researcharticle a stochastic model for malaria...

Research ArticleA Stochastic Model for Malaria Transmission Dynamics

Rachel WaemaMbogo Livingstone S Luboobi and JohnW Odhiambo

Institute of Mathematical Sciences (IMS) Strathmore University Box 59857 00200 Nairobi Kenya

Correspondence should be addressed to Rachel Waema Mbogo rmbogostrathmoreedu

Received 14 September 2017 Accepted 26 December 2017 Published 11 February 2018

Academic Editor Zhidong Teng

Copyright copy 2018 Rachel Waema Mbogo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIVAIDS and tuberculosis) In this paperwe compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness inmalaria transmission dynamics Relationships between the basic reproduction number formalaria transmission dynamics betweenhumans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived undercertain assumptionsThe stochasticmodel is formulated using the continuous-time discrete state Galton-Watson branching process(CTDSGWbp) The reproduction number of deterministic models is an essential quantity to predict whether an epidemic willspread or die out Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control andelimination and mitigation of infectious diseases Analytical and numerical results show some significant differences in modelpredictions between the stochastic and deterministic models In particular we find that malaria outbreak is more likely if thedisease is introduced by infectedmosquitoes as opposed to infected humansThese insights demonstrate the importance of a policyor intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized

1 Introduction

Malaria is an infectious disease caused by the Plasmodiumparasite and transmitted between humans through bites offemale Anopheles mosquitoes Approximately half of theworldrsquos population is at risk of malaria It remains one of themost prevalent and lethal human infections throughout theworld An estimated 40 of the worldrsquos population lives inmalaria endemic areas Most cases and deaths occur in sub-Saharan Africa It causes an estimated 300 to 500 millioncases and 15 to 27million deaths each yearworldwide Africashares 80 of the cases and 90 of deaths [1] According tothe website of the World Health Organization [2] there wereapproximately 214 million new cases of malaria and 438000deaths worldwide in 2015 Most cases were reported in theAfrican region

Recently the incidence of malaria has been rising due todrug resistance Various control strategies have been takento reduce malaria transmissions Since the first mathematicalmodel of malaria transmission was introduced by Ross [3]quite a number of mathematical models have been formu-lated to investigate the transmission dynamics of malaria

Xiao and Zou [4] usedmathematical models to explore a nat-ural concern of possible epidemics caused bymultiple speciesof malaria parasites in one regionThey found that epidemicsinvolving both species in a single region are possible

Li and others [5] considered fast and slow dynamics ofmalariamodel with relapse and analyzed the global dynamicsby using the geometric singular perturbation theory Theysuggested that a treatment should be given to symptomaticpatients completely and adequately rather than to asymp-tomatic patients On the other hand for the asymptomaticpatients their results strongly suggested that to control anderadicate the malaria it is very necessary for governments tocontrol the relapse rate strictly Relapse is when symptomsreappear after the parasites had been eliminated from bloodbut persist as dormant hypnozoites in liver cells [6] Thiscommonly occurs between 8 and 24 weeks and is commonlyseen with P vivax and P ovale infections Other papers alsoconsider the influence of relapse in giving up smoking orquitting drinking please see [7]

Chitnis et al [8] and Li et al [5] assumed that the recov-ered humans have some immunity to the disease and do notget clinically ill but they still harbour low levels of parasite in

HindawiJournal of Applied MathematicsVolume 2018 Article ID 2439520 13 pageshttpsdoiorg10115520182439520

2 Journal of Applied Mathematics

their blood streams and can pass the infection to mosquitoesAfter some period of time they lose their immunity andreturn to the susceptible class Unfortunately Li and othersdid not consider that the recovered humans will return totheir infectious state because of incomplete treatment

Stochasticity is fundamental to biological systems Insome situations the system can be treated as a large numberof similar agents interacting in a homogeneously mixingenvironment and so the dynamics arewell-captured by deter-ministic ordinary differential equations However in manysituations the system can be driven by a small number ofagents or strongly influenced by an environment fluctuatingin space and time [9ndash12]

Stochastic models incorporate discrete movements ofindividuals between epidemiological classes and not averagerates at which individuals move between classes [13ndash15]In stochastic epidemic models numbers in each class areintegers and not continuously varying quantities [13] Asignificant possibility is that the last infected individual canrecover before the disease is transmitted and the infection canonly reoccur if it is reintroduced from outside the population[16] In contrast most deterministic models have the flawthat infections can fall to very low levelsmdashwell below thepoint at which there is only one infected individual only torise up later [17] In addition the variability introduced instochastic models may result in dynamics that differ from thepredictions made by deterministic models [16]

For a large population size and a large number of infec-tious individuals the deterministic threshold1198770 gt 1 providesa good prediction of a disease outbreak However this predic-tion breaks down when the outbreak is initiated by a smallnumber of infectious individuals In this setting Markovchain (MC) models with a discrete number of individuals aremore realistic than deterministic models where the numberof individuals is assumed to be continuous-valued [18]

Motivated by these works in this paper we propose amodel which is an extension of the model formulated byHuo and Qui (2014) who assumed that the pseudorecoveredhumans can recover and return to the susceptible class orrelapse and become infectious again Using the extendedmodel we will formulate the basic reproductive number 1198770and use it to compare the disease dynamics of the determin-istic and stochastic models in order to determine the effect ofrandomness in malaria transmission dynamics

This paper is organized as follows in Section 2 we presenta malaria transmission deterministic model with relapsewhich is an extension of the model in [6] We computethe basic reproduction number 1198770 of the malaria transmis-sion deterministic model using the next-generation matrixapproach The stochastic version of the deterministic modeland its underlying assumptions necessary formodel formula-tion are presented and discussed in Section 3 In this sectionwe also compute the stochastic threshold for disease extinc-tion or invasion by applying the multitype Galton-Watsonbranching process In Section 4 we show the relationshipbetween reproductive number of the deterministicmodel andthe thresholds for disease extinction of the stochastic versionwe also illustrate our results using numerical simulationsWeconclude with a discussion of the results in Section 5

Uninfected host Infected host

Uninfected mosquitoInfected mosquito

Bite

Bite

Figure 1 The mosquito-human transmission cycle An infectiousmosquito bites a susceptible uninfected human and transmitsthe virus via saliva Once the human becomes infectious (usuallyaccompanied by symptoms) the human can transmit the pathogento an uninfected mosquito via the blood the mosquito ingestsSource Figure 1 is reproduced fromManore and Hyman (2016) [20][under the Creative Commons Attribution Licensepublic domain]

2 The Malaria Deterministic Model

The interaction of the Mosquito-host is shown in Figure 1

21 Model Formulation In this section we introduce a deter-ministic model of malaria with relapse an extended form ofthe model in [6] The model proposes a more realistic math-ematical model of malaria in which it is assumed that thepseudorecovered humans interact with infected mosquitoesand may acquire more parasites causing reinfection or dueto incomplete treatment the infection my reoccur (relapse)and return to infectious class Also the pseudorecoveredhumanmay lose their immunity and return to the susceptibleclass Given that humans might get repeatedly infected dueto not acquiring complete immunity then the populationdynamics are assumed to be described by the SIRS modelhence we consider a deterministic compartmental modelwhich divides the total human population size at time 119905denoted by 119873(119905) into susceptible individuals 119878ℎ(119905) (thosewho are not currently harbouring the parasite but are liableto be infected) infectious individuals 119868ℎ(119905) (those alreadyinfected and are able to transmit the disease to mosquitoes)and pseudorecovered individuals119877ℎ(119905) (thosewho are treatedfrom the disease but with partial recovery and hence cantransmit the disease tomosquitoes) Mosquitoes are assumednot to recover from the parasites so the mosquito populationcan be described by the SI model

