Ecoepidemiological Model and Analysis of MSV...

Research ArticleEcoepidemiological Model and Analysis of MSV DiseaseTransmission Dynamics in Maize Plant

Haileyesus Tessema Alemneh 1 Oluwole Daniel Makinde 2 and David Mwangi Theuri3

1Pan African University Institute of Basic Sciences Technology and Innovation Nairobi Kenya2Faculty of Military Science Stellenbosch University Stellenbosch South Africa3Department of Mathematics Jomo Kenyatta University Nairobi Kenya

Correspondence should be addressed to Haileyesus Tessema Alemneh hailatessemagmailcom

Received 1 September 2018 Revised 12 November 2018 Accepted 20 December 2018 Published 20 January 2019

Academic Editor Harvinder S Sidhu

Copyright copy 2019 Haileyesus Tessema Alemneh et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper an ecoepidemiological deterministic model for the transmission dynamics of maize streak virus (MSV) disease inmaize plant is proposed and analysed qualitatively using the stability theory of differential equationsThe basic reproduction numberwith respect to the MSV free equilibrium is obtained using next generation matrix approach The conditions for local and globalasymptotic stability of MSV free and endemic equilibria are establishedThemodel exhibits forward bifurcation and the sensitivityindices of various embedded parameters with respect to the MSV eradication or spreading are determined Numerical simulationis performed and dispalyed graphically to justify the analytical results

1 Introduction

Maize (Zea mays L) is grown globally across temperate andtropical zones spanning all continents [1] It is the mostwidely grown and consumed staple crop in Africa with morethan 100 million Africans depending on it as their mainfood source which is annually planted over an area of 155million hectares [2 3] In Ethiopia maize is the most cerealgown which ranks first in yield per hectare and is grownin all 11 administrative regions [4] It is the primary foodaveraging slightly more than 20 of daily caloric intakeAbout 9 million smallholder farmers grow maize and 75 ofthe maize produced is consumed as food [5 6] The reportsof Ethiopian Commodity Exchange show that three-fourthof maize produced is used for household expenditure onlyabout ten percent is marketed and the remaining is used forseed in-kind expenses for labor and animal feed [6]

Maize production is constrained by many abiotic andbiotic factors of which maize streak disease (MSD) is themajor biotic threat in Ethiopia It is the most destructive anddevastating disease of maize in Sub-Saharan Africa whichis caused by maize streak virus (MSV genus Mastrevirus

family Geminiviridae) [3 7] including Ethiopia [4] MSV isa major constraint to maize and over 80 other crops species[8] including oats wheat sorghum millet finger millet andsugarcane [2 9] MSD is a major threat to cereal cropsamongst smallholder farmers in Sub-Saharan Africa causingup to US $ 480 million losses annually [10] MSD is a viraldisease which has single-component circular ssDNA [2 3]

MSV has been reported to be the most economically sig-nificant causing 100 yield loss if infection occurs in the firstthree weeks of planting maize [3 11] It is irregular in natureand transmitted in a persistent manner by leafhoppers in thegenus Cicadulina [2 5] Globally 22 species of Cicadulinaleafhoppers have been reported of which 18 are found inAfrica Cicadulina mbila is the most predominant vector andthe most important in the epidemiology of the virus [2] fromthe 8 known vectors of MSV in the genus In Ethiopia five ofthese 22 known species of Cicadulina have been recorded [4]

Mathematical modeling is an important tool used inanalyzing the dynamics of infectious diseases Several modelshave been formulated and analyzed to explain the dynamicsof plant disease transmission The authors in [12] investigatedthe impacts of foliar diseases on maize plant population

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2019 Article ID 7965232 14 pageshttpsdoiorg10115520197965232

2 International Journal of Mathematics and Mathematical Sciences

Table 1 Description of parameters of the MSV model (1)

parameter Description1205731 Predation and infection rate of Infected Leafhopper on Susceptible Maize plant1205732 Predation and infection rate of Susceptible Leafhopper on Infected Maize plant119887 Conversion rate of Infected Leafhopper119902 Recruitment rate of Susceptible Leafhopper119870 Carrying capacity119862 Half saturation rate of Susceptible Leafhopper with Infected Maize plant119860 Half saturation rate of Susceptible maize with Infected plant1205831 Death rate of infected maize1205832 Death rate of susceptible leafhopper1205833 Death rate of infected leafhopper119903 Intrinsic growth rate of Maize

dynamics from the developed epidemiological mathematicalmodelThey also applied optimal control theory with chemi-cal cultural and disease resistance as a control interventionIn [13] the authors derived a continuous epidemiologicalmodel of African cassava mosaic virus disease in whichthe dynamics are within a locality of healthy and infectedcassava and of infective and noninfective whitefly vectorsThe authors in [14] reviewed a differential equation modelon plants by adjusting to a specific plant disease modelwhich was a general model of [15] Optimal control the-ory to a continuous deterministic epidemiological Cassavabrown streak disease model with chemical and uprootingas control measures was applied by the author [16] In[17] the authors proposed and analyzed an SI-SEI maizedisease model which is a combination of both the host andthe vector population models and has been formulated tostudy and analyse the dynamics of maize lethal necrosisdisease

Motivated by references [12 16 17] in this paper wepresent a deterministic model to study and analyze thedynamics of MSV in the maize plant population We believethat the results of our research work will be useful indicatingsuitable means of controlling the disease transmission orrather eradicate it This may ensure maximum maize harvestfor farmers for food security The organization of this paperis as follows In Section 2 the mathematical analysis of themodel including determining the invariant region where themodel is mathematically and epidemiologically well posedis presented basic reproduction number R0 is obtained byusing the next generation matrix method the local stabilityof the equilibria is obtained by the Jacobian matrix methodand by the method of Castillo- Chavez the global stability ofthe disease-free equilibrium is investigated we determine theendemic equilibriumpoint of themodel and the local stabilityof the equilibria by the Jacobian matrix method and bythe method of constructing a Lyapunov function the globalstability of the endemic equilibrium is investigated At theend of Section 3 we determine the bifurcation and sensitivityanalysis In Section 4 we give some numerical simulationsto prove our theoretical results with a brief discussion In thefinal section there is a conclusion

2 Model Description and Formulation

The model we introduce consists of two populations themaize population and the leafhopper vector population Bothpopulations have two subclasses susceptible and infected Attime t let S(t) denote the density of the susceptible maize andI(t) denote the density of the infected maize The susceptibleand infected leafhopper vector densities are denoted by H(t)and Y(t) respectively

If there is no leafhopper predator population and nodisease the host population grows logistically with intrinsicgrowth rate 119903 and environmental carrying capacity 119870(119870 gt0) In the presence of the disease in maize population theinfected host population contributes to the susceptible hostpopulation growth towards the carrying capacity 119870(119870 gt 0)The susceptible leafhopper vectors are recruited at rate qand move to infected leafhopper subgroup by eating infectedmaize plant at a rate 1205732 and its natural death rate is 1205832 Theinfected leafhopper has a natural death rate 1205833 The diseasespreads to the susceptible host when it comes in contact withthe MSV pathogen infected leafhopper vector Susceptiblehost plants move to the infected class following contact withinfected leafhopper at a per capita rate 1205731 The host oncebecame infected never recovers and gives no or very low yieldof maize The infected maize plants have death rate 1205831 due tothe disease Furthermore the disease can not be transmittedhorizontally and vertically in both populations and it is notgenetically inherited The predation functional response ofthe leafhopper towards susceptiblemaize is assumed to followMichaelis-Menten kinetics and is modelled using a Hollingtype II [18] functional form with predation and infectioncoefficients 1205731 1205732 and half saturation constants A and C Allthe parameters and their descriptions are listed in Table 1

