Mathematical model analysis for dynamics and control of African cassava mosaic virus and East African cassava mosaic Cameroon virus co-infections

1 Introduction

Cassava is one of the major staple food source for hundreds of millions of people. It is a good source of carbohydrates, vitamin C, potassium, manganese, and folate. Among staple crops, cassava is regarded to be one of the top suppliers of carbohydrates [1]. Due to its high carbohydrates contents, it serve as an essential source of energy for over 500 millions people especially in developing countries [1]. Cassava is also used as animal feed and serves as a key ingredient in various industrial products, including food processing, starch, flour, and even some alcoholic beverages [1]. Cassava starch and flour have the potential to be valuable alternatives to rice, maize, and wheat crops [1]. Statistically, over 250 million tones of fresh cassava roots are produced globally each year, making up 32% of the total production of roots and tubers [1, 2]. Optimal production of cassava has significant economical value and consequently aid in achieving food security globally.

Despite the importance of cassava to humanity, several factors affect its optimal production. Some of the major factors that influence the optimal production of cassava include: ecological and environmental factors such as climate change, soil, pests and diseases, socio-economic factors, technical and management factors, etc. [3].

ACMV is a virus that causes cassava mosaic disease, causing mosaic symptoms (patchy, uneven coloring) on leaves and leading to significant yield losses. The ACMV co-infections, especially with other Cassava Mosaic Geminiviruses (CMGs) like East African cassava mosaic Cameroon virus (EACMCMV) are common and significantly worsen symptoms, leading to severe yield losses, stunted growth, and dramatic leaf distortion [4]. These two viruses are primarily transmitted through infected cassava stem cuttings and by the whitefly vector (Bemisia tabaci) which transmits these viruses. Co-infection enables complex viral interactions and promotes wider spread, resulting in more severe symptoms [4]. The co-infections of EACMCMV and ACMV have been found to be more prevalent in some endemic area in West Africa [4]. The high prevalence of EACMCMV and ACMV co-infections, together with their complex interactions that cause severe yield losses, has motivated the development of an applied mathematical model to investigate the dynamics and possible control of the diseases. The results of our study are expected to aid researcher, farmers and policy makers in taking decisions that can improve the management of the diseases leading to a decrease in disease prevalence and increase yields.

In terms of transmission biology, both ACMV and EACMCMV are transmitted by Bemisia tabaci in a persistent, circulative manner. After acquisition from an infected cassava plant, the virus passes through the gut of the whitefly, circulates within the hemolymph, and reaches the salivary glands, from where it can be transmitted to healthy plants during subsequent feeding. Once acquired, the viruses can be retained by the vector for an extended period without replicating inside the insect body. This persistent transmission mechanism implies that infected whiteflies remain infectious for a substantial duration, thereby enhancing the potential for sustained disease spread in cassava fields.

Even though ACMV and EACMCMV are transmitted by the whitefly Bemisia tabaci, they have some differences. Genetically, ACMV are highly conserved while EACMCMV are highly diverse and recombinants. Epidemiologically, ACMV are widespread and dominant while EACMCMV are still emerging. ACMV have mild to severe mosaic symptoms while EACMCMV often have severe symptoms especially in mixed infection.

Several research projects have examined the dynamics and control of cassava mosaic disease (CMD) using a variety of approaches. Some of the key studies that form part of our literature review are summarized below. For example, Erick et al. [5] employs a mathematical model to investigate the dynamics of cassava mosaic disease in the presence of non-cassava host plants. Their results show that the mortality rate and carrying capacity of whiteflies are the most sensitive parameters, indicating that effective control strategies should focus on reducing whitefly populations.

Another important study by Chapwanya et al. [6] develops a mathematical epidemiological model that incorporates both crop growth and vector dynamics to examine interactions among cassava plants, whitefly vectors, and the virus. Their qualitative analysis explores several epidemiological properties of the model, including the existence and uniqueness of solutions, the disease-free and endemic equilibria, conditions for local and global asymptotic stability, and the presence of a Hopf bifurcation.

Al Basir et al. [7] proposes a mathematical model for CMD dynamics that includes immature and mature vectors, demonstrating that whitefly maturation has a significant effect on disease epidemiology. Similarly, Sangsawang et al. [8] develop a deterministic model with saturated incidence to explain how whitefly vectors influence CMD outbreaks.

A discrete-time model introduced by Reja et al. [9] captures the complex transmission dynamics of CMD via whitefly vectors. Their study shows that effective control of both plant infection rates and vector abundance is crucial for managing the disease, and their findings align with those from continuous-time models.

In another contribution, Jittamai et al. [10] develop a mathematical model for CMD transmission through infected cuttings and whitefly vectors and use it to analyse outbreak dynamics and identify cost-effective control strategies. The study of Fargette et al. [11] also highlights the importance of planting date and plant age in determining the primary spread of ACMV. By fitting and simulating their model using experimental data, they provide valuable insights for improving ACMV control strategies.

A fractional-order model for CMD dynamics based on the Caputo-Fabrizio derivative was proposed by Abdullah et al. [12], who show that reducing whitefly populations is more effective at limiting plant infection than increasing the recovery rate. Complementing this, Minhós et al. [13] developed a stochastic fractional delay model to analyze CMD dynamics, revealing that their approach is useful for identifying critical parameters influencing disease magnitude and the timing of interventions. More studies on CMD and related plant diseases can be found in Collins and Duffy [14], Collins and Duffy [15], Collins and Duffy [16], Kinene et al. [17], Minhós et al. [13], and Santra et al. [18].