The structure of model is shown in Figure 2

22 Variable and Parameter Description for the Model Thevariables for themodel are summarized in description of vari-ables for the malaria transmission model in the Notations

The parameters for the model are described as in descrip-tion of parameters for the malaria transmission model in theNotations

Journal of Applied Mathematics 3

Hum

an p

opul

atio

nM

osqu

ito p

opul

atio

n

1Rℎ

1Iℎ

1RℎRℎ

Iℎ(SℎIℎ)N

Sℎ Iℎ

Sℎ

Sm ImM 1Im

(1SmIℎ + 2SmRℎ)N

2(RℎIm)N

Sm

N

Figure 2 Schematic representation of the mosquito-human malaria transmission dynamics

23 Equations of the Model Assuming that the disease trans-mits in a closed system which translates into the simplifyingassumption of a constant population size with the birth rateequal to death rate that is 120582 = 120583 (see [21]) so that119873(119905) = 119873hence the above assumptions lead to the following system ofdifferential equations which describe the interaction betweenmosquitoes and humans119889119878ℎ (119905)119889119905 = 120582119873 minus 120573119878ℎ119868119898119873 + 1205881119877ℎ minus 120583119878ℎ119889119868ℎ (119905)119889119905 = 120573119878ℎ119868119898119873 + 1205882119877ℎ119868119898119873 minus (120574 + 1205831) 119868ℎ119889119877ℎ (119905)119889119905 = 120574119868ℎ minus (1205881 + 1205882119868119898119873 + 1205831)119877ℎ119889119878119898 (119905)119889119905 = 120575119872 minus 1205721119878119898119868ℎ119873 minus 1205722119878119898119877ℎ119873 minus 120578119878119898119889119868119898 (119905)119889119905 = 1205721119878119898119868ℎ119873 + 1205722119878119898119877ℎ119873 minus 1205781119868119898

(1)

where 119878ℎ 119868ℎ 119877ℎ 119878119898 and 119868119898 represent the number of suscep-tible humans infectious humans recovered humans suscep-tible mosquitoes and infectious mosquitoes respectively

24 Computation of 1198770 Using the Next-Generation MatrixApproach The basic reproduction number 1198770 is defined asthe secondary infections produced by one infective agent

that is introduced into an entirely susceptible populationat the disease-free equilibrium The next-generation matrixapproach is frequently used to compute 1198770 The originalnonlinear system ofODEs including these compartments canbe written as 120597119883119894120597119905 = F minus V where F = (F119894) andV = (V119894) represent new infections and transfer betweencompartments respectively [22ndash24]

The Jacobian matrices ofF(119883) andV(119883) at the disease-free equilibrium 1198640 are respectively

119865 = F (1198640) = ( 0 0 1205730 0 01205721 1205722 0)119881 =V (1198640) = (120574 + 1205831 minus1205882 0minus120574 1205831 + 1205881 + 1205882 00 0 1205781)

(2)

The matrixJ = 119865 minus119881 is the Jacobian matrix evaluated at theDFE

J = 119865 minus 119881 = (minus120574 minus 1205831 1205882 120573120574 minus1205831 minus 1205881 minus 1205882 01205721 1205722 minus1205781) (3)

The inverse matrix of 119881 is119881minus1 =((

1205831 + 1205881 + 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 0120574(120574 + 1205831) (1205831 + 1205881) + 12058311205882 120574 + 1205831(120574 + 1205831) (1205831 + 1205881) + 12058311205882 00 0 11205781))


119865119881minus1 =( 0 0 1205731205780 0 01205741205722 + 1205721 (120583 + 1205881 + 1205882)(120574 + 120583) (120583 + 1205881) + 1205831205882 (120574 + 120583) 1205722 + 12057211205882(120574 + 120583) (120583 + 1205881) + 1205831205882 0)(4)

The matrix FVminus1 is called the next-generation matrixThe (119894 119896) entry of 119865119881minus1 indicates the expected number ofnew infections in compartment 119894 produced by the infectedindividual originally introduced into compartment 119896 Themodel reproduction number 1198770 which is defined as thespectral radius of FVminus1 and denoted by 120588(FVminus1) is givenby 1198770 = 120588 (FV

minus1) = radic 120573 [1205741205722 + 1205721 (1205881 + 1205882 + 1205831)]1205781 [(120574 + 1205831) (1205881 + 1205831) + 12058311205882] (5)

Here 1198770 is associated with disease transmission by infectedhumans as well as the infection of susceptible humans byinfected mosquitoes

Simplifying (5) we find an equivalent form which isdefined as the product of the transmission from mosquito tohuman and from human to mosquito as follows1198770 = radic( 1205731205781)( 1205721(120574 + 1205831) + 1205722(1205881 + 1205882 + 1205831)) (6)

In (6) 1198770 is associated with disease transmission by infectedmosquitoes as well as infection of susceptible mosquitoes byinfected humans The term 1205731205781 represents the number ofinfected humans generated by infectious mosquito in its lifespanThe term 1205721(120574 + 1205831) represents the number of infectedmosquitoes generated by infectious human during the infec-tious period of the individual while the term 1205722(1205881 +1205882 +1205831)represents the number of infected mosquitoes generated bypseudorecovered human during hisher infectious periodSusceptible humans acquire infection following effectivecontacts with infected mosquitoes Susceptible mosquitoesacquire malaria infection from infected humans in two waysnamely by infectious human or pseudorecovered humans

Also from (6) we can rewrite 1198770 as infection betweenmosquitoes and infectious humans or infection betweenmosquitoes and pseudorecovered humans1198770 = radic11987701 + 11987702 (7)

where 11987701 = ( 1205731205781)( 1205721(120574 + 1205831)) (8)

11987702 = ( 1205731205781)( 1205722(1205881 + 1205882 + 1205831)) (9)11987701 in (8) represents the product of individuals generated byinfectious human and infectedmosquito respectively duringtheir infectious time

11987702 in (9) represents the product of individuals generatedby pseudorecovered human and infected mosquito respec-tively during their infectious time

For model (1) the disease dies out if 1198770 lt 1 and the dis-ease persists if 1198770 gt 1 Hence 1198770 lt 1 iff 11987701 lt 1 and 11987702 lt 13 Malaria Stochastic Epidemic Model