With regard to the above considerations we have thecompartmental flow diagram shown in Figure 1 From theflow chart (Figure 1) the model will be governed by thefollowing system of differential equation equations119889119878119889119905 = 119903119878 (1 minus 119878 + 119868119870 ) minus 1205731119878119884119860 + 119878119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868

International Journal of Mathematics and Mathematical Sciences 3

rs (1 minuss+I

K) 1SY

A+S

2HI

C+I

b2HI

C+I

1）

q

2（ 3９

Figure 1 Compartmental diagram for the transmission dynamicsof MSV

119889119867119889119905 = 119902 minus 1205732119868119867119862 + 119868 minus 1205832119867119889119884119889119905 = 1198871205732119868119867119862 + 119868 minus 1205833119884(1)

With the initial condition

119878 (0) = 1198780 ge 0119868 (0) = 1198680 ge 0119867 (0) = 1198670 ge 0119884 (0) = 1198840 ge 0(2)

3 Model Analysis

31 Positivity of Solutions For model (1) to be ecoepidemio-logically meaningful and well posed it is necessary to provethat all solutions of system with positive initial data willremain positive for all times 119905 gt 0 This will be establishedby the following theorem

Theorem 1 Let Ω = (119878 119868119867119884) isin R4 119878(0) gt 0 119868(0) gt0119867(0) gt 0 119884(0) gt 0 Then the solution set (119878(119905) 119868(119905)119867(119905)119884(119905)) of system (1) is positive for all 119905 ge 0Proof From the first equation of the system

119889119878119889119905 = 119903119878 (1 minus 119878 + 119868119870 ) minus 1205731119878119884119860 + 119878 le 119903119878 (1 minus 119878119870) (3)

Then we have 119889119878119878 (1 minus 119878119870) le 119903119889119905 997904rArr119878 (119905) le 119870119878 (0)119890minus119903119905 (119870 minus 119878 (0)) + 119878 (0) (4)

As t approachers infin we obtain 0 le 119878(119905) le 119870 By using thesame procedure we obtained119868 (119905) ge 119868 (0) 119890minus1205831119905 ge 0119867 (119905) ge 119867 (0) 119890minus1205832119905 ge 0119884 (119905) ge 119884 (0) 119890minus1205833119905 ge 0 (5)

Thus the model is meaningful and well posedTherefore it issufficient to study the dynamics of the model inΩ32 Invariant Region Let us determine a region in whichthe solution of model(1) is bounded For this model thetotal maize population is 1198731(119878 119868) = 119878(119905) + 119868(119905) Then afterdifferentiating 1198731 with respect to time and substituting theexpression of 119889119878119889119905 119889119868119889119905 we obtain1198891198731119889119905 = 119903119878 (1 minus 119878 + 119868119870 ) minus 1205831119868 le 119903119878 minus 1205831119868= 119878 (119903 + 1) minus (119878 + 1205831119868) le (119903 + 1) minus 1205721198731 (6)

where = max119878(0)119870 and 120572 = min1 1205831 Then1198891198731119889119905 + 1198891198731 le (119903 + 1) (7)

After solving Eq (7) and evaluating it as 119905 997888rarr infin we got

Ωℎ = (119878 119868) 1205761198772+ 1198731 (119905) le 120572 (119903 + 1) (8)

Similarly for leafhopper population1198732(119867 119884) = 119867(119905) + 119884(119905)we get 1198891198732119889119905 = 119902 minus 1205832119867 minus 1205833119884 le 119902 minus 1198981198732 (9)

Where119898 = min(1205831 1205832) Then1198891198732119889119905 + 1198981198732 le 119902 (10)


ΩV = (119867 119884) 1205761198772+ 1198732 (119905) le 119902119898 (11)


Therefore the feasible solution set for the MSV model givenby

Ω = Ωℎ times ΩV = (119878 119868119867 119884) isin 1198774+ 1198731 (119905)le 120572 (119903 + 1) 1198732 (119905) le 119902119898 (12)

is positively invariant inside which the model is consideredto be epidemiologically meaningful and mathematically wellposed

33 Disease-Free Equilibrium Point (DFE) The disease-freeequilibrium ofmodel (1) is obtained by equating all equationsof the model to zero and then letting 119868 = 0 and 119884 = 0 Thenwe get

1198640 = (119870 0 1199021205832 0) (13)

34 Basic Reproduction Number We compute the basicreproduction number R0 for the model to analyze thestability of the equilibrium points The basic reproduc-tion number R0 measures the expected number of sec-ondary infections that result from one newly infected indi-vidual introduced into a susceptible population [16] Wecalculate the basic reproduction number R0 of the sys-tem by applying the next generation operator approachas laid out by [19] and so it is the spectral radius ofthe next-generation matrix The first step to get R0 isrewriting the model equations starting with newly infectiveclasses

119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868119889119884119889119905 = 1198871205732119868119867119862 + 119868 minus 1205833119884 (14)

Then by the principle of next-generation matrix we obtained

119865 = [[[[[1205731119878119884119860 + 1198781198871205732119868119867119862 + 119868

]]]]] 119881 = [12058311198681205833119884]

(15)

Therefore the basic reproduction number is given as

R0 = radic 11988712057311205732119870119902120583112058321205833119862 (119860 + 119870) (16)

R0 is a threshold parameter that represents the average num-ber of infected vectors and infected hosts caused by a cross-infection of one infectious maize plant and one infectiousleafhopper vector when the other population consists of onlysusceptible population [19] Two generations are required fortransmission of MSV to take place in the maize field that iswhy the square root is found inR0 that is from an infectiousmaize plant to a susceptible leafhopper vector and then froman infectious leafhopper vector to susceptible maize [16] It istoo clear whenR0 is rewritten as follows

R0 = radic 12057311198701205831 (119860 + 119870) 119887120573211990211986212058321205833 = R0ℎ timesR0V (17)

Where

(i) R0ℎ = radic12057311198701205831(119860 + 119870) is themaize plants contribu-tion when they infect the leafhopper and

(ii) R0V = radic119887120573211990211986212058321205833 is the contribution ofthe leafhopper population when it infects maizeplants

35 Local Stability of DFE

Theorem 2 The DFE point is locally asymptotically stable ifR0 lt 1 and unstable ifR0 gt 1Proof To proof this theorem let us first find the Jacobianmatrix of system (1)

119869 =(((((((((((

119903(1 minus 2119878 + 119868119870 ) minus 1205731119860119884(119860 + 119878)2 minus119903119878119870 0 minus 1205731119878119860 + 1198781205731119860119884(119860 + 119878)2 minus1205831 0 minus 1205731119878119860 + 1198780 minus 1198621205732119867(119862 + 119868)2 minus 1205732119868(119862 + 119868) minus 1205832 00 1198871198621205732119867(119862 + 119868)2 1198871205732119868(119862 + 119868) minus1205833

)))))))))))

(18)