Although these studies have made significant contributions to understanding CMD dynamics and control, none have addressed the co-infection of EACMCMV and ACMV, a combination that is increasingly prevalent and associated with severe yield losses. The aim of the present study is to fill this gap by developing a mathematical epidemiological model to investigate the dynamics and control of ACMV and EACMCMV co-infection. The findings of this study are expected to assist researchers, farmers, and policymakers in making informed decisions to improve the management of co-infection, reduce disease prevalence, enhance cassava yield, and ultimately contribute to global food security.

2 Model development

The mathematical model development for cassava plant population dynamics under ACMV and EACMCMV disease co-infection is derived based on the following assumptions. One of the means of transmitting the Cassava Mosaic Virus (CMV) which include ACMV and EACMCMV is through the whitefly vector. This primary vector is the whitefly, Bemisia tabaci, which can transmit multiple viruses in a single bite. Thus, these two virus have the same vector. Another means of transmitting the virus is by using cuttings from an infected plant. Based on these explanations, we consider a total cassava plant population N and partition it into four sub-populations: namely healthy cassava plants denoted by S, cassava plants infected by only ACMV denoted by I₁, cassava plants infected by only EACMCMV denoted by I₂ and cassava plants co-infected by ACMV and EACMCMV denoted by I₁₂. Treatment of cassava plants infected with mosaic virus using thermotherapy and hot water treatment does not guarantee complete recovery as a proportion remain infected after being returned to the normal temperature [19–21]. Based on this, recovered class for the cassava plants is not considered in the model. The mode of transmission of ACMV or EACMCMV is through whitefly vectors carrying the viruses, or by using cuttings from an infected cassava plant. Let M denote the total population of whitefly vectors. The whitefly population M is partitioned into four classes: whitefly vectors not carrying ACMV or EACMCMV; whitefly vectors carrying ACMV (Z₁), whitefly vectors carrying EACMCMV (Z₂), and whitefly vectors carrying both ACMV and EACMCMV (Z₁₂).

The interactions among the cassava and whitefly sub-populations that give rise to the epidemiological dynamics are described as follows. The susceptible cassava population S increase logistically to its carrying capacity K at a growth rate r and decreases through infection as a results of contact with whitefly vectors Z₁, Z₂ or Z₁₂ that carry the virus at a rate β1, β2 or β0* respectively. The infected cassava plants I₁, I₂ or I₁₂ increase through planting infected planting materials at a rate σ1, σ2, or σ12* respectively. The infected cassava plants I₁, I₂ or I₁₂ decreases due to crop failure at a rate μ1, μ2, and μ12* respectively. Cassava plants already infected with ACMV I₁ can be co-infected with EACMCMV through whitefly vector Z₂ at a rate β₁₂* while the cassava plants already infected with EACMCMV I₂ can be co-infected with ACMV through whitefly vector Z₁ at a rate β₂₁*.

Whitefly vectors not carrying ACMV or EACMCMV (Y) increases through recruitment at a rate Λ^* and decrease as they get the virus through contact with infected cassava plants I₁, I₂ or I₁₂ at a rate α1, α2, or α12*. Since the virus do not affect (spread in the body of) the whitefly, both the vectors that carrying the virus and those not carrying the virus undergo regeneration at a rate ξ. Whitefly vectors already carrying ACMV Z₁ can in addition carry EACMCMV at a rate α₁₂* while the whitefly vectors already carrying EACMCMV Z₂ can in addition to it carry ACMV at a rate α₂₁*.

To consider control measures, we consider planting resistant cassava varieties and good management practices like using disease free planting materials, weed control, crop rotation and removal of crop residues decreases infections. Use of appropriate insecticides reduces the vectors population at a rate γ^*. Use of modern technology (i.e., use of AI powered drones for fertilization, tracking of weather, and irrigation, etc.) enhances growth of healthy cassava plants S at a rate ϕ. Use of disease free-planting materials reduces the planting of infected cassava stem by a proportion θ. Use of cassava resistant varieties reduces contact rates βi, αi, βij* or αij* by a proportion ρ_*i*, ρ_*ij* respectively. Based of these formulations, the following ACMV and EACMCMV co-infection model with control measures is obtained.

dSdt=ϕrS(1-SK)-(1-ρ1)β1Z1S-(1-ρ2)β2Z2S - (1-ρ0)β0Z12S, dI1dt=(1-ρ1)β1Z1S+(1-θ)σ1I1-(1-ρ12)β12I1Z2 - μ1I1, dI2dt=(1-ρ2)β2Z2S+(1-θ)σ2I2-(1-ρ21)β21I2Z1 - μ2I2, dI12dt=(1-ρ0)β0Z12S+(1-θ)σ12I12+(1-ρ12)β12I1Z2 + (1-ρ21)β21I2Z1-μ12I12, dYdt=Λ-(1-ρ1)α1I1Y-(1-ρ2)α2I2Y - (1-ρ0)α0I12Y-(γ+ξ)Y, dZ1dt=(1-ρ1)α1I1Y-(1-ρ12)α12Z1I2-(γ+ξ)Z1, dZ2dt=(1-ρ2)α2I2Y-(1-ρ21)α21Z2I1-(γ+ξ)Z2,dZ12dt=(1-ρ0)α0I12Y+(1-ρ12)α12Z1I2+(1-ρ21)α21Z2I1 - (γ+ξ*)Z12.(1)

The description of variables and parameters are presented in Table 1, 2.

Table 1. Description of variables for Equation 1.

Table 2. Description of parameters for Equation 1.