For the mosquito-human malaria dynamics since time iscontinuous and the disease states are discrete we derivethe stochastic version of the deterministic model (1) usinga continuous-time discrete state Galton-Watson branchingprocess (CTDSGWbp) which is a type of stochastic processThemalaria CTDSGWbpmodel is a time-homogeneous pro-cess with the Markov property The model takes into accountrandom effects of individual birth and death processes thatis demographic variability A stochastic process is defined bythe probabilitieswithwhich different events happen in a smalltime interval Δ119905 In our model there are two possible events(production and deathremoval) for each population Thecorresponding rates in the deterministic model are replacedin the stochastic version by the probabilities that any of theseevents occur in a small time interval of length Δ119905 [25 26]31 Model Formulation Let time be continuous 119905 isin [0infin)and let 119878ℎ(119905) 119868ℎ(119905) 119877ℎ(119905) 119878119898(119905) and 119868119898(119905) represent discreterandom variables for the number of susceptible humansinfectious humans recovered humans susceptible mosqui-toes and infectiousmosquitoes respectively with finite space119878ℎ (119905) 119868ℎ (119905) 119877ℎ (119905) 119878119898 (119905) 119868119898 (119905)isin 0 1 2 3 119867 (10)

where 119867 is positive and represents the maximum size of thepopulations

If a disease emerges from one infectious group with1198770 gt 1 and if 119894 infective agents are introduced into a whol-ly susceptible population then the probability of a majordisease outbreak is approximated by 1 minus (11198770)119894 while theprobability of disease extinction is approximately (11198770)119894 [13]However this result does not hold if the infection emanatesfrom multiple infectious groups [27] For multiple infectiousgroups the stochastic thresholds depend on two factorsnamely the number of individuals in each group and theprobability of disease extinction for each group Furtherthe persistence of an infection into a wholly susceptiblepopulation is not guaranteed by having 1198770 greater than one

For CTDSGWbpmodels the transition from one state toa new state may occur at any time 119905 If the process begins in


Table 1 State transitions and rates for the CTDSGWbp malaria model

Event Populationcomponents at 119905 Population components at119905 + Δ119905 Transition probabilities

Birth of humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ + 1 119868ℎ 119877ℎ 119878119898 119868119898) 120582Δ119905Death of susceptiblehumans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ minus 1 119868ℎ 119877ℎ 119878119898 119868119898) 120583119878ℎΔ119905Infection of humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ minus 1 119868ℎ + 1 119877ℎ 119878119898 119868119898) 120573(119878ℎ119873)119868119898Δ119905Recovery rate of humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ + 1 119868ℎ 119877ℎ minus 1 119878119898 119868119898) 1205881119877ℎΔ119905Relapse rate of humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ + 1 119877ℎ minus 1 119878119898 119868119898) 1205882119877ℎΔ119905Treatment rate (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ minus 1 119877ℎ + 1 119878119898 119868119898) 120574119868ℎΔ119905Death of infected humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ minus 1 119877ℎ 119878119898 119868119898) 1205831119868ℎΔ119905Death of recovered humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ 119877ℎ minus 1 119878119898 119868119898) 1205831119877ℎΔ119905Birth of mosquitoes (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ 119877ℎ 119878119898 + 1 119868119898) 120575Δ119905Death of susceptiblemosquitoes (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ 119877ℎ 119878119898 minus 1 119868119898) 120578119878119898Δ119905Infection of mosquitoesfrom humans (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ 119877ℎ 119878119898 minus 1 119868119898 + 1) (1205721119868ℎ + 1205722119877ℎ) (119878119898119873)Δ119905Death of infectedmosquitoes (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898) (119878ℎ 119868ℎ 119877ℎ 119878119898 119868119898 minus 1) 1205781119868119898Δ119905a state 119866(0) after a random time period 120591 it transits to a newstate 119866(120591) The process remains in state 119866(120591) for a randomtime 119905 after which it moves through to the new state 119866(119901)with 119901 = 120591 + 119905 [27] This process continues throughout themodelThe state transitions and rates for the stochasticmodelare presented in Table 1

32The Branching Process Approximation We use branchingprocess to analyze the malaria dynamics near the disease-free equilibrium Since infectious human pseudorecoveredhumans and infectious mosquitoes are the only sources ofinfection we apply the multitype branching process in thethree variables 119868ℎ(119905)119877ℎ(119905) and 119868119872(119905)The susceptible humansand mosquitoes are assumed to be at the disease-free state[28] We use the multitype Galton-Watson branching pro-cess (GWbp) to determine disease invasion and extinctionprobabilities More review on the GWbp branching theoryprocess can be accessed through [18 27 28] We now definethe offspring pgfs for the three variables where each offspringpgf has a general form119866119894 (1199041 1199042 119904119899)= infinsum

119896119899=0

infinsum119896119899minus1=0

sdot sdot sdot infinsum1198961=0

119875119894 (1198961 1198962 119896119899) 11990411989611 11990411989622 sdot sdot sdot 119904119896119899119899 (11)

where 119875119894(1198961 1198962 119896119899) = prob(1198831198941 = 1198961 1198831198942 = 1198962 119883119894119899 =119896119899) is the probability that one infected individual of type 119894gives birth to 119896119895 individuals of type 119895 see [29]

For one malaria infectious human there are three possi-ble events infection of a mosquito recovery of the infectioushuman or death of the infectious human The offspring pgffor infectious humans define the probabilities associated withthe ldquobirthrdquo of secondary infectious mosquito or the ldquodeathrdquoof the initial infectious human given that the process started

with only one infectious human that is 119868ℎ(0) = 1 119877ℎ(0) = 0and 119868119872(0) = 0

The offspring pgf for 119868ℎ is given by1198661 (1199041 1199042 1199043) = infinsum1198961=0

infinsum1198962=0

infinsum1198963=0

1198751 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (12)

where 1198751(1198961 1198962 1198963) = prob(11988311 = 1198961 11988312 = 1198962 11988313 = 1198963) isthe probability that one infectious human through infectionproduces another infectious human 1198961 or a pseudorecoveredhuman 1198962 or an infectious mosquito 1198963

Similarly the offspring pgf for 119877ℎ is given by1198662 (1199041 1199042 1199043) = infinsum1198961=0

infinsum1198962=0

infinsum1198963=0

1198752 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (13)

where 1198752(1198961 1198962 1198963) = prob(11988321 = 1198961 11988322 = 1198962 11988323 = 1198963)is the probability that one pseudorecovered human throughinfection produces an infectious human 1198961 or another pseu-dorecovered human 1198962 or an infectious mosquito 1198963

Lastly the offspring pgf for 119868119898 is given by1198663 (1199041 1199042 1199043) = infinsum1198961=0

infinsum1198962=0

infinsum1198963=0

1198753 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (14)

where 1198753(1198961 1198962 1198963) = prob(11988331 = 1198961 11988332 = 1198962 11988333 = 1198963)is the probability that one infectious mosquito through infec-tion produces an infections human 1198961 or a pseudorecoveredhuman 1198962 or another infectious mosquito 1198963

The power towhich 119904119895 is raised is the number of infectiousindividuals generated from one infectious individual If anindividual recovers or dies then no new infections aregenerated hence (1199040119895)

The offspring pgfs for 119868ℎ 119877ℎ and 119868119898 are used to calculatethe expected number of offsprings produced by a single


infectious humanor by pseudorecovered humanor infectiousmosquito They are also used to calculate the probability ofdisease extinction

The specific offspring pgfs for 119868ℎ 119877ℎ and 119868119898 are definedusing the rates in description of parameters for the malariatransmissionmodel in theNotations when the initial suscep-tible populations are near disease-free equilibrium 119878ℎ(0) asymp 119873and 119878119898(0) asymp 119872

From (12) the offspring pgf for infectious human given119868ℎ(0) = 1 119877ℎ(0) = 0 and 119868119872(0) = 0 is given by1198661 (1199041 1199042 1199043) = 120572111990411199043 + 120574 + 12058311205721 + 120574 + 1205831 (15)