Evaluating Eq (18) at the disease-free equilibrium 1198640 =(119870 0 1199021205832 0) we get

119869 =(((((((((((

minus119903 minus119903 0 minus 1205731119870119860 + 1198700 minus1205831 0 1205731119870119860 + 1198700 minus12057321199021198621205832 minus1205832 00 11988712057321199021198621205832 0 minus1205833

)))))))))))

(19)

From the Jacobian matrix we obtained a characteristic poly-nomial as (minus120582 minus 119903) (minus120582 minus 1205832) (1205822 + 1198861120582 + 1198862) = 0 (20)

Where 1198861 = 1205831 + 12058331198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) (21)

From Eq (20) we see thatminus120582 minus 119903 997904rArr 1205821 = minus119903 lt 0minus120582 minus 1205832 997904rArr 1205822 = minus1205832 lt 0 (22)

From the last expression that is1205822 + 1198861120582 + 1198862 = 0 (23)

we applied Routh-Hurwitz criteria and by the principle Eq(23) has strictly negative real root if 1198861 gt 0 and 1198862 gt 0 Clearlywe see that 1198861 gt 0 because it is the sum of positive parametersand also 1198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) = 1 minusR

20 gt 0 (24)

Hence the DFE is locally asymptotically stable ifR0 lt 136 Global Stability of DFE To investigate the global stabilityof DFE we used technique implemented by Castillo-Chavezand Song [20] as done in the paper [16] Thus we rewrite ourmodel (1) in the form 119889119883119889119905 = 119865 (119883119885) 119889119885119889119905 = 119866 (119883119885) 119866 (119883 0) = 0

(25)

where 119883 = (119878119867) isin 1198772 denotes uninfected populations and119885 = (119868 119884) isin 1198772 denotes the infected population 1198640 = (119883lowast 0)represents the disease-free equilibrium of this system 1198640 is aglobally asymptotically stable equilibrium for the model if itsatisfies conditions (i) and (ii) below

(i) For 119889119883119889119905 = 119865(119883 0) 119883lowast is globally asymptoticallystable

(ii) 119889119885119889119905 = 119863119885119866(119883lowast 0)119885 minus 119866(119883119885) 119866(119883119885) ge 0 119891119900119903119886119897119897 (119883119885) isin Ωwhere119863119885119866(119883lowast 0) is an M-matrix (the off diagonal elementsare nonnegative) and is also the Jacobian of G(XZ) taken in(I Y) and evaluated at (119883lowast 0) = (119870 1199021205832 0 0) If system (25)satisfies the above conditions then the following theoremholds

Theorem3 The equilibrium point 1198640 = (119883lowast 0) of system (25)is globally asymptotically stable if R0 le 1 and conditions (i)and (ii) are satisfied

Proof From system (1) we can get 119865(119883119885) and 119866(119883119885)119865 (119883119885) = (119903119904 (1 minus 119878 + 119868119870 ) minus 1205731119878119884119860 + 119878119902 minus 1205732119868119867119862 + 119868 minus 1205832119867 )119866 (119883119885) = ( 1205731119878119884119860 + 119878 minus 12058311198681198871205732119868119867119862 + 119868 minus 1205833119884)

(26)

Now we consider the reduced system 119889119883119889119905 = 119865(119883 0) fromcondition (i) 119889119878119889119905 = 119903119904 (1 minus 119878119870) 119889119867119889119905 = 119902 minus 1205832119867 (27)

119883lowast = (119870 1199021205832) is a globally asymptotically stable equilibriumpoint for the reduced system 119889119883119889119905 = 119865(119883 0) This can beverified from the solution of the first equation in Eq (27) weobtain S(t) = 119878(0)119870119890minus119903119905(119870 minus 119878(0)) + 119878(0) which approachesK as 119905 997888rarr infin and from the second equation of Eq (27)we get 119867(119905) = 1199021205832 + (119867(0) minus 1199021205832)119890minus1205832119905 which approaches1199021205832 as 119905 997888rarr infin We note that this asymptomatic dynamicsis independent of the initial conditions in Ω therefore theconvergence of the solutions of the reduced system (27) isglobal in Ω Now we compute

119863119885119866(119883lowast 0) = ( minus1205831 1205731119870119860 + 11987011988712057321199021198621205832 minus1205833 ) (28)

Then 119866(119883119885) can be written as119866 (119883119885) = 119863119885119866(119883lowast 0) 119885 minus 119866 (119883119885) (29)


and we want to show 119866(119883119885) ge 0 which is obtained as

119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)

Here119870 ge 119878 and 1199021205832 ge 119867 Hence it is clear that 119866(119883119885) ge 0for all (XZ) isin Ω Therefore this proves that DFE is globallyasymptotically stable whenR0 le 137 The Endemic Equilibrium Point In the presence of MSD119878(119905) ge 0 119868(119905) ge 0119867(119905) ge 0 119884(119905) ge 0 the model has anequilibrium point called endemic equilibrium point denotedby 119864lowast = (119878lowast 119868lowast119867lowast 119884lowast) 119864lowast is the steady state solutionwhere MSD persist in the population of maize plants It canbe obtained by equating each equation of the model equal tozero that is 119889119878119889119905 = 119889119868119889119905 = 119889119867119889119905 = 119889119884119889119905 = 0 (31)

Then we obtain

119868lowast = 119903119878lowast (R20120583112058321205833119862119860 minus 119878lowast (11988712057311205732119902 minusR20120583112058321205833))R201205832112058321205833119862119860 + 119903119878lowast (11988712057311205732119902 minusR20120583112058321205833)119867lowast = 119902119888 (1205831 + 119903119878lowast119870) + 119902119903119878lowast (1 minus 119878lowast119870)1205732119903119878lowast (1 minus 119904119870) + 12057321205832 (1205831 + 119903119878lowast119870)119884lowast = 1199021198881205833 (1 minus 12058321205731 (1205831 + 119903119878lowast119870) + 119903119878lowast (1 minus 119878lowast119870)1205732119903119878lowast (1 minus 119878lowast119870) + 1205832 (1205831 + 119903119878lowast119870) )

(32)

and 119878lowast is the positive root of the equation11987611198784 + 11987621198783 + 11987631198782 + 1198764119878 + 1198765 = 0 (33)

where

1198761 = minus12058311205833120573211990321198762 = 1205831120583312057321199032 (1205832 minus 119860 + 2119870) + 12057311199021198881199032 (1205831 minus 1205732)1198763 = 1205831120583212058331205732119903 (119860119903 minus 119870119903 + 1205831119870)minus 12057311205732119902119888119903 (1205831119870 minus 119870119903 minus 1205832119903)minus 12057311205831119902119888119903 (1205731119903 minus 119870 + 119870119903)+ 1205831120583312057321198701199032 (2119860 minus 1)1198764 = 120573112057321205831119902119888119903119870 (119870 + 21205832) minus 120573112058321119902119888119903119870 (21205731 + 119870)minus 120583112058331205732119903119870119860 (119903119870 + 1205832119903)+ 1205831120583212058331205732119903119870 (1205831119860 minus 119870)1198765 = 120583211198702 (1205731119902119888 (1205732120583212057311205831) minus 120583212058331205732119903119860)

(34)

38 Local Stability of Endemic Equilibrium

Theorem4 Theendemic equilibrium119864lowast of system (1) is locallyasymptotically stable in Ω if the following conditions hold forR0 gt 1