3 Analyses and results

3.1 Model transformation

The mathematical Equation 1 consists of both insect vector populations and cassava plant populations, each measured in different units. Therefore, to facilitate analysis, the model is non-dimensionalized by converting all variables and parameters into dimensionless quantities. This helps to reveal hidden relationships and dependencies between variables and parameters, thereby providing deeper insight into the model dynamics [22, 23]. By rescaling Equation 1 such that τ=rt, s=SK, i1=I1K, i2=I2K,i12=I12K, y=YM0, z1=Z1M0, z2=Z2M0, z12=Z12M0, β1=β1M0r, β2=β2M0r, β0=β0M0r, β12=β12M0r, β21=β21M0r, α1=α1Kr, α2=α2Kr, α0=α0Kr, α12=α12Kr, α21=α21Kr, Λ=ΛrM0, μ1=μ1r, μ2=μ2r, μ12=μ12r, ξ=ξr, γ=γr, M0=Y(0)+Z1(0)+Z2(0)+Z12(0) the dimensionless version of Equation 1 is obtained as follows:

dsdτ=ϕs(1-s)-(1-ρ1)β1z1s-(1-ρ2)β2z2s -(1-ρ0)β0z12s, di1dτ=(1-ρ1)β1z1s-(1-ρ12)β12i1z2-(μ1-(1-θ)σ1)i1, di2dτ=(1-ρ2)β2z2s-(1-ρ21)β21i2z1-(μ2-(1-θ)σ2)i2, di12dt=(1-ρ0)β0z12s+(1-ρ12)β12i1z2+(1-ρ21)β21i2z1 - (μ12-(1-θ)σ12)i12, dydτ=Λ-(1-ρ1)α1i1y-(1-ρ2)α2i2y-(1-ρ0)α0i12y - (γ+ξ)y, dz1dτ=(1-ρ1)α1i1y-(1-ρ12)α12z1i2-(γ+ξ)z1, dz2dτ=(1-ρ2)α2i2y-(1-ρ21)α21z2i1-(γ+ξ)z2,dz12dt=(1-ρ0)α0i12y+(1-ρ12)α12z1i2+(1-ρ21)α21z2i1 - (γ+ξ)z12.(2)

For simplicity, the following assumptions are introduced where appropriate: δ₁ = (1−ρ₁)β₁, δ₂ = (1−ρ₂)β₂, δ₃ = (1−ρ₀)β₀, δ₄ = (1−ρ₁₂)β₁₂, δ₅ = (1−ρ₂₁)β₂₁, δ₆ = μ₁−(1−θ)σ₁, δ₇ = μ₂−(1−θ)σ₂, δ₈ = μ₁₂−(1−θ)σ₁₂, η₁ = (1−ρ₁)α₁, η₂ = (1−ρ₂)α₂, η₃ = (1−ρ₀)α₀, η₄ = (1−ρ₁₂)α₁₂, η₅ = (1−ρ₂₁)α₂₁, η₆ = γ+ξ.

3.2 ACMV sub-model

The ACMV only sub-model is a special case of the Equation 2 where ACMV is the only disease affecting the cassava plants. A qualitative analysis of this special case, which is crucial for understanding ACMV dynamics, is presented in this section. The ACMV only sub-model is obtained by setting i₂ = i₁₂ = z₂ = z₁₂ = 0 in the Equation 2, to obtain

dsdτ=ϕs(1-s)-δ1z1s,di1dτ=δ1z1s-δ6i1,dydτ=Λ-η1i1y-η6y,dz1dτ=η1i1y-η6z1.(3)

The analysis of Equation 3 describes the cassava plant dynamics in the presence of ACMV only.

3.2.1 Equilibrium points of the ACMV Equation 3

The ACMV Equation 3 has two important equilibrium points namely: the disease free equilibrium (DFE) and endemic equilibrium (EE). The DFE is the equilibrium point of the model when there is no ACMV disease in the system while the EE is the equilibrium point of the model when there is ACMV disease in the system. The analytical description of the DFE of Equation 3 is given by

(s0, i10, y0, z10)=(1, 0, Λη6, 0).(4)

Similarly, the analytical representation of the EE of Equation 3 is given by Equation 5

(s,i1,y,z1)=(−b1+(b1)2−4a1c12a1,A3z1y, y0−z1,A1(1−s)),(5)

where a1=δ1A1, b1=δ1(y0-A1), c1=-A2A3, A1=ϕδ1, A2=δ6=μ1-(1-θ)σ1>0, A3=η6η1.

3.2.2 The basic reproduction number of the ACMV Equation 3

Epidemiologically, the basic reproduction number of an infection, denoted by R0, represents the expected number of cases generated by one case in a population where all individuals are susceptible to infection [24]. The value of the basic reproduction number indicates whether an outbreak will be eradicated or will persist [24, 25]. The basic reproduction number for the Equation 3 is computed using the next generation matrix method of Van den Driessche and Watmough [24] and is given by Equation 6

R1=δ1η1s0y0δ6η6.(6)

This can be re-written as Equation 7

(R1)2=δ1η1s0y0δ6η6=(1-ρ1)β1(1-ρ1)α1s0y0(μ1-(1-θ)σ1)(γ+ξ).(7)

3.2.3 Stability analysis of the ACMV Equation 3

The stability analysis of a model about its equilibrium point describes the dynamics of the model about the equilibrium point. The stability analysis of the ACMV sub-model is summarized in the following theorems.

Theorem 1. The Equation 3 is locally asymptotically stable about the DFE (Equation 15) provided R1<1.

Proof. To prove Theorem 1, it suffices to show that all the eigenvalues of the Jacobian of Equation 3 about the DFE point (Equation 4) have negative real part. The Jacobian of Equation 3 about the DFE (Equation 4) is given by Equation 8

J0=(-ϕ00-δ10-δ60δ10-y0η1-η600y0η10-ϕ4).(8)

The eigenvalues of the Jacobian are

λ1=-ϕ,λ2=-η6,λ3=-(δ6+η6)-(δ6+η6)2+4δ6η6(R12-1)2,λ4=-(δ6+η6)+(δ6+η6)2+4δ6η6(R12-1)2.