For 1198661 one infectious human dies or is treated with proba-bility (1205831 + 120574)(1205721 + 120574 + 1205831) this means an infectious humandies before infecting a susceptiblemosquitoThe term 120574(1205721+120574+1205831) represents the probability that the infectious human istreated and moves to the pseudorecovered class this resultsin ldquo11988311 = 0 11988312 = 01 and 11988313 = 0 though there ismovement of an infectious human to pseudorecovered statedue to partial treatment (this is not new offspring)rdquo Theinfectious human infects amosquitowith probability1205721(1205721+120574 + 1205831) This means an infectious human infects a susceptiblemosquito and remains infectious which results in ldquo11988311 = 111988312 = 0 and 11988313 = 1rdquo Note the term 11990411199043 in (15) meansone infectious human generates one infectious mosquito (1199043raised to power one) and remains infectious (1199041 raised topower one)

For one pseudorecovered human there are four eventsinfection of a mosquito relapse to infected class successfultreatment of the pseudorecovered human or death of therecovered host Similarly from (13) the offspring pgf forrecovered humans given 119868ℎ(0) = 0 119877ℎ(0) = 1 and 119868119872(0) = 0is given by 1198662 (1199041 1199042 1199043) = 120572211990421199043 + 1205881 + 1205882 + 12058311205722 + 1205881 + 1205882 + 1205831 (16)

For 1198662 one pseudorecovered host dies or relapses or is fullytreated with probability (1205881 + 1205882 + 1205831)(1205722 + 1205881 + 1205882 + 1205831) orinfects a mosquito with probability 1205722(1205722 + 1205881 + 1205882 + 1205831)This means a pseudorecovered human infects a susceptiblemosquito and remains infectious which results in ldquo11988321 = 011988322 = 1 and 11988323 = 1rdquo Note the term 11990421199043 in (16) meansone recovered human generates one infectious mosquito (1199043raised to power one) and remains infectious (1199042 raised topower one)

For one infectious mosquito there are only two eventsinfection of a susceptible human or death of the mosquito

From (14) the offspring pgf for infected mosquito given119868ℎ(0) = 0 119877ℎ(0) = 0 and 119868119872(0) = 1 is given by

1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)

For1198663 one infectious mosquito dies with probability 1205781(120573+1205781) or infects a human with probability 120573(120573 + 1205781) Thismeans an infectious mosquito infects a susceptible humanand remains infectious which results in ldquo11988331 = 1 11988332 = 0and11988333 = 1rdquo Note the term 11990411199043 in (17) means one infectiousmosquito generates one infectious human (1199041 raised to powerone) and remains infectious (1199043 raised to power one)

33 The Relationship between 1198770 and the Stochastic Threshold1198780 The offspring pgfs evaluated at (1 1 1) gives the expec-tation matrix with elements119898119895119894 Below are the offspring pgfsevaluated at (1 1 1)12059711986611205971199041 = 1205721120572 + 120574 + 1205831 12059711986611205971199042 = 12057412059711986611205971199043 = 1205721120572 + 120574 + 1205831 12059711986621205971199041 = 120588212059711986621205971199042 = 12057221205722 + 1205881 + 1205882 + 1205831 12059711986621205971199043 = 12057221205722 + 1205881 + 1205882 + +1205831 12059711986631205971199041 = 120573120573 + 1205781 12059711986631205971199042 = 012059711986631205971199043 = 120573120573 + 1205781

(18)

The expectation matrix of the offspring pgfs evaluated at(1 1 1) is given by

M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)


The entries 11989811 11989821 and 11989831 represent the expected num-ber of infectious humans pseudorecovered humans andinfectious mosquitoes respectively produced by one infec-tious human Similarly the entries 11989812 11989822 and 11989832 rep-resent the expected number of infectious humans pseu-dorecovered humans and infectious mosquitoes respec-tively produced by one pseudorecovered human Lastly theentries 11989813 11989823 and 11989833 represent the expected numberof infectious humans pseudorecovered humans and infec-tious mosquitoes respectively produced by one infectiousmosquito

The matrix C = W(M minus 119868) is the Jacobian matrixevaluated for the stochastic version

C =W (M minusI)= (minus120574 minus 1205831 120574 12057211205882 minus1205831 minus 1205881 minus 1205882 1205722120573 0 minus1205781) (20)

where W = diag(120574 + 1205831 + 1205721 1205831 + 1205722 + 1205881 + 1205882 120573 + 1205781) is adiagonal matrix andI is the identity matrix

From (3) and (20) we show that

F minusV =W (M minusI)119879= (minus120574 minus 1205831 1205882 120573120574 minus1205831 minus 1205881 minus 1205882 01205721 1205722 minus1205781) (21)

The spectral radius of matrix M obtained by finding theeigenvalues of matrixM is given by1198780 = 120588 (M) = max120573 (120574 + 1205831) + (2120573 + 1205781) 1205721(120573 + 1205781) (120574 + 1205831 + 1205721) 12057221205722 + 1205881 + 1205882 + 1205831 (22)

Taking the first expression of 1198780 = 120588(M) in (24) we have1198780 = 120588 (M) = 120573 (120574 + 1205831 + 1205721) + 1205721 (120573 + 1205781)(120573 + 1205781) (120574 + 1205831 + 1205721)= ( 120573120573 + 1205781) + ( 1205721120574 + 1205831 + 1205721) (23)

This gives the probability of malaria transmission by eitherinfectious human or by infectious mosquito From (23)1198780 = 120588 (M) = 120573 (120574 + 1205831 + 1205721) + 1205721 (120573 + 1205781)(120573 + 1205781) (120574 + 1205831 + 1205721) (24)

The probability of disease extinction is one if 120588(M) lt 1Hence from (24) we have120573 (120574 + 1205831 + 1205721) + 1205721 (120573 + 1205781)lt (120573 + 1205781) (120574 + 1205831 + 1205721) (25)

Expanding and simplifying the inequality we have1205731205721 lt 1205781 (120574 + 1205831) (26)

which reduces to( 1205731205781)( 1205721(120574 + 1205831)) lt 1 997904rArr11987701 lt 1 (27)

When the infection is between infectious humans and infec-tious mosquitoes (27) is true The result in (27) agrees withthe deterministic reproduction number for disease elimi-nation Hence we conclude that the probability of diseaseelimination in the CTDSGWbp model is one iff120588 (M) lt 1 997904rArr1198770 lt 1 (28)

34 Deriving Probability of Disease Extinction 1198750 UsingBranching Process Approximation To find the probability ofextinction (no outbreak) we compute the fixed points of thesystem (1199021 1199022 1199023) isin (0 1) of the offspring pgfs for the threeinfectious stages that is we solve 119866119868ℎ = 1199021 119866119877ℎ = 1199022 and119866119868119898 = 1199023 The solutions of these systems are (1 1 1) and(1199021 1199022 1199023) see [19] Equating 1198661(1199041 1199042 1199043) in (15) to 1199021 thenletting 1199041 = 1199021 and 1199043 = 1199023 and solving for 1199021 we have1199021 = (120573 + 1205781) (120574 + 1205831)120573 (120574 + 1205831 + 1205721)= 120574 + 12058311205721 + 120574 + 1205831 + 12057211205721 + 120574 + 1205831 ( 111987701) (29)