119903 (1 minus 2119878lowast + 119868lowast119870 ) gt 1205731119860119884lowast(119860 + 119878lowast)2 + 1205831minus (1205831 + 1205832) (119862 + 119868lowast) gt 1205732119868lowast11988711986212057311205732119878lowast119867lowast(119860 + 119878lowast) (119862 + 119868lowast)2 gt 12058311205832(35)

Proof Let us first obtain the Jacobian matrix of system (1)

119869 =((((((((((

119903(1 minus 2119878 + 119868119870 ) minus 1205731119860119884(119860 + 119878)2 minus119903119878119870 0 minus 1205731119878119860 + 1198781205731119860119884(119860 + 119878)2 minus1205831 0 1205731119878119860 + 1198780 minus 1198621205732119867(119862 + 119868)2 minus 1205732119868(119862 + 119868) minus 1205832 00 1198871198621205732119867(119862 + 119868)2 1198871205732119868(119862 + 119868) minus1205833

))))))))))

(36)

Evaluating this at the endemic equilibrium 119864lowast =(119878lowast 119868lowast 119867lowast 119884lowast) we get119869 = (11986011 11986012 11986013 1198601411986021 11986022 11986023 1198602411986031 11986032 11986033 1198603411986041 11986042 11986043 11986044) (37)

where 11986011 = 119903(1 minus 2119878lowast + 119868lowast119870 ) minus 1205731119860119884lowast(119860 + 119878lowast)2 11986012 = minus119903119878lowast119870 11986013 = 0


11986014 = minus 1205731119878lowast119860 + 119878lowast11986021 = 1205731119860119884lowast(119860 + 119878lowast)2 11986022 = minus120583111986023 = 011986024 = 1205731119878lowast119860 + 119878lowast11986031 = 011986032 = minus 1198621205732119867lowast(119862 + 119868lowast)2 11986033 = minus 1205732119868lowast(119862 + 119868lowast) minus 120583211986034 = 011986041 = 011986042 = 1198871198621205732119867lowast(119862 + 119868lowast)2 11986043 = 1198871205732119868lowast(119862 + 119868lowast) 11986044 = minus1205833(38)

The characteristic equation of the Jacobian matrix is given by

1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)

where119860 = minus (11986011 + 11986022 + 11986033 + 11986044)119861 = 1198601111986022 + 1198601111986033 + 1198602211986044+ (11986011 + 11986033) (11986022 + 11986044) + 1198602111986012+ 1198604211986024119862 = 1198603311986044 (11986011 + 11986022) + 1198601111986022 (11986033 + 11986044)+ 1198602411986042 (11986011 + 11986033)+ 1198601211986021 (11986033 + 11986044) minus 119860241198603211986043minus 119860211198601411986042119863 = 1198601111986033 (1198602211986044 minus 1198602411986042)minus 1198602111986014 (1198603211986043 minus 1198604211986033)+ 11986011119860241198604311986032 minus 11986021119860121198603311986044

(40)

The sufficient conditions for 119860 gt 0 119862 gt 0119863 gt 0119860119861 gt119862 119860119861119862 gt 1198622 + 1198602119863 are as follows


Thus according to the Routh Hurwitz criterion the charac-teristic equation (39) will have negative roots or imaginaryroots with negative real part for R0 gt 1 the endemicequilibrium 119864lowast is locally asymptotically stable

39 Global Stability of Endemic Equilibrium

Theorem 5 If R0 gt 1 the endemic equilibrium 119864lowast of themodel (1) is globally stable

Proof To establish the global stability of the endemic equilib-rium 119864lowast we consider the following Lyapunov function

119881 = (119878 minus 119878lowast)22 + 1205901 (119868 minus 119868lowast)22 + 1205902 (119867 minus 119867lowast)22+ 1205903 (119884 minus 119884lowast)22 (42)

where 1205901 1205902 1205903 gt 0 are to be chosen properly such that(119889119881119889119905)(119864lowast) = 0 and119881(119878 119868119867 119884) gt 0 for all (119878 119868119867 119884)119864lowastBy direct calculation the derivative of 119881 along the

solution curve of system (1) yields

119889119881119889119905 = (119878 minus 119878lowast) 119889119878119889119905 + 1205901 (119868 minus 119868lowast) 119889119868119889119905+ 1205902 (119867 minus 119867lowast) 119889119867119889119905 + 1205903 (119884 minus 119884lowast) 119889119884119889119905 (43)

Now substituting equations of model (1) we get

= (119878 minus 119878lowast) [119903119878 (1 minus 119878 + 119868119870 ) minus 1205731119878119884119860 + 119878]+ 1205901 (119868 minus 119868lowast) [ 1205731119878119884119860 + 119878 minus 1205831119868]+ 1205902 (119867 minus 119867lowast) [119902 minus 1205732119868119867119862 + 119868 minus 1205832119867]+ 1205903 (119884 minus 119884lowast) [1198871205732119868119867119862 + 119868 minus 1205833119884](44)


By rearranging we obtain

119889119881119889119905 = minus (119878 minus 119878lowast)2 [119903 (minus1 + 119878 + 119868119870 ) + 1205731119884119860 + 119878]minus 1205901 (119868 minus 119868lowast)2 [ minus1205731119878119884119868 (119860 + 119878) + 1205831]minus 1205902 (119867 minus 119867lowast)2 [minus 119902119867 + 1205732119868119862 + 119868 + 1205832]minus 1205903 (119884 minus 119884lowast)2 [minus 1198871205732119868119867119884 (119862 + 119868) + 1205833](45)

We now choose

1205901 = 119868 (119860 + 119878)1205831119868 (119860 + 119878) minus 1205731119878119884 1205902 = 119867 (119862 + 119868)1205832119867(119862 + 119868) minus 119902 (119862 + 119868) + 12057321198681198671205903 = 119884 (119862 + 119868)1205833119884 (119862 + 119868) minus 1198871205732119868119867(46)

Thus (119889119881119889119905)(119878 119868119867 119884) le 0 and an endemic equilibriumpoint is globally stable Also 119889119881119889119905 = 0 if and only if 119878 =119878lowast 119868 = 119868lowast119867 = 119867lowast 119884 = 119884lowast Therefore the largest compactinvariant set in (119878lowast 119868lowast119867lowast 119884lowast) isin Ω 119889119881119889119905 = 0 is thesingleton 119864lowast where 119864lowast is the endemic equilibrium of thesystem (1) By LaSallersquos invariant principle [21] it implies that119864lowast is globally asymptotically stable inΩ310 Bifurcation Analysis A bifurcation is a qualitativechange in the nature of the solution trajectories due to aparameter change The point at which this change takesplace is called a bifurcation point At the bifurcation pointa number of equilibrium points or their stability propertiesor both change We investigate the nature of the bifurcationby using the method introduced in [22] which is based onthe use of the center manifold theory in [22]

Theorem 6 (Castillo-Chavez amp Song [22]) Let us consider ageneral system of ODErsquos with a parameter 120601119889119909119889119905 = 119891 (119909 120601) 119891 119877119899 times 119877 997888rarr 119877119899 119891 isin 1198622 (119877119899 times 119877) (47)

where 119909 = 0 is an equilibrium point for the system in Eq (47)That is 119891(0 120601) equiv 0 for all 120601 Assume the following1198721 119860 = 119863119909119891(0 0) = ((120597119891120597119909119895)(0 0)) is the linearization

matrix of the system given by (47) around the equilibrium 0with120601 evaluated at 0 Zero is a simple eigenvalue of A and othereigenvalues of A have negative real parts