Obviously, λ₁, λ₂, λ₃ are negative. However, λ₄ < 0 if R1<1. Thus, we conclude that the Equation 3 is locally asymptotically stable about the DFE (Equation 4) provided R1<1.

The implication of Theorem 1 is that when ACMV is the only disease present, eradication is possible if the initial number of infected plants and whitefly vectors (cassava plants i₁(0) and whitefly vectors z₁(0)) is sufficiently small (within the neighborhood of the DFE) and R1<1. However, if the initial infected population is large, the actual dynamics of the model about the DFE require the investigation of the global stability of the model. The global stability analysis of Equation 3 is presented in Theorem 2.

Theorem 2. The Equation 3 is globally asymptotically stable about the DFE (Equation 4) provided R1<1.

The proof of Theorem 2 will be established using a stability result in Castillo-Chavez et al. [26] which is stated in Lemma 1 below Equation 9.

Lemma 1. Castillo-Chavez et al. [26] Consider a model system written in the form

dZ1dt=F(Z1,Z2)dZ2dt=G(Z1,Z2), G(Z1,0)=0,(9)

where Z1∈Rm and Z2∈Rn. Z0=(Z1*,0) denotes the disease-free equilibrium of the system. Assume that

• (H1) For dZ1dt=F(Z1,0), Z1* is globally asymptotically stable;
• (H2) G(Z₁, Z₂) = AZ₂−Ĝ(Z₁, Z₂), Ĝ(Z₁, Z₂)≥0 for (Z₁, Z₂)∈Ω, where the Jacobian A=∂G∂Z2(Z1,0) *is an M–matrix (the off diagonal elements of A are non-negative) and Ω is the region where the model makes biological sense.*

*Then the disease-free equilibrium Z₀ is globally asymptotically stable provided that* R0<1 [26].

Proof. To use Lemma 1, it suffices to show that the disease free part of Equation 3 is globally stable (condition (H1)) and the disease part of Equation 3 satisfies conditions H2 of the Lemma. From Equation 3, let Z₁ = (s, y), Z₂ = (i₁, z₁). So, disease free part of Equation 3 is given by

dZ1dτ=F(Z1,0)=(s(1-s)Λ-η6y)(10)

whereas the disease part of Equation 3 is

dZ2dt=G(Z1,Z2)=(δ1z1s-δ6i1η1i1y-η6z1).(11)

Clearly, disease free part (Equation 10) is made up of a set of linear ordinary differential equations and its exact solution can be obtained as s(τ)=11+A4e-τ, y(τ)=Λη6+A5e-η6τ, where A4=1-s(0)s(0) and A5=y(0)-Λη6. As τ → ∞, s(τ)→s0=1,y(τ)→y0=Λη6. Thus, the disease free part of Equation 3 is globally asymptotically stable.

For the disease part (Equation 11), we compute the Jacobian of Equation 11 about the disease free equilibrium as

A=(-δ6δ1s0η1y0-η6),(12)

where A is an M–matrix with all its off diagonal elements non-negatives. From Equations 11, 12, the term Ĝ(Z₁, Z₂) is determined as Equation 13

Ĝ(Z1,Z2)=(δ1z1(s0-s)η1i1(y0-y)).(13)

Clearly, Ĝ(Z₁, Z₂) ≥ 0, since s⁰ ≥ s and y⁰ ≥ y. This shows that condition (H₂) holds and hence the proof is complete.

The epidemiological implication of this results is that ACMV disease could possibly be eradicated irrespective of the size of the infected cassava plants and whitefly vectors carrying ACMV at the initial stage of the outbreak, provided the control measures are effective to keep the associated basic reproduction number R1 below unity.

3.3 EACMCMV sub-model

The EACMCMV sub-model is a special case of the Equation 2 where EACMCMV is the only disease affecting the cassava plants. Qualitative analysis of this special case is crucial for understanding the dynamics of EACMCMV is considered in this section. The EACMCMV sub-model is obtained by setting i₁ = i₁₂ = z₁ = z₁₂ = 0 in Equation 2, to obtain

dsdτ=ϕs(1-s)-δ2z2s,di2dτ=δ2z2s-δ7i2,dydτ=Λ-η2i2y-η6y,dz2dτ=η2i2y-η6z2.(14)

The analysis of Equation 14 will reveal the cassava plants dynamics in the presence of EACMCMV and control measures only.

3.3.1 Equilibrium points of the EACMCMV Equation 14

The EACMCMV Equation 14 has two important equilibrium points namely: the DFE and EE. The analytical description of the DFE of Equation 14 is given by

(s0, i20, y0, z20)=(1, 0, Λη6, 0).(15)

Similarly, the analytical representation of the EE of Equation 14 is given by Equation 16

(s, i2, y, z2)= (-b1+(b1)2-4a1c12a1, B3z1y, y0-z1, B1(1-s*)),(16)

where a1=δ2B1, b1=δ2(y0-B1), c1*=-B2B3, B1=ϕδ2, B2=μ2-(1-θ)σ2>0, B3=η6η2.

3.3.2 The basic reproduction number of the EACMCMV Equation 14

The basic reproduction number for the Equation 14 is determined using the next generation matrix method of Van den Driessche and Watmough [24] and is given by Equation 17

R2=δ2η2s0y0δ7η6.(17)

This can be re-written as Equation 18

(R2)2=δ2η2s0y0δ7η6=(1-ρ2)β2s0(μ2-(1-θ)σ2).(1-ρ2)α2y0(γ+ξ).(18)

3.3.3 Stability analysis of the EACMCMV Equation 3

The stability analysis of the EACMCMV sub-model is summarized in the following theorems.

