b) and apply Taylor’s series expansion of In+1,
(15)
In the following expression,
(16)
We obtain as follows:
(17)
By applying h → 0, we obtain as follows:
(18)
Similarly, taking Eq. (3c) and applying Taylor’s series expansion of Tn+1, after proper simplifications, we obtain as follows:
(19)
Applying h → 0, we obtain as follows:
(20)
Similarly, taking Eq. (3e) and applying Taylor’s series expansion of Vn+1, we obtain as follows:
(21)
Applying h → 0, we obtain as follows:
(22)
Taking Eq. (3f) and applying Taylor’s series expansion of Wn+1, we obtain as follows:
(23)
Applying h → 0, we obtain as follows:
(24)
Hence, our numerical scheme is consistent with the system of Eqs. (3a)–(3f).

We gather information from the source as mentioned in Table I and solve the model using the non-standard finite difference method. Some of the parameter values are based on reality, as mentioned in Ref. 12, from where we got maximum data. For initial conditions, let S(0) = 100, I(0) = 30, T(0) = 25, R(0) = 10, V(0) = 600, and W(0) = 100. Following the NSFD technique, we graphically represent the findings and demonstrate the impact of vertical transmission and cure rate. At the disease-free equilibrium, everyone seems to be in the susceptible compartment, and there is no ongoing transmission of the disease. Verifying that the numerical solutions yield the expected behavior at this point helps ensure that the model accurately represents the disease dynamics in the absence of infection. Disease-free points and endemic points are critical in stability analysis. For example, the disease-free equilibrium is often considered a reference point for stability analysis, and understanding the behavior of the system near this point is crucial.

TABLE I.

Values of parameters.

ParametersDescriptionValues/daysResources
b1 Human’s recruitment rate 20 12  
b2 Vector’s recruitment rate 100 12  
δ1 Death rate in humans by disease 0.01 12  
δ2 Death rate in vectors by disease 0.21 12  
β1 Transmission rate from human to human 0.000 01 12  
β2 Transmission rate from vector to population 0.0012 12  
β3 Transmission probability from 0.001 12  
  humans to vector population   
μ1 Mortality rate in humans by nature 0.000 039 12  
μ2 Mortality rate in mosquitoes by nature 0.1 12  
α Treatment parameter depends on delay tactics [0 − 100] Fitted 
γ Recovery rate due to treatment 0.4 12  
η Natural recovery rate 0.01 12  
ϵ1 Newborn baby’s infected by parents 0.001 12  
ϵ2 Vertical transmission in mosquitoes 0.002 12  
δ The rate of mortality of humans at the susceptible point 0.000 039 Assumed 
τ Time delay using pre-cautionary measurements τ ≥ 0 Assumed 
ParametersDescriptionValues/daysResources
b1 Human’s recruitment rate 20 12  
b2 Vector’s recruitment rate 100 12  
δ1 Death rate in humans by disease 0.01 12  
δ2 Death rate in vectors by disease 0.21 12  
β1 Transmission rate from human to human 0.000 01 12  
β2 Transmission rate from vector to population 0.0012 12  
β3 Transmission probability from 0.001 12  
  humans to vector population   
μ1 Mortality rate in humans by nature 0.000 039 12  
μ2 Mortality rate in mosquitoes by nature 0.1 12  
α Treatment parameter depends on delay tactics [0 − 100] Fitted 
γ Recovery rate due to treatment 0.4 12  
η Natural recovery rate 0.01 12  
ϵ1 Newborn baby’s infected by parents 0.001 12  
ϵ2 Vertical transmission in mosquitoes 0.002 12  
δ The rate of mortality of humans at the susceptible point 0.000 039 Assumed 
τ Time delay using pre-cautionary measurements τ ≥ 0 Assumed 
View Large

Case 01: Simulation at disease free equilibrium (RVF by E1) without applying delay tactics

The graphs (a)–(f) in Fig. 3 demonstrate the graphical solution of the system (SITRVW) at Rift Valley fever disease free points with the initial data of the model where the system touches the disease free points E1. When there is no delay, the reproduction number’s value is 0.4461 < 1, which also combines the behavior of the model as presented in these graphs.

FIG. 3.
Time series plots (a)–(f) of the model’s equations at the disease-free equilibrium.
View largeDownload slide

Time series plots (a)–(f) of the model’s equations at the disease-free equilibrium.