1198722 Matrix A has a nonnegative right eigenvector 119908 and aleft eigenvector V corresponding to the zero eigenvalue Let 119891119896be the 119896119905ℎ component of 119891 and119886 = 119899sum

119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)

The local dynamics of (52) around 0 are totally determined by119886 and 119887(i) 119886 gt 0 119887 gt 0 When 120601 lt 0 with |120601| ≪ 1 0 is

locally asymptotically stable and there exists a positiveunstable equilibrium when 0 lt 120601 ≪ 1 0 is unstableand there exists a negative locally asymptotically stableequilibrium

(ii) 119886 lt 0 119887 lt 0 When 120601 lt 0 with |120601| ≪ 1 0 isunstable when 0 lt 120601 ≪ 1 0 is locally asymptoticallystable equilibrium and there exists a positive unstableequilibrium

(iii) 119886 gt 0 119887 lt 0 When 120601 lt 0 with |120601| ≪ 1 0 is unstableand there exists a locally asymptotically stable negativeequilibriumwhen 0 lt 120601 ≪ 1 0 is stable and a positiveunstable equilibrium appears

(iv) 119886 lt 0 119887 gt 0 When 120601 changes from negative to positive0 changes its stability from stable to unstable Cor-respondingly a negative unstable equilibrium becomespositive and locally asymptotically stable

In particular if119886 lt 0 and 119887 gt 0 then the bifurcation is forwardif 119886 gt 0 and 119887 gt 0 then the bifurcation is backward Using thisapproach the following result may be obtained

Theorem 7 The model in system (1) exhibits forward bifurca-tion at 1198770 = 1Proof We prove using center manifold theorem [22] thepossibility of bifurcation at R0 = 1 Let 119878 = 1199091 119868 = 1199092119867 = 1199093 and 119884 = 1199094 In addition using vector notation 119909 =(1199091 1199092 1199093 1199094)119879 and 119889119909119889119905 = 119865(119909) with 119865 = (1198911 1198912 1198913 1198914)119879then model in system (1) is rewritten in the form1198891199091119889119905 = 1199031199091 (1 minus 1199091 + 1199092119870 ) minus 120573111990911199094119860 + 11990911198891199092119889119905 = 120573111990911199094119860 + 1199091 minus 120583111990921198891199093119889119905 = 119902 minus 120573211990921199093119862 + 1199092 minus 120583211990931198891199094119889119905 = 119887120573211990921199093119862 + 1199092 minus 12058331199094

(49)

We consider the predation and transmission rate 1205731 asbifurcation parameters so thatR0 = 1 if1205731 = 120573lowast1 = 120583112058321205833119862 (119860 + 119870)1198871205732119902119870 (50)


Thedisease-free equilibrium is given by (1199091 = 119870 1199092 = 0 1199093 =1199021205832 1199094 = 0) Then the linearizion matrix of Eq (49) at adisease-free Equilibrium is given by

119869 =(((((((((((

minus119903 minus119903 0 minus 120573lowast1119870119860 + 1198700 minus1205831 0 120573lowast1119870119860 + 1198700 minus120573lowast2 1199021198621205832 minus1205832 00 119887120573lowast2 1199021198621205832 0 minus1205833

)))))))))))

(51)

The right eigenvector 119908 = (1199081 1199082 1199083 1199084)119879 associated withthis simple zero eigenvalue can be obtained from 119869119908 = 0Thesystem becomes

minus1199031199081 minus 1199031199082 minus 120573lowast1119870119860 + 1198701199084 = 0minus12058311199082 + 120573lowast1119870119860 + 1198701199084 = 0120573lowast2 11990211986212058321199082 minus 12058321199083 = 0119887120573lowast2 1199021198621205832 1199082 minus 12058331199084 = 0(52)

From Eq (52) we obtain

1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)

Here we have taken into account the expression for 120573lowast1 Nextwe compute the left eigenvector V = (V1 V2 V3 V4) associatedwith this simple zero eigenvalue can be obtained from V119869 = 0and the system becomes minus119903V1 = 0minus119903V1 minus 1205831V2 minus 120573lowast2 1199021198621205832 V3 + 119887120573lowast2 1199021198621205832 V4 = 0minus1205832V3 = 0minus 120573lowast1119870119860 + 119870V1 + 120573lowast1119870119860 + 119870V2 minus 1205833V4 = 0

(54)

From equation of Eq (54) we obtain

V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)

Here we have taken into account the expression for 120573lowast1 whereV2 is calculated to ensure that the eigenvectors satisfy thecondition V119908 = 1 Since the first and third components ofV are zero we do not need the derivatives of 1198911 and 1198913 Fromthe derivatives of 1198912 and 1198914 the only ones that are nonzeroare the following1205972119891212059711990911205971199094 = 1205972119891212059711990941205971199091 = 120573lowast1119860(119860 + 119909lowast1 )2 1205972119891412059711990921205971199093 = 1205972119891412059711990931205971199092 = 11988712057321198621205972119891412059711990922 = minus2 1198871205732119862119909lowast3(119862 + 119909lowast2 )3 1205972119891212059711990941205971205731 = 119909lowast1119860 + 119909lowast1

(56)

and all the other partial derivatives of 1198912 and 1198914 are zero Thedirection of the bifurcation at R0 = 1 is determined by thesigns of the bifurcation coefficients a and b obtained fromthe above partial derivatives given respectively by119886 = V211990841199081 120573lowast1119860(119860 + 119909lowast1 )2 + V211990811199084 120573lowast1119860(119860 + 119909lowast1 )2+ V411990831199082 1198871205732119862 + V411990831199082 1198871205732119862+ V411990821199082(minus2 1198871205732119862119909lowast3(119862 + 119909lowast2 )3)= 2V211990841199081 120573lowast1119860(119860 + 119909lowast1 )2 + 2V411990831199082 1198871205732119862minus 2V411990821199082 1198871205732119862119909lowast3(119862 + 119909lowast2 )3

(57)

Hence119886 = (minus21205831V211990822) [119860 (1 + 1205831119903)119870 (119860 + 119870) minus 12057321198621205832 + 1205831119862 ] lt 0 (58)

119887 = V21199084 119909lowast1119860 + 119909lowast1 = V21199084 119870119860 + 119870 gt 0 (59)

As the coefficient 119887 is always positive and the sign of thecoefficient 119886 is negative MSD model exhibits a forwardbifurcation and there exists at least one stable endemicequilibrium when R0 gt 1 Using expression for 119868lowast inthe endemic equilibrium we plotted a forward bifurcationdiagram in Figure 7 We used a set of estimated and assumedparameters in Table 3


Table 2 Sensitivity indices table

parameter symbol Sensitivity indices1205731 051205732 05119887 05119902 05119870 039998 times 10minus41205831 -051205832 -051205833 -05119862 -05119860 minus039998 times 10minus4311 Sensitivity Analysis of Model Parameters We carried outsensitivity analysis on the basic parameters to check andidentify parameters that can impact the basic reproductivenumber Sensitivity analysis notifies us on how significanteach parameter is to disease transmission To go throughsensitivity analysis we followed the approach defined by [23]like in [24] This technique develops a formula to obtainthe sensitivity index of all the basic parameters defined asfollows