Theorem 3. The Equation 14 is locally asymptotically stable about the DFE (Equation 15) provided R2<1.

The proof of Theorem 3 can be established using a similar approach used in the proof of Theorem 1.

The implication of Theorem 3 is that for the situation when only EACMCMV is present in the system, the disease can be eradicated if the initial infected cassava plants i₂(0) and its associated whitefly z₂(0)) is small (within the neighborhood of the DFE) and provided the control measures are effective enough to keep associated basic reproduction number R2 below unity. However, if the initial infected population is large, the actual dynamics of the model about the DFE required the investigation of the global stability of the model. The global stability analysis of Equation 3 is presented in Theorem 4.

Theorem 4. The Equation 14 is globally asymptotically stable about the DFE (Equation 15) provided R2<1.

The proof of Theorem 4 can be established using a similar approach used in the proof of Theorem 2.

The implication of this result is that EACMCMV can be eradicated regardless of the initial number of infected cassava plants and whitefly vectors carrying EACMCMV, provided the control measures are effective to keep the associated basic reproduction number R2<1.

3.4 Analysis of the ACMV and EACMCMV co-infection Equation 2

The dynamical system analysis of the ACMV and EACMCMV co-infection Equation 2 is presented in this section. This analysis is crucial for understanding the dynamics and control of ACMV and EACMCMV co-infection.

3.4.1 The basic reproduction number where there is co-infection of ACMV and EACMCMV but they do not appear in the system independently

The basic reproduction number of the situation where there is co-infection of ACMV and EACMCMV but they do not appear in the system as independent diseases is calculated. This number (R3) is required to calculate the basic reproduction number (R0) for the entire co-infection Equation 2 as can be seen in section 3.4.3 below. The calculation steps are the same as above for R1 and R2 and result in Equation 19

R3=δ3η3s0y0δ8η6.(19)

3.4.2 Equilibrium points of the ACMV and EACMCMV co-infection Equation 2

The ACMV and EACMCMV co-infection Equation 2 has important equilibrium points namely: DFE and EE. The analytical representation of the DFE of the ACMV and EACMCMV co-infection Equation 2 is given by

(s0,i10,i20,i120,y0,z10,z20,z120)=(1, 0, 0, 0, Λη6, 0, 0, 0).(20)

The EE of the ACMV and EACMCMV co-infection Equation 2 is complex to represent analytically, so will considered numerically.

3.4.3 Basic reproduction number of the ACMV and EACMCMV co-infection Equation 2

The basic reproduction number of the ACMV and EACMCMV co-infection Equation 2 is computed using the next generation matrix method of Van den Driessche and Watmough [24] and is given by Equations 21, 22

R0=max{R1, R2, R3},(21)

where

R1=δ1η1s0y0δ6η6, R2=δ2η2s0y0δ7η6, R3=δ3η3s0y0δ8η6.(22)

This shows that the basic reproduction number of the full co-infection model exceeds those of the individual ACMV and EACMCMV sub-models. An increase in basic reproduction number implies an increase in severity of the infection. This supports the findings that co-infection by ACMV and EACMCMV leads to greater severity of infections and consequently results in greater reductions in crop yields compared to a single infection by either virus [27].

3.4.4 Stability analysis of the ACMV and EACMCMV co-infection Equation 2

Theorem 5. For R0<1, the ACMV and EACMCMV co-infection Equation 2 is globally asymptotically stable about the disease free equilibrium point (Equation 20) if δ3z12(s0-s)>δ4i1z2+δ5i2z1 and η3i12(y0-y)>η4i2z1+η5i1z2 otherwise not globally asymptotically stable.

The proof of Theorem 5 will be established using Lemma 1.

Proof: To prove Theorem 5, it is sufficient to verify how the model satisfies the conditions of the global stability conditions of Lemma 1 [26]. From the ACMV and EACMCMV co-infection Equation 2, let Z₁ = (s, y), Z₂ = (i₁, i₂, i₁₂, z₁, z₂, z₁₂). So, disease free part of ACMV and EACMCMV co-infection Equation 2 is given by Equation 23

dZ1dt=F(Z1,0)=(ϕs(1-s)Λ-η6y)(23)

whereas the disease part of ACMV and EACMCMV co-infection Equation 2 is

dZ2dt=G(Z1,Z2)=(δ1z1s-δ4i1z2-δ6i1δ2z2s-δ5i2z1-δ7i2δ3z12s+δ4i1z2+δ5i2z1-δ8i12η1i1y-η4i2z1-η6z1η2i2y-η5i1z2-η6z2η3i12y+η4i2z1+η5i1z2-η6z12).(24)

The Jacobian of the disease part (Equation 24) about the disease free equilibrium is

A=(-δ60δ1s00000-δ700δ2s0000-δ800δ3s0η1y000-η6000η2y000-η6000η3y000-η6),(25)

where A is an M–matrix with all its off diagonal elements non-negatives. From Equations 24, 25, the term Ĝ(Z₁, Z₂) is determined as

Ĝ(Z1,Z2)=(δ4i1z2+δ1z1(s0-s)δ5i2z1+δ2z2(s0-s)-δ4i1z2-δ5i2z1+δ3z12(s0-s)η4i2z1+η1z1(y0-y)η5i1z2+η2z2(y0-y)-η4i2z1-η5i1z2+η3i12(y0-y)).(26)

Clearly, Ĝ(Z₁, Z₂) is positive if δ3z12(s0-s)>δ4i1z2+δ5i2z1 and η3i12(y0-y)>η4i2z1+η5i1z2 otherwise it is negative. This shows how condition (H₂) holds and hence completes the proof. Therefore, the ACMV and EACMCMV co-infection Equation 2 is globally asymptotically stable if z12(s0-s)>δ4i1z2+δ5i2z1 and η3i12(y0-y)>η4i2z1+η5i1z2 otherwise it is not globally asymptotically stable about the disease free equilibrium point (Equation 20).