Equating1198662(1199041 1199042 1199043) in (16) to 1199022 letting 1199042 = 1199022 and 1199043 = 1199023and solving for 1199022 we have1199022= (120573 + 1205781) 1205721 (1205831 + 1205881 + 1205882)minus1205781 (120574 + 1205831) 1205722 + 1205721 (1205731205722 + (120573 + 1205781) (1205831 + 1205881 + 1205882)) (30)Equating 1198663(1199041 1199042 1199043) in (17) to 1199023 then letting 1199041 = 1199021 and1199043 = 1199023 and solving for 1199023 we have1199023 = 1205781 (120574 + 1205831 + 1205721)(120573 + 1205781) 1205721 = 1205781120573 + 1205781 + 120573120573 + 1205781 ( 111987701) (31)

The expression for 1199021 in (29) has a biological interpretationBeginning from one infectious human there is no outbreakif the infectious human recovers or dies with probability (120574 +1205831)(1205721 + 120574 + 1205831) or if there is no successful transmissionto a susceptible mosquito with probability (1205721(1205721 + 120574 +1205831))(111987701) This implies that if there is successful contactthen the probability of successful transmission from infec-tious human to susceptible mosquito is 1 minus 111987701

The expression for 1199023 in (31) has a biological interpreta-tion Beginning from one infectious mosquito there is nooutbreak if the infectious mosquito dies with probability


Table 2 Model parameter values

Parameter Description Units Parameter values Source120582 Birth rate of humans Per day 0000039 [5]120583 Death rate of susceptible humans Per day 0000039 [5]120573 Infection rate of humans Per day 002 [19]1205831 Death rate of infected humans Per day 000039 [5]1205881 Recovery rate of humans Per day 001 [19]1205882 Relapse rate of humans Per day 0002 [5]120574 Treatment rate Per day 0037 [5]1205831 Death rate of recovered humans Per day 000034 [5]120575 Birth rate of mosquitoes Per day 0143 [5]120578 Death rate of susceptible mosquitoes Per day 0143 [5]1205721 Infection rate from infectious human Per day 0072 [5]1205722 Infection rate from recovered human Per day 00072 [5]1205781 Death rate of infected mosquitoes Per day 0143 [19]

1205781(120573 + 1205781) or if there is no successful transmission to asusceptible human with probability (120573(120573 + 1205781))(111987701)

There are some other important relationships if diseasetransmission is by infectious mosquitoes as well as infectionof susceptible mosquitoes by infectious humans then1199021 lowast 1199023 = 111987701 (32)

From (32) we see that in both mosquito and human pop-ulations the probability of no successful transmission frominfectious human to susceptible mosquito and from infectedmosquito to susceptible human is 111987701

If disease transmission is by infectiousmosquitoes as wellas infection of susceptible mosquitoes by pseudorecoveredhumans then 1199022 lowast 1199023 = 111987702 (33)

From (33) we see that in both mosquito and human pop-ulations the probability of no successful transmission frompseudorecovered humans to susceptible mosquito and frominfected mosquito to pseudorecovered humans is 111987702

To compute the probability of disease extinction and of anoutbreak for our malaria model we recall that for multipleinfectious groups the stochastic thresholds depend on twofactors namely the number of initial individuals in eachgroup and the probability of disease extinction for eachgroup Using 1199021 1199022 and 1199023 in (29)ndash(31) and assuming initialindividuals for infectious humans pseudorecovered humansand infected mosquitoes are 119868ℎ(0) = ℎ0 119877ℎ(0) = 1199030 and119868119898(0) = 1198980 respectively Then the probability of malariaclearance is given by1198750 = 119902ℎ01 lowast 11990211990302 lowast 11990211989803 = ((120573 + 1205781) (120574 + 1205831)120573 (120574 + 1205831 + 1205721) )ℎ0lowast ( 1205721 (120573 + 1205781) (1205831 + 1205881 + 1205882)minus12057811205722 (120574 + 1205831 + 1205721) + 1205721 (120573 + 1205781) (1205722 + 1205831 + 1205881 + 1205882))1199030lowast (1205781 (120574 + 1205831 + 1205721)(120573 + 1205781) 1205721 )1198980

(34)

minus06 minus04 minus02 0 02 04 06 08

1

1

2

1

2

1

Figure 3 Tornado plot for parameters that influence 11987704 Numerical Simulations

In this section we illustrate numerically the disease dynamicsofmodel (1) using parameter values in Table 2The numericalsimulations are done using Maple codes

41 Effects of Model Parameters on 1198770 Using parameter val-ues in Table 2 we identified how different input parametersaffect the reproduction number 1198770 of our model as shown inFigure 3

From Figure 3 we see that increase in 120574 1205881 1205831 and 1205781 willdecrease 1198770 Also decrease in 120573 1205721 1205722 and 1205882 will decrease1198770

From the Tornado plot the infection of susceptible hu-mans by infected mosquitoes (denoted by 120573) is a major factorin the malaria transmission dynamics Reducing 120573 wouldreduce 1198770 significantly hence reducing the possibility ofdisease outbreak Vector control is the main way to preventand reducemalaria transmission If coverage of vector controlinterventions within a specific area is high enough then ameasure of protection will be conferred across the commu-nity WHO recommends protection for all people at risk of


R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010

(a) Effects of relapse on 1198770

R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010

(b) Effects of treatment rate on 1198770

R0

10

09

08

07

06

1

0 002 004 006 008 010

(c) Effects of recovery rate on 1198770

Figure 4 Effects of relapse and recovery rate on 1198770malaria with effective malaria vector control which agreeswith the information from Figure 3 Two forms of vectorcontrol insecticide-treatedmosquito nets and indoor residualspraying are effective in a wide range of circumstancesAnother factor which affects the malaria transmission sig-nificantly is the death rate of infected mosquitoes (denotedby 1205781) Increasing 1205781 would decrease 1198770 hence reducing thechances of disease outbreak

411 Graphical Representation of Parameters Effects on 1198770Figure 4 shows the effects of relapse and recover rates on thebasic reproduction number From the simulations we findthat 1198770 is increasing with increase in relapse rate while itis decreasing with increase in recovery rate To control and

eradicate the malaria epidemic it is important and necessaryfor governments of endemic countries to decrease the relapserate and increase the treatment and recovery rate

42 Probability of Disease Extinction Using parameter valuesin Table 2 we compute numerically the probability of diseaseextinction 1198750 and of an outbreak 1minus1198750 for our malaria modelusing different initial values for the infectious classes

The probability of disease extinction is high if the diseaseemerges from infected humans It is very low if the infectionemerges from infected mosquitoes However as the initialnumber of infected humans grows largely there is a highprobability of disease outbreak as illustrated in Table 3 Theprobability of disease extinction is significantly low if the


Table 3 Probability of disease extinction 1198750 and of an outbreak 1 minus 1198750 for the malaria modelℎ0 1199030 1198980 1198750 1 minus 1198750 11987701 11987702 11987701 0 0 06838 03162 11987701 gt 1 11987702 lt 1 1198770 gt 10 1 0 09834 00166 11987701 gt 1 11987702 lt 1 1198770 gt 10 0 1 01365 08635 11987701 gt 1 11987702 lt 1 1198770 gt 11 1 0 06724 03276 11987701 gt 1 11987702 lt 1 1198770 gt 11 1 1 00918 09082 11987701 gt 1 11987702 lt 1 1198770 gt 110 0 0 00223 09777 11987701 gt 1 11987702 lt 1 1198770 gt 10 10 0 08459 01541 11987701 gt 1 11987702 lt 1 1198770 gt 10 0 10 0 1 11987701 gt 1 11987702 lt 1 1198770 gt 110 10 0 00188 09812 11987701 gt 1 11987702 lt 1 1198770 gt 110 10 10 0 1 11987701 gt 1 11987702 lt 1 1198770 gt 1disease emerges from infected mosquitoes Therefore thedisease dynamics for model system (1) at the beginning ofthe epidemics are being driven by initial number of infectedmosquitoes