FIG. 3.
Time series plots (a)–(f) of the model’s equations at the disease-free equilibrium.
View largeDownload slide

Time series plots (a)–(f) of the model’s equations at the disease-free equilibrium.

Close modal

Case 02: Simulation at disease existing equilibrium (RVF by E2) without delay

Here, Fig. 4 represents graphs (a)–(f) for the model’s (SITRVW) equations’ response at Rift Valley fever disease existing points with initial data of the model; therefore, the system touches on disease existing points E2. In the absence of tactics, the value of the reproduction number is 1.1599 > 1, which also combines the behavior of the model as presented in these graphs.

FIG. 4.
Time series plots (a)–(f) of the model’s equations at the disease-existing equilibrium.
View largeDownload slide

Time series plots (a)–(f) of the model’s equations at the disease-existing equilibrium.

FIG. 4.
Time series plots (a)–(f) of the model’s equations at the disease-existing equilibrium.
View largeDownload slide

Time series plots (a)–(f) of the model’s equations at the disease-existing equilibrium.

Close modal

Case 03: Simulations at RVF existing equilibrium with time delay effect

This section examines the impact on the system’s equations by putting the endemic equilibrium in the model for Rift Valley fever, utilizing artificial time delay variables to get a good effect. Figure 5 plots (a)–(c) represent how human vulnerability rises when delay techniques are used. On the other hand, we can see that Rift Valley fever patients’ infectivity falls and even approaches to zero.

FIG. 5.
Combined behavior of the system (a) at τ = 0.4, (b) at τ = 0.5, and (c) at τ = 0.9.
View largeDownload slide

Combined behavior of the system (a) at τ = 0.4, (b) at τ = 0.5, and (c) at τ = 0.9.

FIG. 5.
Combined behavior of the system (a) at τ = 0.4, (b) at τ = 0.5, and (c) at τ = 0.9.
View largeDownload slide

Combined behavior of the system (a) at τ = 0.4, (b) at τ = 0.5, and (c) at τ = 0.9.

Close modal

Case 04: Comparison graph of the reproduction number and artificial delay term of the model

Figure 6 displays the fact that the increases in delay strategy can overcome the spread of Rift Valley fever, as needed.

FIG. 6.
Comparison graph of the reproduction number and artificial delay term of the model.
View largeDownload slide

Comparison graph of the reproduction number and artificial delay term of the model.

FIG. 6.
Comparison graph of the reproduction number and artificial delay term of the model.
View largeDownload slide

Comparison graph of the reproduction number and artificial delay term of the model.

Close modal

Hence, by numerical investigation, we can clearly analyze that the researchers in this study apply mathematical techniques to inquire how delay strategies have an effect on the behavior of the SITRVW model.

Vector-borne ailments are diseases that are spread by vectors to humans and animals. This research developed a delay epidemically model for vector-borne illness dynamics of transmission, including transmissions in both vertical and horizontal directions and a disease delay method. The model’s equilibrium points and fundamental reproduction are discovered. As the reproduction number increases, the sickness spreads into the population, and the disease dies off as it falls. Although the susceptible population increases, due to delay tactics, it does not have an effect on human beings because they remain safe due to these tactics. By using conditions on the threshold number, it has been shown that both equilibrium points are locally and globally stable. However, the global behavior of these equilibrium points is studied using the theory of Lyapunov function. The effect of other parameters can be easily understood by the sensitivity analysis of our reproductive number.

The NSFD scheme is dynamically consistent, and the results proved that the scheme preserves the properties of our mathematical model. The most effective methods for the delay factors are an increase in values of constant precautions and mosquito repellent. These parameters reduce the unconditional spread of disease stability through our convergence analysis. Furthermore, we conducted an in-depth investigation of how the delay factor affected the number of reproductions, the infected class, and the recovered class. Based on our findings, we were able to control the dynamics of the virus by applying various delay-causing techniques, such as mosquito replication, which has been shown to be the most efficient method to stop the disease from spreading. Humanity and the healthcare industry will benefit from this investigation’s findings. In conclusion, delayed epidemically mathematical models are an important addition to understanding virus dynamics. In the future, we intend to expand this work to include fractal fractional analysis of this model.