Definition 8 The normalized forward sensitivity index of avariable 119892 that depends differentiably on a parameter 119901 isdefined as Λ119892119875 = 120597119892120597119901 times 119901119892 (60)

for 119901 represents all the basic parameters Here we haveR0 = radic11988712057311205732119870119902120583112058321205833119862(119860 + 119870) For example the sensi-tivity index ofR0 to 1205731 isΛR0

1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)

And we do this in a similar fashion for the remainingparameters

312 Interpretation of Sensitivity Indices The sensitivityindices of the basic reproductive number with respect tomain parameters are found in Table 2 Those parametersthat have positive indices (1198871 1205731 1205732 119870 119886119899119889 119902) show thatthey have great impact on expanding the disease in thecommunity if their values are increasing Due to the reasonthat the basic reproduction number increases as their valuesincrease it means that the average number of secondarycases of infection increases in the community And also thoseparameters in which their sensitivity indices are negative(119862119860 1205831 1205832 119886119899119889 1205833) have an effect of minimizing the burdenof the disease in the community as their values increase whilethe others are left constant And also as their values increasethe basic reproduction number decreases which leads tominimizing the endemicity of the disease in the community

Table 3 Parameter values for the MSV model

parameter symbol Value 119889119886119910minus1 Source1205731 045 [2]1205732 004 [2]119902 002 Assumed119870 10 000 Assumed1205831 0008 Assumed1205832 00303 [3]1205833 00303 [3]119887 045 Assumed119860 04 Assumed119862 06 Assumed119903 0005 Assumed

4 Numerical Simulation

Numerical simulations of the model (1) are carried out inorder to illustrate some of the analytical results of the studyA set of reasonable parameter values is given in Table 3Theseparameter values were obtained from literature and some ofthem were assumed We used S(0) = 1000 I(0) = 20 H(0) =100 Y(0) = 0 as initial values and parameter values in Table 3for simulation of MSVmodel in addition to parameter valuesin Table 3

From the left-hand side of Figure 2 the susceptible maizepopulation decelerates exponentially to acquire endemicequilibrium level as they die due to infected leafhopper vectorpopulationThe infected maize population assumes paraboliccurve as it increases exponentially to a certain maximumpoint before exponential deceleration to the certain endemiclevel

From the right-hand side of Figure 2 the susceptiblevectors decrease exponentially due to natural death andacquisition of infestation from severely infected maize andMSV from the environment and finally acquire the endemicequilibrium level and the infected vectors form a paraboliccurve as they do raise and drop exponentially to the endemic-ity level

Figure 3 shows the simulation of infected maize andsusceptible maize for different value of 1205731 We can see fromthe figure that increasing the infection and predation rateof infected leafhopper on susceptible maize 1205731 the basicreproduction number increases which leads to an increasein the number of infective maize and on the other hand thenumber of susceptible maize population decrease

Figure 4 shows the simulation of infected maize andinfected leafhopper for different value of 1205732 We can seefrom the figure that increasing the infection and predationrate of susceptible leafhopper on infected maize 1205732 thebasic reproduction number increases which will lead to anincrease in the number of the infected maizes as well as thenumber of infected leafhoppers

Figure 5 shows simulation of infected leafhopper andsusceptible leafhopper for a different value of 1205832 and 1205833We can see that increasing the death rate of the leafhopperpopulation reduces the basic reproduction number Due to


1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0

Susceptible MaizeInfected Maize

Susceptible LeaopperInfected Leaopper

Figure 2 Simulation results of susceptible and infected population of maize and leafhopper whenR0 gt 1

22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2

Time (months)0 05 1 15 2

Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085

Figure 3 Simulation of infected maize and susceptible maize with different value of 1205731

the indirect relation ofR0 and 1205832 and 1205833 it leads to a decreasein the population of the leafhopper

Figure 6 shows simulation of infected maize for differentvalue of 1205831 We can see that increasing the death rate ofinfected and infectious maize 1205831 reduces the reproductionnumberThis leads to a decrease in the infection rate ofmaize

5 Discussions and Conclusions

In this paper we have proposed and analysed an ecoepi-demiological mathematical model of MSV We considered aHolling type II functional response which is biologically real-istic We showed that the system was uniformly bounded and


22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)

Figure 4 Simulation of infected maize and infected leafhopper with different value of 12057327

6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)

Figure 5 Simulation of infected leafhopper and susceptible leafhopper with different value of 1205832 and 1205833positive We found the disease-free and endemic equilibriumpoints and their local and global stability analysis has beeninvestigated The bifurcation analysis of the model is shownThe model analysis also shows the sensitivity of parametersto the disease persistence and dying out

Finally analytical results were confirmed by numericalsimulation with realistic parameter values We showed thatincreasing the infection and predation rates 1205731 and1205732 makes

an increase of basic reproduction number which leads to theincrease of the number of infected maize population How-ever increasing death rate of infected maize and leafhopperpopulation decreases the reproduction number which in turnmeans that the disease dies out from the maize populationThus from the results of this paper control interventionstrategies reduces the disease infection of maize populationThe model shows that the spread of the disease largely


24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03

0 05 1 15 2Time (months)

Figure 6 Simulation of the MSV model with different value of 1205831006

004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE

Stable EndemicEquilibrium

Unstable DFE

Figure 7 Forward bifurcation of MSV model (1)

depends on the infection and predation rates 1205731 and 1205732therefore efforts should be made to minimize the contact ofinfected maize and susceptible leafhopper and MSV infectedmaize should be treated either using insecticide chemical toreduce the infection rate of leafhoppers and it should be donebefore the arrival of leafhopper or uprooting and burning theinfected maize from the fieldThis implies that to get the bestand cost-effective control strategy we should apply optimalcontrol theory Thus we come next with a paper applying theoptimal and cost-effective strategies to identify the best andcost-effective strategy for this model

Data Availability

The data supporting this deterministic model are fromprevious published articles and they have been duly cited inthis paper These published articles are cited in Table 3 andrelevant places in this paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by Pan African University Instituteof Basic Sciences Technology and Innovation (PAUSTI) Wewould like to express our appreciation for the support

References

[1] F Mmboyi S Mugo M Murenga and L Ambani ldquoMaizeproduction and improvement in sub-saharan africardquo in AfricanBiotechnology Stakeholders Forum (ABSF) vol 25 p 2013Nairobi Kenya 2010

[2] N A Bosque-Perez ldquoEight decades of maize streak virusresearchrdquoVirus Research vol 71 no 1-2 pp 107ndash121 2000

[3] O E V Magenya J Mueke and C Omwega ldquoSignificance andtransmission of maize streak virus disease in Africa and optionsfor management A reviewrdquo African Journal of Biotechnologyvol 7 no 25 pp 4897ndash4910 2008

[4] T Mesfin J Den Hollander and P G Markham ldquoCicadulinaspecies and maize streak virus in Ethiopiardquo International Jour-nal of Pest Management vol 37 no 3 pp 240ndash244 1991

[5] Y Mazengia ldquoSmallholders commercialization of maize pro-duction in guangua district northwestern ethiopiardquo WorldScientific News vol 58 pp 65ndash83 2016