The implication of this results indicate whether or not it will be difficult to eradicate both ACMV and EACMCMV simultaneously by lowering the basic reproduction number below unity. For instance, if δ3z12(s0-s)<δ4i1z2+δ5i2z1 and η3i12(y0-y)<η4i2z1+η5i1z2, it will be difficult to eradicate the disease by only lowering the basic reproduction number below unity. This could explain why co-infection ACMV and EACMCMV is more severe and difficult to control.

4 Numerical simulations

In this section, numerical simulations of the co-infection of ACMV and EACMCMV Equation 2 is presented. Insights gained from this study can inform broader global strategies aimed at combating food insecurity. The parameter values used for the numerical simulations are given in Table 3. As shown in the Table 3, most of the parameter values are extracted from the literature while the remaining are assumed from a statistically realistic range. For instance, the realistic parameter value for effectiveness of resistance variety (ρ₁), is 0 ≤ ρ ≤ 1, where 0 means that the resistance variety has no effect and 1 means that the resistant variety is 100% effective. Based, on this explanation, 0.1 ≤ ρ ≤ 0.9 is taken as parameter range for the numerical simulation. By a similar reasoning, the parameter values for the remaining parameters are obtained.

Table 3. Parameter values used in the simulations.

4.1 Numerical simulation of ACMV Equation 3

The numerical simulation of the ACMV Equation 3 which is a situation where the cassava plants is being affected by ACMV only is presented in this subsection.

Figure 1 is a plot illustrating the dynamics of ACMV Equation 3 when R1=0.4<1. The figure shows that when R1<1, the healthy cassava (s) increases to its carrying capacity while the cassava infected by ACMV (i₁) decreases to zero. Similarly, the whitefly vectors not carrying ACMV (y) increases to its carrying capacity while the whitefly infected by ACMV (z₁) decreases to zero. This shows that ACMV disease could possibly be eradicated leading to increased healthy cassava plants and ACMV free whitefly vectors by using appropriate control measures that are effective to keep the associated basic reproduction number R1 below unity. This results confirms Theorem 2 which state that the Equation 3 is globally asymptotically stable about the DFE (Equation 4) provided R1<1.

Figure 2 is a plot illustrating the dynamics of ACMV Equation 3 when R1≈1. The figure shows that when R1 is approximately one, the dynamics of the healthy cassava (s), infected cassava (i₁), whitefly vectors not carrying ACMV (y) and whitefly vectors carrying ACMV (z₁) remain unchanged (stable) for a long time. So for R1≈1, the dynamics of cassava and whitefly vectors remain unchanged/stable for the duration of the study period. This is not recommended especially if the infected population is high at the initial stage of the outbreak. Thus, control measures that will reduce the R1 below one are strongly recommended to ensure disease eradication as shown in Figure 1.

Figure 3 is a plot illustrating the dynamics of ACMV Equation 3 when R1>1. The figure shows that when R1=3.5>1, the dynamics of the ACMV Equation 3 undergoes a cyclic dynamics. These complex dynamics illustrate the possible dynamics that can occur at the endemic stage of an outbreak when there are many infected cassava plants and whitefly vectors carrying ACMV, interacting in the system. This situation is undesirable and can result in substantial cassava yield losses if not controlled. Thus, control measures that will reduce the R1 below one are strongly recommended to ensure disease eradication.

The numerical illustration of the dynamics of ACMV Equation 3 in Figures 1–3 confirm that R1 is a bifurcation quantity [24–26]. Thus, estimating the value of R1 using real data together with our model can give accurate cassava mosaic disease prediction. This will be carried out in our future studies as real data becomes available.

4.2 Numerical simulation of EACMCMV Equation 14

Numerical simulation of the EACMCMV Equation 14 where the cassava plants are affected by EACMCMV only is presented in this subsection.

Figure 4 is a plot illustrating the dynamics of EACMCMV Equation 14 when R2=0.4<1. When R2<1, the healthy cassava (s) increase to carrying capacity while the cassava infected by EACMCMV (i₂) decreases to zero. Similarly, the whitefly vectors not carrying EACMCMV (y) increases to its carrying capacity while the whitefly vectors carrying EACMCMV (z₂) decrease to zero. This shows that implementing appropriate control measures that keep the associated basic reproduction number R2 below unity result in EACMCMV disease eradication irrespective of the initial infected populations. This results support Theorem 3 which state that the EACMCMV Equation 14 is globally asymptotically stable about the DFE (Equation 4) provided R2<1.

Figure 5 is a plot illustrating the dynamics of EACMCMV Equation 14 when R2≈1. The figure shows that when R2 is approximately one, the dynamics of the healthy cassava (s), infected cassava (i₂), whitefly vectors not carrying EACMCMV (y) and whitefly vectors carrying EACMCMV (z₂) remain unchanged (stable) indefinitely. So for R2≈1, the dynamics of cassava and whitefly vectors remain unchanged/stable which is also not recommended especially if the infected population is high at the onset of the outbreak. Thus, again control measures that reduce the R2 below one is recommended to ensure disease eradication.

Figure 6 is a plot illustrating the dynamics of EACMCMV Equation 14 when R2>1. The figure reveals that when R2=3.5>1, the dynamics of the EACMCMV Equation 14 undergoes a cyclic dynamics. These complex dynamics illustrate a possible dynamics that can occur at the endemic stage of EACMCMV outbreak when there are many infected cassava plants and whitefly vectors carrying EACMCMV, interacting in the system. This situation can lead to high loss of cassava yield if not controlled. Thus, control measures that will reduce the R2 below one are strongly recommended to ensure EACMCMV disease eradication.