We acknowledge the support of Universiti Sains Malaysia (USM) and we thank the reviewers for their valuable feedback.

The authors have no conflicts to disclose.

Shah Zeb: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal). Siti Ainor Mohd Yatim: Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Muhammad Rafiq: Supervision (equal); Visualization (equal); Writing – review & editing (equal). Waheed Ahmad: Data curation (equal); Formal analysis (equal); Methodology (equal). Ayesha Kamran: Visualization (equal); Writing – review & editing (equal). Md. Fazlul Karim: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

1.
R. M.
Murithi
,
P.
Munyua
,
P. M.
Ithondeka
,
J. M.
Macharia
,
A.
Hightower
,
E. T.
Luman
, and
M. K.
Njenga
, “
Rift Valley fever in Kenya: History of epizootics and identification of vulnerable districts
,”
Epidemiol. Infect.
139
(
3
),
372
–
380
(
2011
).
https://doi.org/10.1017/S0950268810001020
Google Scholar
Crossref
Search ADS
PubMed
 
2.
T.
Ikegami
 and
S.
Makino
, “
Rift valley fever vaccines
,”
Vaccine
27
(
Suppl 4
),
D69
–
D72
(
2009
).
https://doi.org/10.1016/j.vaccine.2009.07.046
Google Scholar
Crossref
Search ADS
PubMed
 
3.
G. D.
Maganga
,
A. L.
Abessolo Ndong
,
C. S.
Mikala Okouyi
,
S.
Makiala Mandanda
,
N.
N’Dilimabaka
,
A.
Pinto
,
E.
Agossou
 et al, “
Serological evidence for the circulation of rift valley fever virus in domestic small ruminants in southern Gabon
,”
Vector-Borne Zoonotic Dis.
17
(
6
),
443
–
446
(
2017
).
https://doi.org/10.1089/vbz.2016.2065
Google Scholar
Crossref
Search ADS
PubMed
 
4.
R.
Daubney
,
J. R.
Hudson
, and
P. C.
Garnham
, “
Enzootic hepatitis or Rift Valley fever. An undescribed virus disease of sheep cattle and man from East Africa
,”
J. Pathol. Bacteriol.
34
(
4
),
545
–
579
(
1931
).
https://doi.org/10.1002/path.1700340418
Google Scholar
Crossref
Search ADS
 
5.
M.
De Filette
,
S.
Ulbert
,
M.
Diamond
, and
N. N.
Sanders
, “
Recent progress in West Nile virus diagnosis and vaccination
,”
Vet. Res.
43
,
16
(
2012
).
https://doi.org/10.1186/1297-9716-43-16
Google Scholar
Crossref
Search ADS
PubMed
 
6.
A. M.
Kilpatrick
, “
Globalization, land use, and the invasion of West Nile virus
,”
Science
334
(
6054
),
323
–
327
(
2011
).
https://doi.org/10.1126/science.1201010
Google Scholar
Crossref
Search ADS
PubMed
 
7.
Regional Office of Africa
, “
World Health Organization African Region
,” Retrieved from https://www.afro.who.int/ (
2023
).
8.
B. H.
Bird
,
M. L.
Khristova
,
P. E.
Rollin
,
T. G.
Ksiazek
, and
S. T.
Nichol
, “
Complete genome analysis of 33 ecologically and biologically diverse Rift Valley fever virus strains reveals widespread virus movement and low genetic diversity due to recent common ancestry
,”
J. Virol.
81
(
6
),
2805
–
2816
(
2007
).
https://doi.org/10.1128/JVI.02095-06
Google Scholar
Crossref
Search ADS
 
9.
S.
Akash
,
M. R.
Islam
, and
M. M.
Rahman
, “
Rift Valley fever (RVF): A re-emerging zoonotic disease, pathogenesis, epidemiology, current status, and future perspective - correspondence
,”
Int. J. Surg.
109
(
3
),
587
–
588
(
2023
).
https://doi.org/10.1097/JS9.0000000000000172
Google Scholar
Crossref
Search ADS
PubMed
 
10.
H.
Chopra
,
K.
Dhama
, and
T. B.
Emran
, “
Re-emergence of Rift Valley fever: An update and preventive measures
,”
Int. J. Surg.
109
(
1
),
68
–
70
(
2023
).
https://doi.org/10.1097/JS9.0000000000000132
Google Scholar
Crossref
Search ADS
PubMed
 