[6] K Schneider and L Anderson ldquoYield gap and productivitypotential in ethiopian agriculture Staple grains pulsesrdquo EvansSchool Policy Analysis and Research (EPAR) no 98 p 24 2010

[7] D P Martin and D N Shepherd ldquoThe epidemiology economicimpact and control of maize streak diseaserdquo Food Security vol1 no 3 pp 305ndash315 2009

[8] D N Shepherd D P Martin E Van Der Walt K Dent AVarsani and E P Rybicki ldquoMaize streak virus An old andcomplex emerging pathogenrdquoMolecular Plant Pathology vol 11no 1 pp 1ndash12 2010

[9] P Mylonas T Yonow and D J Kriticos Cicadulina mbila(naude) 2014

[10] K Charles ldquoMaize streak virus A review of pathogen occur-rence biology and management options for smallholder farm-ersrdquo African Journal of Agricultural Research vol 9 no 36 pp2736ndash2742 2014

[11] M D Alegbejo S O Olojede B D Kashina and M E AboldquoMaize streakmastrevirus inAfrica Distribution transmissionepidemiology economic significance and management strate-giesrdquo Journal of Sustainable Agriculture vol 19 no 4 pp 35ndash462002

[12] O C Collins and K J Duffy ldquoOptimal control of maizefoliar diseases using the plants population dynamicsrdquo ActaAgriculturae Scandinavica Section BmdashSoil amp Plant Science vol66 no 1 pp 20ndash26 2016

[13] J Holt M J Jeger J M Thresh and G W Otim-NapeldquoAn epidemiological model incorporating vector populationdynamics applied to African cassava mosaic virus diseaserdquoJournal of Applied Ecology vol 34 no 3 pp 793ndash806 1997

[14] M J Jeger J Holt F van den Bosch and L V MaddenldquoEpidemiology of insect-transmitted plant viruses Modellingdisease dynamics and control interventionsrdquoPhysiological Ento-mology vol 29 no 3 pp 291ndash304 2004


[15] M J Jeger F van Den Bosch L V Madden and J HoltldquoA model for analysing plant-virus transmission characteris-tics and epidemic developmentrdquo IMA Journal of MathematicsApplied in Medicine and Biology vol 15 no 1 pp 1ndash18 1998

[16] T Kinene L S Luboobi B Nannyonga and G G Mwanga ldquoAmathematical model for the dynamics and cost effectiveness ofthe current controls of cassava brown streak disease in ugandardquoJournal of Mathematical and Computational Science vol 5 no4 pp 567ndash600 2015

[17] A William D Kuznetsov and L S Luboobi ldquoA mathematicalmodel for themlnd dynamics and sensitivity analysis in amaizepopulationrdquoAsian Journal ofMathematics and Applications vol2017 2017

[18] O D Makinde ldquoSolving ratio-dependent predator-prey systemwith constant effort harvesting using Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 186 no 1pp 17ndash22 2007

[19] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquo Mathematical Biosciences vol 180pp 29ndash48 2002

[20] C Castillo-Chavez S Blower P DriesscheD Kirschner and AYakubu Mathematical approaches for emerging and reemerginginfectious diseases models methods and theory vol 1 2002

[21] J P LaSalle ldquoThe stability of dynamical systems society forindustrial and applied mathematicsrdquo in Proceedings of theConference Series in Applied Mathematics 1976

[22] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[23] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of complex models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[24] O DMakinde andKO Okosun ldquoImpact of chemo-therapyonoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Table 1 Description of parameters of the MSV model (1)

parameter Description1205731 Predation and infection rate of Infected Leafhopper on Susceptible Maize plant1205732 Predation and infection rate of Susceptible Leafhopper on Infected Maize plant119887 Conversion rate of Infected Leafhopper119902 Recruitment rate of Susceptible Leafhopper119870 Carrying capacity119862 Half saturation rate of Susceptible Leafhopper with Infected Maize plant119860 Half saturation rate of Susceptible maize with Infected plant1205831 Death rate of infected maize1205832 Death rate of susceptible leafhopper1205833 Death rate of infected leafhopper119903 Intrinsic growth rate of Maize

dynamics from the developed epidemiological mathematicalmodelThey also applied optimal control theory with chemi-cal cultural and disease resistance as a control interventionIn [13] the authors derived a continuous epidemiologicalmodel of African cassava mosaic virus disease in whichthe dynamics are within a locality of healthy and infectedcassava and of infective and noninfective whitefly vectorsThe authors in [14] reviewed a differential equation modelon plants by adjusting to a specific plant disease modelwhich was a general model of [15] Optimal control the-ory to a continuous deterministic epidemiological Cassavabrown streak disease model with chemical and uprootingas control measures was applied by the author [16] In[17] the authors proposed and analyzed an SI-SEI maizedisease model which is a combination of both the host andthe vector population models and has been formulated tostudy and analyse the dynamics of maize lethal necrosisdisease

Motivated by references [12 16 17] in this paper wepresent a deterministic model to study and analyze thedynamics of MSV in the maize plant population We believethat the results of our research work will be useful indicatingsuitable means of controlling the disease transmission orrather eradicate it This may ensure maximum maize harvestfor farmers for food security The organization of this paperis as follows In Section 2 the mathematical analysis of themodel including determining the invariant region where themodel is mathematically and epidemiologically well posedis presented basic reproduction number R0 is obtained byusing the next generation matrix method the local stabilityof the equilibria is obtained by the Jacobian matrix methodand by the method of Castillo- Chavez the global stability ofthe disease-free equilibrium is investigated we determine theendemic equilibriumpoint of themodel and the local stabilityof the equilibria by the Jacobian matrix method and bythe method of constructing a Lyapunov function the globalstability of the endemic equilibrium is investigated At theend of Section 3 we determine the bifurcation and sensitivityanalysis In Section 4 we give some numerical simulationsto prove our theoretical results with a brief discussion In thefinal section there is a conclusion

2 Model Description and Formulation

The model we introduce consists of two populations themaize population and the leafhopper vector population Bothpopulations have two subclasses susceptible and infected Attime t let S(t) denote the density of the susceptible maize andI(t) denote the density of the infected maize The susceptibleand infected leafhopper vector densities are denoted by H(t)and Y(t) respectively

If there is no leafhopper predator population and nodisease the host population grows logistically with intrinsicgrowth rate 119903 and environmental carrying capacity 119870(119870 gt0) In the presence of the disease in maize population theinfected host population contributes to the susceptible hostpopulation growth towards the carrying capacity 119870(119870 gt 0)The susceptible leafhopper vectors are recruited at rate qand move to infected leafhopper subgroup by eating infectedmaize plant at a rate 1205732 and its natural death rate is 1205832 Theinfected leafhopper has a natural death rate 1205833 The diseasespreads to the susceptible host when it comes in contact withthe MSV pathogen infected leafhopper vector Susceptiblehost plants move to the infected class following contact withinfected leafhopper at a per capita rate 1205731 The host oncebecame infected never recovers and gives no or very low yieldof maize The infected maize plants have death rate 1205831 due tothe disease Furthermore the disease can not be transmittedhorizontally and vertically in both populations and it is notgenetically inherited The predation functional response ofthe leafhopper towards susceptiblemaize is assumed to followMichaelis-Menten kinetics and is modelled using a Hollingtype II [18] functional form with predation and infectioncoefficients 1205731 1205732 and half saturation constants A and C Allthe parameters and their descriptions are listed in Table 1