The numerical illustration of the dynamics of EACMCMV Equation 14 in Figures 4–6 also confirm that R2 is a bifurcation quantity [24–26]. Thus, estimating the value of R2 using real data together with our Equation 14 will give accurate EACMCMV disease prediction. This will be carried out in our future studies as real data becomes available. The above analysis revealed a similarity on the dynamics of EACMCMV Equation 14 and ACMV Equation 3. A possible explanation for the similarity could be because the two viruses are transmitted by the same whitefly vector. In the next section, we shall examined the dynamics of the co-infection of the two viruses: ACMV and EACMCMV.

4.3 Numerical simulation of ACMV and EACMCMV co-infection Equation 2

The numerical simulation of the ACMV and EACMCMV co-infection Equation 2 which is a situation where the cassava plants is being affected by ACMV and EACMCMV simultaneously is presented in this subsection.

Figure 7 is a plot illustrating the dynamics of co-infection ACMV and EACMCMV Equation 2 when R0<1. The figure reveals that when R0<1, the healthy cassava (s) increases to its carrying capacity while the cassava plants infected by ACMV (i₁), the cassava plants infected by EACMCMV (i₂) and cassava plants co-infected by ACMV and EACMCMV (z₁₂) decreases to zero. This shows that implementing appropriate control measures that are effective to keep the associated basic reproduction numbers R0 below unity can leads to total disease eradication. Therefore, implementation of effective control measures that will keep the R0 below unity is strongly recommended for possible disease eradication for a cassava plants that are co-infected by ACMV and EACMCMV.

Figures 8, 9 are plots illustrating the endemic dynamics of ACMV and EACMCMV co-infection Equation 2 when R0>1. From the figures, it can be observed that when R0>1, both cassava plants and whitefly undergoes an endemic cyclic dynamics. These figures reveal the type of interaction that exists among the cassava plants infected by ACMV, EACMCMV, or co-infected by ACMV and EACMCMV. This type of complex dynamics among the various cassava plants and whitefly vectors could explain why it is difficult to control ACMV and EACMCMV co-infection. Therefore, every effort should be made to keep the R0 below unity so that the disease should be eradicated. Next, we investigate numerically the impact of various control measures on the endemic dynamics of ACMV and EACMCMV co-infection Equation 2.

Figure 10 is a graphical illustration of the impact of use of modern technology (ϕ) on the dynamics of ACMV and EACMCMV co-infection Equation 2. The figure shows that use of modern technology (ϕ) does not have significant impact to enhance cassava productivity at endemic stage of the outbreak nor reduces infections.

A possible explanation for this result is that once the outbreak reaches the endemic stage, with high levels of infected plants and whitefly vectors carrying the viruses, productivity enhancing technologies alone cannot improve yields unless the disease is first controlled. Thus, it is recommended that infection eradication measures be implemented before introducing productivity enhancing interventions.

Figure 11 is a graphical illustration of the impact of use of resistant varieties (ρ₁) on the dynamics of ACMV and EACMCMV co-infection Equation 2. The figure shows that use of resistant varieties to reduce infections due to ACMV only have positive impact in reducing the cassava plants infected by ACMV (i₁), but do not have significant positive impact in reducing infections due to EACMCMV diseases or co-infection. Specifically, effective implementation of the use of resistant varieties leads to eradication of the cassava plants infected by ACMV within a short period. Thus, it is recommended to use resistant varieties to reduce infections in cassava plants infected with ACMV only.

Figure 12 is a graphical illustration of the impact of use of resistant varieties (ρ₂) on the dynamics of ACMV and EACMCMV co-infection Equation 2. The figure shows that use of resistant varieties to reduce infections due to EACMCMV only have positive impact in reducing the cassava plants infected by EACMCMV (i₂), but do not have significant positive impact on reducing infections due to ACMV diseases or co-infections. Specifically, effective implementation of the use of resistant varieties leads to eradication of the cassava plants infected by EACMCMV within a short period. Thus, the use of resistant varieties to reduce infections in cassava plants infected with EACMCMV only is recommended.

Figure 13 is a graphical illustration of the impact of use of two different type of resistant varieties (ρ₁ and ρ₂) on the dynamics of ACMV and EACMCMV co-infection Equation 2. The figure shows that use of resistant varieties to simultaneously reduce infection due to ACMV or EACMCMV have positive impact in reducing the cassava plants infected by ACMV or EACMCMV, but do not have significant positive impact on reducing co-infected cassava plants. Specifically, effective implementation of the use of resistant varieties leads to eradication of the cassava plants infected by ACMV or EACMCMV within a short period. Thus, the use of multiple types of resistant varieties simultaneously to reduce infections in cassava plants infected with ACMV or EACMCMV is strongly recommended.

Figure 14 is a graphical illustration of the impact of removal rate of vectors using control measures (γ) on the dynamics of ACMV and EACMCMV co-infection Equation 2. The figure shows that removing of whitefly vectors using appropriate insecticides have significant impact in reducing cassava infections due to ACMV or EACMCMV. Specifically, effective implementation of vector removal can lead to eradication of infections due to ACMV, EACMCMV or both. Thus, the whitefly vector removal using appropriate insecticides is strongly recommended for ACMV and EACMCMV disease eradication.