11.
A.
Lashari
,
K.
Hattaf
,
G.
Zaman
, and
X.-Z.
Li
, “
Backward bifurcation and optimal control of a vector borne disease
,”
Appl. Math. Inf. Sci.
7
(
1
),
301
–
309
(
2013
).
https://doi.org/10.12785/amis/070138
Google Scholar
Crossref
Search ADS
 
12.
Abdullah
,
A.
Seadawy
, and
W.
Jun
, “
New mathematical model of vertical transmission and cure of vector-borne diseases and its numerical simulation
,”
Adv. Differ. Equ.
2018
(
1
),
66
.
https://doi.org/10.1186/s13662-018-1516-z
Crossref
Search ADS
 
13.
EFSA Panel on Animal Health and Welfare EFSA AHAW Panel
,
S. S.
Nielsen
,
J.
Alvarez
,
D. J.
Bicout
,
P.
Calistri
,
K.
Depner
 et al, “
Rift Valley Fever—Assessment of effectiveness of surveillance and control measures in the EU
,”
EFSA J.
18
(
11
),
e06292
(
2020
).
https://doi.org/10.2903/j.efsa.2020.6292
Google Scholar
Crossref
Search ADS
PubMed
 
14.
J. M.
Gachohi
,
M. K.
Njenga
,
P.
Kitala
, and
B.
Bett
, “
Modelling vaccination strategies against Rift Valley fever in livestock in Kenya
,”
PLoS Neglected Trop. Dis.
10
(
12
),
e0005049
(
2016
).
https://doi.org/10.1371/journal.pntd.0005049
Google Scholar
Crossref
Search ADS
 
15.
H.
Gaff
,
C.
Burgess
,
J.
Jackson
,
T.
Niu
,
Y.
Papelis
, and
D.
Hartley
, “
Mathematical model to assess the relative effectiveness of Rift Valley fever countermeasures
,”
Int. J. Artif. Life Res.
2
(
2
),
1
–
18
(
2011
).
https://doi.org/10.4018/jalr.2011040101
Google Scholar
Crossref
Search ADS
 
16.
D.
Adongo
,
K. R.
Fister
,
H.
Gaff
, and
D.
Hartley
, “
Optimal control applied to Rift Valley fever
,”
Nat. Resour. Model.
26
(
3
),
385
–
402
(
2013
).
https://doi.org/10.1111/nrm.12006
Google Scholar
Crossref
Search ADS
 
17.
F.
Chamchod
,
C.
Cosner
,
R. S.
Cantrell
,
J. C.
Beier
, and
S.
Ruan
, “
Transmission dynamics of Rift Valley fever virus: Effects of live and killed vaccines on epizootic outbreaks and enzootic maintenance
,”
Front. Microbiol.
6
,
1568
(
2016
).
https://doi.org/10.3389/fmicb.2015.01568
Google Scholar
Crossref
Search ADS
PubMed
 
18.
C. X.
Yang
 and
L. F.
Nie
, “
Modelling the use of impulsive vaccination to control Rift Valley fever virus transmission
,”
. Differ. Equ.
2016
(
1
),
134
.
https://doi.org/10.1186/s13662-016-0835-1
Crossref
Search ADS
 
19.
C.
Castillo-Chavez
,
Z.
Feng
, and
W.
Huang
, “
On the computation of RO and its role on global stability
,” in
Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction
, edited by
P. C.
Castillo-Chavez
,
S.
Blower
,
P.
Driessche
,
D.
Kirschner
, and
A.-A.
Yakubu
 (
Springer
,
Berlin
,
2002
), p.
229
.
Google Scholar
 
20.
M.
Naveed
,
D.
Baleanu
,
M.
Rafiq
,
A.
Raza
,
A.
Hassan Soori
, and
N.
Ahmed
, “
Dynamical behavior and sensitivity analysis of a delayed Coronavirus epidemic model
,”
Comput. Mater. Continua
65
(
1
),
225
–
241
(
2020
).
https://doi.org/10.32604/cmc.2020.011534
Google Scholar
Crossref
Search ADS
 