A female mosquito will continue to bite and draw blooduntil her abdomen is full If she is interrupted before sheis full she will fly to the next person After feeding themosquito rests for two or three days before laying her eggsand then it is ready to bite again Since mosquitoes do notrecover from the infection then a single infected mosquito iscapable of biting and infecting so many susceptible humansin their lifespan whichmay in turn infect so many uninfectedmosquitoes there by reducing the probability of diseaseclearance and increasing the probability of a major diseaseoutbreak Moreover mosquitoes are the reservoir host forthe parasites that cause malaria and it takes a long time forthem to bemalaria-free hence the high probability ofmalariaoutbreak if the parasite is introduced by infected mosquito

Table 3 depicts that at the beginning of malaria outbreakany policy or intervention to control the spread of malariashould focus on controlling the infectedmosquito populationas well as the infected humans If more effort to controlthe disease is only focused on the infected humans then itis very unlikely that malaria will be eliminated This is aninteresting insight from the stochastic threshold that couldnot be provided by the deterministic threshold

43 Numerical Simulation of Malaria Model Using param-eter values in Table 2 we numerically simulate the behaviorof model (1) Initial conditions are 119878ℎ(0) = 99 119868ℎ(0) = 1119877ℎ(0) = 0 119878119898(0) = 999 and 119868119898(0) = 1431 Numerical SimulationWhen1198770 lt 1 FromFigure 5 theanalysis shows that when 1198770 lt 1 and 1198780 lt 1 then theprobability of disease extinction is 1198750 = 09476 ≃ 1 althoughthis agrees with (30) which points out that the probability ofdisease elimination in the CTDSGWbp model is one iff1198780 lt 1 997904rArr1198770 lt 1 (35)

There is still a small probability 1 minus 1198750 = 00524 of diseaseoutbreak 1198770 has been widely used as a measure of disease

dynamics to estimate the effectiveness of control measuresand to inform on disease management policy However fromthe analysis in Figure 5 it is evident that 1198770 can be flawed anddisease can persist with 1198770 lt 1 depending on the kind ofdisease being modeled

432 Numerical Simulation When 1198770 gt 1 Using parametervalues in Table 2 we numerically simulate the behavior ofmodel (1) when 1198770 gt 1 Initial conditions are 119878ℎ(0) = 99119868ℎ(0) = 1 119877ℎ(0) = 0 119878119898(0) = 999 and 119868119898(0) = 1

The analysis from Figure 6 suggests that when 1198770 gt 1there is still some probability of disease extinction (1198750 =03713) Although the probability is low there is still a chanceto clear the disease Therefore from the analysis in Figure 6disease can be eliminated with 1198770 gt 1 and hence the 1198770threshold should not be the only parameter to consider inquantifying the spread of a disease

433 Effects of Relapse onHumanPopulation Figure 7 showschanging effects of relapse on the human populations

To control and eradicate themalaria epidemic it is impor-tant and necessary for governments of endemic countries todecrease the relapse rate as can be seen in Figure 7

5 Discussions and Recommendations

In this study we investigated the transmission dynamicsof malaria using CTDSGWbp model The disease dynamicextinction thresholds from the stochastic model were com-pared with the corresponding deterministic threshold Wederived the stochastic threshold for disease extinction 1198780 andshowed the relationship that exists between 1198770 and 1198780 interms of disease extinction and outbreak in both determin-istic and stochastic models

Our analytical and numerical results showed that bothdeterministic and stochastic models predict disease extinc-tion when 1198770 lt 1 and 1198780 lt 1 However the predictions bythese models are different when 1198770 gt 1 In this case deter-ministicmodel predicts with certainty disease outbreak whilethe stochastic model has a probability of disease extinction atthe beginning of an infection Hence with stochastic modelsit is possible to attain a disease-free equilibrium even when1198770 gt 1 Also we noticed that initial conditions do not affectthe deterministic threshold while the stochastic thresholds


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)

(a) Human population dynamics when 1198770 lt 1 and 1198780 lt 1

Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)

(b) Mosquito population dynamics when 1198770 lt 1 and 1198780 lt 1

Figure 5 Malaria dynamics when 1198770 = 04564 and 1198780 = 05661 and 1198750 = 09476

Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)

(a) Human population dynamics when 1198770 gt 1

Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)

(b) Mosquito population dynamics when 1198770 gt 1

Figure 6 Malaria dynamics when 1198770 = 24629 and 1198750 = 03713are affected Thus the dynamics of the stochastic model arehighly dependent on the initial conditions and should not beignored

The probabilities of disease extinction for different initialsizes of infected humans and infected mosquitoes were

approximated numerically The results indicate that theprobability of eliminating malaria is high if the diseaseemerges from infected human as opposed to when it emergesfrom infected mosquito at the beginning of the diseaseThe analysis has shown that any policy or intervention to


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)

(a) Human population dynamics when relapse rate is increased 1205882 = 04

Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)

(b) Human population dynamics when relapse rate is reduced 1205882 =0001

Figure 7 Changing effects of relapse on the human populations

control the spread of malaria at the beginning of an outbreakmust focus not only on infected humans but also on con-trolling the infected mosquito population Also our resultsstrongly suggest that to eradicate malaria governments ofendemic countries should increase the recovery rate andreduce the relapse rate

In conclusion to achieve the WHO vision 2030 strate-gic global targets which include (i) reducing malaria caseincidence by at least 90 by 2030 and (ii) reducing malariamortality rates by at least 90 by 2030 the governments ofendemic countries should embark on increasing the coverageof vector control interventions and reduce themalaria relapserate as well as controlling the infected mosquito population

For futurework the study can be extended by consideringvertical transmission of themosquitoes Another extension ofthe study can be the inclusion of immigration and emigrationof individuals in both populations in order to investigate theeffect of movement on the disease transmission dynamicson the two thresholds Also the study can be extended byconsidering the effect of climatic conditions such as rainfallpatterns temperature and humidity on the disease transmis-sion dynamics

Notations

Description of Variables for the Malaria Transmission Model119878ℎ Susceptible humans119868ℎ Infectious humans

119877ℎ Pseudorecovered humans119878119898 Susceptible mosquito119868119898 Infectious mosquito

Description of Parameters for the Malaria TransmissionModel120582 Natural birth rate of humans120583 Natural death rate of humans1205831 Disease induced death rate of humans120573 Transmission rate from an infectiousmosquito to a susceptible human1205881 Recovery rate1205882 Relapse rate120574 Treatment rate120575 Natural birth rate of mosquitoes120578 Natural death rate of mosquitoes1205781 Parasite induced death rate of mosquitoes1205721 Transmission rate from an infectioushuman to a susceptible mosquito1205722 Transmission rate from a pseudorecoveredhuman to a susceptible mosquito119873 The total size of human population119872 The total size of mosquito population