With regard to the above considerations we have thecompartmental flow diagram shown in Figure 1 From theflow chart (Figure 1) the model will be governed by thefollowing system of differential equation equations119889119878119889119905 = 119903119878 (1 minus 119878 + 119868119870 ) minus 1205731119878119884119860 + 119878119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868


rs (1 minuss+I

K) 1SY

A+S

2HI

C+I

b2HI

C+I

1）

q

2（ 3９




119878 (0) = 1198780 ge 0119868 (0) = 1198680 ge 0119867 (0) = 1198670 ge 0119884 (0) = 1198840 ge 0(2)

3 Model Analysis









Ωℎ = (119878 119868) 1205761198772+ 1198731 (119905) le 120572 (119903 + 1) (8)


Where119898 = min(1205831 1205832) Then1198891198732119889119905 + 1198981198732 le 119902 (10)


ΩV = (119867 119884) 1205761198772+ 1198732 (119905) le 119902119898 (11)






1198640 = (119870 0 1199021205832 0) (13)


119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868119889119884119889119905 = 1198871205732119868119867119862 + 119868 minus 1205833119884 (14)


119865 = [[[[[1205731119878119884119860 + 1198781198871205732119868119867119862 + 119868

]]]]] 119881 = [12058311198681205833119884]

(15)


R0 = radic 11988712057311205732119870119902120583112058321205833119862 (119860 + 119870) (16)



Where





119869 =(((((((((((


)))))))))))

(18)



119869 =(((((((((((


)))))))))))

(19)


Where 1198861 = 1205831 + 12058331198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) (21)




20 gt 0 (24)


(25)






(26)







119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)


Then we obtain


(32)


where


(34)





119869 =((((((((((


))))))))))

(36)






1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018









rs (1 minuss+I

K) 1SY

A+S

2HI

C+I

b2HI

C+I

1）

q

2（ 3９




119878 (0) = 1198780 ge 0119868 (0) = 1198680 ge 0119867 (0) = 1198670 ge 0119884 (0) = 1198840 ge 0(2)

3 Model Analysis









Ωℎ = (119878 119868) 1205761198772+ 1198731 (119905) le 120572 (119903 + 1) (8)


Where119898 = min(1205831 1205832) Then1198891198732119889119905 + 1198981198732 le 119902 (10)


ΩV = (119867 119884) 1205761198772+ 1198732 (119905) le 119902119898 (11)






1198640 = (119870 0 1199021205832 0) (13)


119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868119889119884119889119905 = 1198871205732119868119867119862 + 119868 minus 1205833119884 (14)


119865 = [[[[[1205731119878119884119860 + 1198781198871205732119868119867119862 + 119868

]]]]] 119881 = [12058311198681205833119884]

(15)


R0 = radic 11988712057311205732119870119902120583112058321205833119862 (119860 + 119870) (16)



Where





119869 =(((((((((((


)))))))))))

(18)



119869 =(((((((((((


)))))))))))

(19)


Where 1198861 = 1205831 + 12058331198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) (21)




20 gt 0 (24)


(25)






(26)







119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)


Then we obtain


(32)


where


(34)





119869 =((((((((((


))))))))))

(36)






1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018













1198640 = (119870 0 1199021205832 0) (13)


119889119868119889119905 = 1205731119878119884119860 + 119878 minus 1205831119868119889119884119889119905 = 1198871205732119868119867119862 + 119868 minus 1205833119884 (14)


119865 = [[[[[1205731119878119884119860 + 1198781198871205732119868119867119862 + 119868

]]]]] 119881 = [12058311198681205833119884]

(15)


R0 = radic 11988712057311205732119870119902120583112058321205833119862 (119860 + 119870) (16)



Where





119869 =(((((((((((


)))))))))))

(18)



119869 =(((((((((((


)))))))))))

(19)


Where 1198861 = 1205831 + 12058331198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) (21)




20 gt 0 (24)


(25)






(26)







119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)


Then we obtain


(32)


where


(34)





119869 =((((((((((


))))))))))

(36)






1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










119869 =(((((((((((


)))))))))))

(19)


Where 1198861 = 1205831 + 12058331198862 = 12058311205833 minus 119887120573112057321198701199021205832119862 (119860 + 119870) (21)




20 gt 0 (24)


(25)






(26)







119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)


Then we obtain


(32)


where


(34)





119869 =((((((((((


))))))))))

(36)






1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










119866 (119883119885) = (1205731119884( 119870119860 + 119870 minus 119878119860 + 119878)1198871205732119868 ( 1199021198621205832 minus 119867119862 + 119868) ) (30)


Then we obtain


(32)


where


(34)





119869 =((((((((((


))))))))))

(36)






1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











1205824 + 1198601205823 + 1198611205822 + 119862120582 + 119863 = 0 (39)


(40)
















We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











We now choose







119896119894119895=1

V119896119908119894119908119895 1205972119891119896120597119909119894120597119909119895 (0 0)119887 = 119899sum119896119894=1

V119896119908119894 1205972119891119896120597119909119894120597120601 (0 0) (48)








(49)




119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










119869 =(((((((((((


)))))))))))

(51)




1199081 = minus(1 + 1205831119903 )11990821199082 = 1199082 gt 01199083 = 12057321199021198621205832111990821199084 = 1198871205732119902119862120583212058331199082(53)


(54)


V1 = V3 = 0V4 = 119862120583112058321198871205732119902 V2 (55)


(56)


(57)









1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018













1205731= 120597R01205971205731 times 1205731

R0= 12radic11988712057311205732119870119902120583112058321205833 (119860 + 119870)sdot 1198871205732119870119902120583112058321205833119862 (119860 + 119870) 1205731R0 = 12 ge 0 (61)













1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018









1000

900

800

700

600

500

400

300

200

100

0 10 20 30 40 50 60Time (days)

0 10 20 30 40 50 60Time (days)

100

80

60

40

20

0




22

215

21

205

20

Infe

ctio

us M

aize

0 05 1 15 2


Time (months)

1000

9998

9996

9994

9992

999

9988

9986

9984

9982

Susc

eptib

le M

aize

1=0651=0451=085

1=0651=0451=085







22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018









22

215

225

21

205

20

Infe

ctio

us M

aize

Infe

ctio

us L

ea

oppe

r

6

5

4

3

2

1

2=0062=0042=008

2=0062=0042=008

0 05 1 15 2

Time (months)0

005 1 15 2

Time (months)


6

5

4

3

2

1

0

Infe

cted

Lea

op

per

100

90

80

70

60

50

40

30

20

Susc

eptib

le L

ea

oppe

r

3=04033=003033=0903

2=04032=003032=0903

0 05 1 15 2


Time (months)





24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018









24

23

22

21

20

19

18

17

16

Infe

ctio

us M

aize

1=0091=00081=03



004

002

Ilowast

0

0

05 1 15 2

R0

Stable DFE


Unstable DFE



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018


























Journal of












Journal of







Volume 2018






Volume 2018








Ecoepidemiological Model and Analysis of MSV...

Documents

Transcript of Ecoepidemiological Model and Analysis of MSV...