21.
A.
Raza
,
A.
Ahmadian
,
M.
Rafiq
,
S.
Salahshour
,
M.
Naveed
,
M.
Ferrara
, and
A. H.
Soori
, “
Modeling the effect of delay strategy on transmission dynamics of HIV/AIDS disease
,”
Adv. Differ. Equ.
2020
(
1
),
663
.
https://doi.org/10.1186/s13662-020-03116-8
Crossref
Search ADS
PubMed
 
22.
M.
Naveed
,
M.
Rafiq
,
A.
Raza
,
N.
Ahmed
,
I.
Khan
,
K.
Sooppy Nisar
, and
A.
Hassan Soori
, “
Mathematical analysis of novel Coronavirus (2019-nCov) delay pandemic model
,”
Comput. Mater. Continua
64
(
3
),
1401
–
1414
(
2020
).
https://doi.org/10.32604/cmc.2020.011314
Google Scholar
Crossref
Search ADS
 
23.
A. I. K.
Butt
,
W.
Ahmad
,
M.
Rafiq
,
N.
Ahmad
, and
M.
Imran
, “
Computationally efficient optimal control analysis for the mathematical model of Coronavirus pandemic
,”
Expert Syst. Appl.
234
,
121094
(
2023
).
https://doi.org/10.1016/j.eswa.2023.121094
Google Scholar
Crossref
Search ADS
 
24.
W.
Ahmad
,
A. I. K.
Butt
,
N.
Akhtar
,
M.
Rafiq
,
M.
Gohar
,
Z.
Idrees
, and
N.
Ahmad
, “
Developing computationally efficient optimal control strategies to eradicate Rubella disease
,”
Phys. Scr.
99
(
3
),
035202
(
2024
).
https://doi.org/10.1088/1402-4896/ad1fc0
Google Scholar
Crossref
Search ADS
 
25.
W.
Ahmad
,
M.
Rafiq
,
A. I. K.
Butt
 et al, “
Analytical and numerical explorations of optimal control techniques for the bi-modal dynamics of Covid-19
,”
Nonlinear Dyn.
112
,
3977
–
4006
(
2024
).
https://doi.org/10.1007/s11071-023-09234-8
Google Scholar
Crossref
Search ADS
 
26.
M. M.
Alqarni
,
A.
Nasir
,
M. A.
Alyami
,
A.
Raza
,
J.
Awrejcewicz
,
M.
Rafiq
,
N.
Ahmed
,
T.
Sumbal Shaikh
, and
E. E.
Mahmoud
, “
A SEIR epidemic model of whooping cough-like infections and its dynamically consistent approximation
,” in
Complexity
, edited by
S. A.
Cheong
 (
Hindawi Limited
,
2022
), Vol.
2022
, pp.
1
–
13
.
Google Scholar
Crossref
Search ADS
 
27.
K. F.
Gurski
, “
A simple construction of nonstandard finite-difference schemes for small nonlinear systems applied to SIR models
,”
Comput. Math. Appl.
66
(
11
),
2165
–
2177
(
2013
).
https://doi.org/10.1016/j.camwa.2013.06.034
Google Scholar
Crossref
Search ADS
 
28.
X. J.
Yang
 and
F.
Gao
, “
A new technology for solving diffusion and heat equations
,”
Therm. Sci.
21
,
133
–
140
(
2017
).
https://doi.org/10.2298/tsci160411246y
Google Scholar
Crossref
Search ADS
 
29.
X. J.
Yang
, “
A new integral transform with an application in heat-transfer problem
,”
Therm. Sci.
20
,
S677
–
S681
(
2016
).
https://doi.org/10.2298/tsci16s3677y
Google Scholar
Crossref
Search ADS
 
30.
Abdullah
,
A. R.
Seadawy
, and
W.
Jun
, “
Mathematical methods and solitary wave solutions of three-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma and its applications
,”
Results Phys.
7
,
4269
–
4277
(
2017
).
https://doi.org/10.1016/j.rinp.2017.10.045
Google Scholar
Crossref
Search ADS
 
31.
X. J.
Yang
, “
A new integral transform operator for solving the heat-diffusion problem
,”
Appl. Math. Lett.
64
,
193
–
197
(
2017
).
https://doi.org/10.1016/j.aml.2016.09.011
Google Scholar
Crossref
Search ADS
 
32.
X. J.
Yang
, “
New integral transforms for solving a steady heat transfer problem
,”
Therm. Sci.
21
(
suppl 1
),
S79
–
S87
(
2017
).
https://doi.org/10.2298/tsci17s1079y
Google Scholar
Crossref
Search ADS
 