Conflicts of Interest

All authors declared no conflicts of interest regarding thepublication of this paper


Authorsrsquo Contributions

Rachel Waema Mbogo produced the first draft of themanuscript All authors critically reviewed the paper andapproved the final version

References

[1] WHO World Malaria World Health Organization GenevaSwitzerland 2012


[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometryrdquo in Proceedings of the Royal SocietyA vol 92 pp 204ndash230 1916

[4] Y Xiao and X Zou ldquoCan multiple malaria species co-persistrdquoSIAM Journal onAppliedMathematics vol 73 no 1 pp 351ndash3732013

[5] J Li Y Zhao and S Li ldquoFast and slow dynamics of malariamodel with relapserdquo Mathematical Biosciences vol 246 no 1pp 94ndash104 2013

[6] H-F Huo and G-M Qiu ldquoStability of a mathematical modelof malaria transmission with relapserdquo Abstract and AppliedAnalysis vol 2014 Article ID 289349 9 pages 2014

[7] H-F Huo and C-C Zhu ldquoInfluence of relapse in a givingup smoking modelrdquo Abstract and Applied Analysis vol 2013Article ID 525461 12 pages 2013

[8] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[9] L J S Allen ldquoStochastic population and epidemic mod-elspersistence and extinctionrdquo in Mathematical BiosciencesInstitute lecture series Stochastic in Biological systems SpringerInternational Publishing Switzerland 2015

[10] T Britton ldquoStochastic epidemic models A surveyrdquoMathemati-cal Biosciences vol 225 no 1 pp 24ndash35 2010

[11] Y Cai Y Kang M Banerjee and W Wang ldquoA stochastic epi-demic model incorporating media coveragerdquo Communicationsin Mathematical Sciences vol 14 no 4 pp 893ndash910 2016

[12] Y Cai Y Kang and W Wang ldquoA stochastic SIRS epidemicmodel with nonlinear incidence raterdquo Applied Mathematics andComputation vol 305 pp 221ndash240 2017

[13] M Maliyoni F Chirove H Gaff and K S Govinder ldquoAstochastic tick-borne disease model exploring the probabilityof pathogen persistencerdquo Bulletin of Mathematical Biology vol79 no 9 pp 1999ndash2021 2017

[14] M S Bartlett ldquoThe relevance of stochastic models for large-scale epidemiological phenomenardquo Journal of Applied Statisticsvol 13 no 13 pp 2ndash8 1964

[15] M S Bartlett Stochastic population models Methuen LondonUK 1960

[16] L J S Allen ldquoAn Introduction to stochastic Epidemic mod-elsrdquo in Mathematical Epidemiology Springer Berlin Germany2008

[17] L J S Allen and P Van den Driessche ldquoThe basic reproductionnumber in some discrete-time epidemic modelsrdquo Journal ofDifference Equations and Applications vol 14 pp 11ndash27 2008

[18] L J Allen and P van den Driessche ldquoRelations between deter-ministic and stochastic thresholds for disease extinction incontinuous- and discrete-time infectious disease modelsrdquoMathematical Biosciences vol 243 no 1 pp 99ndash108 2013

[19] L J S Allen ldquoA primer on stochastic epidemic models Formu-lation numerical simulation and analysisrdquo Infectious DiseaseModelling vol 243 pp 1ndash15 2017

[20] C Manore and M Hyman ldquoMathematical models for fightingzika virusrdquo SIAM News vol 2016 pp 1ndash5 2016

[21] S Olaniyi M A Lawal and O S Obabiyi ldquoStability and sen-sitivity analysis of a deterministic epidemiological model withpseudo-recoveryrdquo IAENG International Journal of AppliedMathematics vol 46 no 2 pp 1ndash8 2016

[22] L Xue and C Scoglio ldquoNetwork-level reproduction numberand extinction threshold for vector-borne diseasesrdquoMathemat-ical Biosciences and Engineering vol 12 no 3 pp 565ndash584 2015

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

[24] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[25] W R Mbogo L S Luboobi and J W Odhiambo ldquoStochasticmodel for in-host HIV dynamics with therapeutic interven-tionrdquo ISRN Biomathematics vol 2013 Article ID 103708 11pages 2013

[26] W R Mbogo L S Luboobi and J W Odhiambo ldquoMathemat-ical model for HIV and CD4+ cells dynamics in vivordquo Interna-tional Journal of Pure and AppliedMathematics vol 6 no 2 pp83ndash103 2013

[27] L J S Allen and G E Lahodny Jr ldquoExtinction thresholdsin deterministic and stochastic epidemic modelsrdquo Journal ofBiological Dynamics vol 6 no 2 pp 590ndash611 2012

[28] J Lahodny R Gautam and R Ivanek ldquoEstimating the proba-bility of an extinction ormajor outbreak for an environmentallytransmitted infectious diseaserdquo Journal of Biological Dynamicsvol 9 no 1 pp 128ndash155 2015

[29] L J S Allen ldquoBranching processesrdquo in Encyclopaedia of Theo-retical Ecology University of California Press Calif USA 2012

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


their blood streams and can pass the infection to mosquitoesAfter some period of time they lose their immunity andreturn to the susceptible class Unfortunately Li and othersdid not consider that the recovered humans will return totheir infectious state because of incomplete treatment

Stochasticity is fundamental to biological systems Insome situations the system can be treated as a large numberof similar agents interacting in a homogeneously mixingenvironment and so the dynamics arewell-captured by deter-ministic ordinary differential equations However in manysituations the system can be driven by a small number ofagents or strongly influenced by an environment fluctuatingin space and time [9ndash12]

Stochastic models incorporate discrete movements ofindividuals between epidemiological classes and not averagerates at which individuals move between classes [13ndash15]In stochastic epidemic models numbers in each class areintegers and not continuously varying quantities [13] Asignificant possibility is that the last infected individual canrecover before the disease is transmitted and the infection canonly reoccur if it is reintroduced from outside the population[16] In contrast most deterministic models have the flawthat infections can fall to very low levelsmdashwell below thepoint at which there is only one infected individual only torise up later [17] In addition the variability introduced instochastic models may result in dynamics that differ from thepredictions made by deterministic models [16]

For a large population size and a large number of infec-tious individuals the deterministic threshold1198770 gt 1 providesa good prediction of a disease outbreak However this predic-tion breaks down when the outbreak is initiated by a smallnumber of infectious individuals In this setting Markovchain (MC) models with a discrete number of individuals aremore realistic than deterministic models where the numberof individuals is assumed to be continuous-valued [18]

Motivated by these works in this paper we propose amodel which is an extension of the model formulated byHuo and Qui (2014) who assumed that the pseudorecoveredhumans can recover and return to the susceptible class orrelapse and become infectious again Using the extendedmodel we will formulate the basic reproductive number 1198770and use it to compare the disease dynamics of the determin-istic and stochastic models in order to determine the effect ofrandomness in malaria transmission dynamics

This paper is organized as follows in Section 2 we presenta malaria transmission deterministic model with relapsewhich is an extension of the model in [6] We computethe basic reproduction number 1198770 of the malaria transmis-sion deterministic model using the next-generation matrixapproach The stochastic version of the deterministic modeland its underlying assumptions necessary formodel formula-tion are presented and discussed in Section 3 In this sectionwe also compute the stochastic threshold for disease extinc-tion or invasion by applying the multitype Galton-Watsonbranching process In Section 4 we show the relationshipbetween reproductive number of the deterministicmodel andthe thresholds for disease extinction of the stochastic versionwe also illustrate our results using numerical simulationsWeconclude with a discussion of the results in Section 5