5 Discussion

Cassava is one of the essential staple food crop especially in developing countries. Its nutritional values include a high source of carbohydrate and vitamin C but low in protein and fat. It is also serves as animal feed and is also used as a raw material in starch-based industries. In general, cassava production plays a significant role is achieving and sustaining food security globally. Although cassava offers many benefits, its optimal production is hindered by several factors, particularly pests and diseases. Two of the major diseases that affects cassava yield are ACMV and EACMCMV. The co-infection of these two viruses can result in more severe disease and can lead to up to 100% cassava yield losses. Unfortunately, there is no effective treatment for these viruses that leads to permanent recovery for infected cassava plants, but there are control measures for management of the diseases to reduce infection and improve yields. In this paper, we utilize a mathematical model to study the dynamics and control of ACMV and EACMCMV co-infections.

A mathematical epidemiological model for cassava plants under the attack of ACMV and EACMCMV co-infection is developed. The model also considers possible control measures for management of ACMV and EACMCMV co-infection. Qualitative analysis of the model aimed at improving our understanding of the dynamics of the model and best approach to manage the infections are conducted systematically.

Firstly, we considered a special case of the ACMV and EACMCMV co-infection model which is a situation where ACMV or EACMCMV is the only disease affecting the system. The epidemiological features of these sub-models are determined and analyzed accordingly. For instance, the disease free equilibrium, endemic equilibrium and basic reproduction number of the sub-models are determined. The disease free equilibrium was shown to be globally asymptotically stable when the basic reproduction number is less than one. This show that ACMV or EACMCMV can be eradicated irrespective of the initial infected population provided the associated basic reproduction number is less than one. The existence of an endemic equilibrium when the basic reproduction number is greater than one suggest persistence of the disease when the basic reproduction number is greater than unity. When the associated basic reproduction number is approximately one, the disease persist and remains stable for a long period. Furthermore, when the associated basic reproduction number is greater than one, each disease persists but undergoes complex cyclic dynamics illustrating the nature of dynamics between the cassava plants and whitefly vectors that cause the infections. Based on these findings, it is strongly recommended that effective control measures that will decrease the associated basic reproduction number below one is implemented for possible disease eradication.

Secondly, the general case of the ACMV and EACMCMV co-infection model was considered analytically. The epidemiological features of this model such as the disease free equilibrium and basic reproduction number are determined and analyzed accordingly. The conditions under which the model can be globally asymptotically stable are determined. The results illustrate the severity ACMV and EACMCMV co-infection, which is greater than either disease on its own, and consequently the difficulty that can be encountered in trying to eradicate the combined disease system. When the associated basic reproduction number is greater than one, the ACMV and EACMCMV co-infection disease system persists and also undergoes complex cyclic dynamics illustrating the nature of dynamics between the cassava plants and whitefly vectors that cause the infections.

The impact of the control measures for improved management of the endemic dynamics of the ACMV and EACMCMV co-infection model are analyzed. For instance, the use of modern technology (ϕ) does not have a significant impact in enhancing cassava productivity at the endemic stage of an outbreak nor does it reduce infections. A possible explanation for this result is that when an outbreak reaches the endemic stage, with high populations of infected cassava plants and whitefly vectors carrying the viruses, implementing modern technologies to enhance cassava productivity without first controlling the disease will not yield positive results. The abundant whitefly vectors will continue to infect healthy cassava plants, ultimately leading to low productivity. Thus, it is recommended that control measures that eradicate infections are implemented before enhancing the cassava productivity through modern technologies.

The use of resistant varieties to reduce infections due to ACMV only have positive impact in reducing the cassava plants infected by ACMV (i₁), but do not have significant positive impact in reducing infections due to EACMCMV diseases or co-infection. Specifically, effective implementation of the use of resistant varieties leads to eradication of the cassava plants infected by ACMV within a short period. Thus, it is recommended to use resistant varieties to reduce infections in cassava plants infected with ACMV only. Similar results were obtained when resistant varieties is used to reduce infections due to EACMCMV only.

Next, we discovered that using two different resistant varieties to simultaneously reduce infection due to ACMV or EACMCMV have positive impact in reducing the cassava plants infected by ACMV or EACMCMV. Specifically, effective implementation of the use of these resistant varieties leads to eradication of the cassava plants infected by ACMV or EACMCMV within a short period. Thus, the use of multiple types of resistant varieties simultaneously to reduce infections in cassava plants infected with ACMV or EACMCMV is strongly recommended.

Furthermore, we discover that removing of whitefly vectors using appropriate insecticides have significant impact in reducing cassava infections due to ACMV or EACMCMV. Specifically, effective implementation of vector removal can lead to eradication of infections due to ACMV, EACMCMV or both. Thus, the whitefly vector removal using appropriate insecticides is strongly recommended for ACMV and EACMCMV disease eradication.

This results of this study have revealed how ACMV and EACMCMV co-infection can be managed to reduce disease infection and enhance cassava yield. Effective implementation of the results of this study by the farmers and policy makers is expected to aid disease management and in achieving food security globally.

6 Conclusions

This research offers an unique perspective in that it presents the first mathematical epidemiological model that explicitly accounts for the co-infection dynamics of ACMV and EACMCMV, two cassava mosaic viruses whose simultaneous presence is shown to potentially result in more severe losses than situations without co-infection.

We incorporate dual infection pathways, shared vector transmission, and interactions between cassava and whitefly populations, allowing for a more realistic representation of disease progression in endemic regions. The study further integrates multiple control measures, including resistant varieties, vector removal, disease-free planting materials, and modern agricultural technologies, providing a comprehensive framework for assessing their individual and combined effects. By revealing how co-infection fundamentally alters system behavior and control effectiveness, this research fills a critical gap and offers new insights for designing effective cassava disease management strategies.

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Footnotes

Edited by: Haitao Song, Shanxi University, China

Reviewed by: Mathias Tembo, Zambia Agriculture Research Institute (ZARI), Zambia

Geofrey Sikazwe, University of Dar es Salaam, Tanzania