33.
A. R.
Seadawy
, “
Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods
,”
Eur. Phys. J. Plus
132
,
518
(
2017
).
https://doi.org/10.1140/epjp/i2017-11755-6
Google Scholar
Crossref
Search ADS
 
34.
D.
Lu
,
A. R.
Seadawy
, and
M.
Arshad
, “
Bright–dark solitary wave and elliptic function solutions of unstable nonlinear Schrödinger equation and their applications
,”
Opt. Quantum Electron.
50
,
23
(
2018
).
https://doi.org/10.1007/s11082-017-1294-y
Google Scholar
Crossref
Search ADS
 
35.
D.
Kumar
,
A. R.
Seadawy
, and
A. K.
Joardar
, “
Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology
,”
Chin. J. Phys.
56
,
75
–
85
(
2018
).
https://doi.org/10.1016/j.cjph.2017.11.020
Google Scholar
Crossref
Search ADS
 
36.
A. R.
Seadawy
 and
K.
El-Rashidy
, “
Traveling wave solutions for some coupled nonlinear evolution equations
,”
Math. Comput. Modell.
57
,
1371
–
1379
(
2013
).
https://doi.org/10.1016/j.mcm.2012.11.026
Google Scholar
Crossref
Search ADS
 
37.
A. R.
Seadawy
, “
Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron-positron plasma
,”
Physica A
455
,
44
–
51
(
2016
).
https://doi.org/10.1016/j.physa.2016.02.061
Google Scholar
Crossref
Search ADS
 
38.
D.
Lu
,
A. R.
Seadawy
, and
M.
Arshad
, “
Applications of extended simple equation method on unstable nonlinear Schrödinger equations
,”
Optik
140
,
136
–
144
(
2017
).
https://doi.org/10.1016/j.ijleo.2017.04.032
Google Scholar
Crossref
Search ADS
 
39.
A. R.
Seadawy
, “
Solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili dynamic equation in dust-acoustic plasmas
,”
Pramana
89
,
49
(
2017
).
https://doi.org/10.1007/s12043-017-1446-4
Google Scholar
Crossref
Search ADS
 
40.
A. R.
Seadawy
, “
The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions
,”
Optik
139
,
31
–
43
(
2017
).
https://doi.org/10.1016/j.ijleo.2017.03.086
Google Scholar
Crossref
Search ADS
 
41.
A. R.
Seadawy
, “
Three-dimensional nonlinear modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma
,”
Comput. Math. Appl.
71
,
201
–
212
(
2016
).
https://doi.org/10.1016/j.camwa.2015.11.006
Google Scholar
Crossref
Search ADS
 
42.
K. W.
Blayneh
 and
S. R.
Jang
, “
A discrete SIS-model for a vector-transmitted disease
,”
Appl. Anal.
85
(
10
),
1271
–
1284
(
2006
).
https://doi.org/10.1080/00036810600841498
Google Scholar
Crossref
Search ADS
 
43.
See
https://www.who.int/news-room/fact-sheets/detail/rift-valley-fever
for more information about Rift Valley fever
.
44.
Z. I.
Coulibaly
,
S.
Gowelo
,
I.
Traore
,
R. B.
Mbewe
,
W.
Ngulube
,
E. A.
Olanga
,
A. J.
DePina
,
A.
Sanou
,
S.
Coleman
, and
J. A.
Tangena
, “
Strengthening adult mosquito surveillance in Africa for disease control: learning from the present
,”
Curr. Opin. Insect Sci.
60
,
101110
(
2023
).
https://doi.org/10.1016/j.cois.2023.101110
Google Scholar
Crossref
Search ADS
PubMed
 
45.
C. J. A.
Bradshaw
,
B.
Leroy
,
C.
Bellard
,
D.
Roiz
,
C.
Albert
,
A.
Fournier
,
M.
Barbet-Massin
,
J. M.
Salles
,
F.
Simard
, and
F.
Courchamp
, “
Massive yet grossly underestimated global costs of invasive insects
,”
Nat. Commun.
7
,
12986
(
2016
).
https://doi.org/10.1038/ncomms12986
Google Scholar
Crossref
Search ADS
PubMed
 
© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).