Uninfected host Infected host

Uninfected mosquitoInfected mosquito

Bite

Bite

Figure 1 The mosquito-human transmission cycle An infectiousmosquito bites a susceptible uninfected human and transmitsthe virus via saliva Once the human becomes infectious (usuallyaccompanied by symptoms) the human can transmit the pathogento an uninfected mosquito via the blood the mosquito ingestsSource Figure 1 is reproduced fromManore and Hyman (2016) [20][under the Creative Commons Attribution Licensepublic domain]

2 The Malaria Deterministic Model

The interaction of the Mosquito-host is shown in Figure 1

21 Model Formulation In this section we introduce a deter-ministic model of malaria with relapse an extended form ofthe model in [6] The model proposes a more realistic math-ematical model of malaria in which it is assumed that thepseudorecovered humans interact with infected mosquitoesand may acquire more parasites causing reinfection or dueto incomplete treatment the infection my reoccur (relapse)and return to infectious class Also the pseudorecoveredhumanmay lose their immunity and return to the susceptibleclass Given that humans might get repeatedly infected dueto not acquiring complete immunity then the populationdynamics are assumed to be described by the SIRS modelhence we consider a deterministic compartmental modelwhich divides the total human population size at time 119905denoted by 119873(119905) into susceptible individuals 119878ℎ(119905) (thosewho are not currently harbouring the parasite but are liableto be infected) infectious individuals 119868ℎ(119905) (those alreadyinfected and are able to transmit the disease to mosquitoes)and pseudorecovered individuals119877ℎ(119905) (thosewho are treatedfrom the disease but with partial recovery and hence cantransmit the disease tomosquitoes) Mosquitoes are assumednot to recover from the parasites so the mosquito populationcan be described by the SI model

The structure of model is shown in Figure 2

22 Variable and Parameter Description for the Model Thevariables for themodel are summarized in description of vari-ables for the malaria transmission model in the Notations

The parameters for the model are described as in descrip-tion of parameters for the malaria transmission model in theNotations


Hum

an p

opul

atio

nM

osqu

ito p

opul

atio

n

1Rℎ

1Iℎ

1RℎRℎ

Iℎ(SℎIℎ)N

Sℎ Iℎ

Sℎ

Sm ImM 1Im


2(RℎIm)N

Sm

N



(1)





119865 = F (1198640) = ( 0 0 1205730 0 01205721 1205722 0)119881 =V (1198640) = (120574 + 1205831 minus1205882 0minus120574 1205831 + 1205881 + 1205882 00 0 1205781)

(2)




1205831 + 1205881 + 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 0120574(120574 + 1205831) (1205831 + 1205881) + 12058311205882 120574 + 1205831(120574 + 1205831) (1205831 + 1205881) + 12058311205882 00 0 11205781))


119865119881minus1 =( 0 0 1205731205780 0 01205741205722 + 1205721 (120583 + 1205881 + 1205882)(120574 + 120583) (120583 + 1205881) + 1205831205882 (120574 + 120583) 1205722 + 12057211205882(120574 + 120583) (120583 + 1205881) + 1205831205882 0)(4)


minus1) = radic 120573 [1205741205722 + 1205721 (1205881 + 1205882 + 1205831)]1205781 [(120574 + 1205831) (1205881 + 1205831) + 12058311205882] (5)





where 11987701 = ( 1205731205781)( 1205721(120574 + 1205831)) (8)













119896119899=0



119875119894 (1198961 1198962 119896119899) 11990411989611 11990411989622 sdot sdot sdot 119904119896119899119899 (11)





infinsum1198962=0

infinsum1198963=0

1198751 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (12)



infinsum1198962=0

infinsum1198963=0

1198752 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (13)



infinsum1198962=0

infinsum1198963=0

1198753 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (14)













1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)



(18)


M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)




























(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018









Hum

an p

opul

atio

nM

osqu

ito p

opul

atio

n

1Rℎ

1Iℎ

1RℎRℎ

Iℎ(SℎIℎ)N

Sℎ Iℎ

Sℎ

Sm ImM 1Im


2(RℎIm)N

Sm

N



(1)





119865 = F (1198640) = ( 0 0 1205730 0 01205721 1205722 0)119881 =V (1198640) = (120574 + 1205831 minus1205882 0minus120574 1205831 + 1205881 + 1205882 00 0 1205781)

(2)




1205831 + 1205881 + 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 1205882(120574 + 1205831) (1205831 + 1205881) + 12058311205882 0120574(120574 + 1205831) (1205831 + 1205881) + 12058311205882 120574 + 1205831(120574 + 1205831) (1205831 + 1205881) + 12058311205882 00 0 11205781))


119865119881minus1 =( 0 0 1205731205780 0 01205741205722 + 1205721 (120583 + 1205881 + 1205882)(120574 + 120583) (120583 + 1205881) + 1205831205882 (120574 + 120583) 1205722 + 12057211205882(120574 + 120583) (120583 + 1205881) + 1205831205882 0)(4)


minus1) = radic 120573 [1205741205722 + 1205721 (1205881 + 1205882 + 1205831)]1205781 [(120574 + 1205831) (1205881 + 1205831) + 12058311205882] (5)





where 11987701 = ( 1205731205781)( 1205721(120574 + 1205831)) (8)













119896119899=0



119875119894 (1198961 1198962 119896119899) 11990411989611 11990411989622 sdot sdot sdot 119904119896119899119899 (11)





infinsum1198962=0

infinsum1198963=0

1198751 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (12)



infinsum1198962=0

infinsum1198963=0

1198752 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (13)



infinsum1198962=0

infinsum1198963=0

1198753 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (14)













1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)



(18)


M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)




























(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018









119865119881minus1 =( 0 0 1205731205780 0 01205741205722 + 1205721 (120583 + 1205881 + 1205882)(120574 + 120583) (120583 + 1205881) + 1205831205882 (120574 + 120583) 1205722 + 12057211205882(120574 + 120583) (120583 + 1205881) + 1205831205882 0)(4)


minus1) = radic 120573 [1205741205722 + 1205721 (1205881 + 1205882 + 1205831)]1205781 [(120574 + 1205831) (1205881 + 1205831) + 12058311205882] (5)





where 11987701 = ( 1205731205781)( 1205721(120574 + 1205831)) (8)













119896119899=0



119875119894 (1198961 1198962 119896119899) 11990411989611 11990411989622 sdot sdot sdot 119904119896119899119899 (11)





infinsum1198962=0

infinsum1198963=0

1198751 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (12)



infinsum1198962=0

infinsum1198963=0

1198752 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (13)



infinsum1198962=0

infinsum1198963=0

1198753 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (14)













1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)



(18)


M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)




























(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018













119896119899=0



119875119894 (1198961 1198962 119896119899) 11990411989611 11990411989622 sdot sdot sdot 119904119896119899119899 (11)





infinsum1198962=0

infinsum1198963=0

1198751 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (12)



infinsum1198962=0

infinsum1198963=0

1198752 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (13)



infinsum1198962=0

infinsum1198963=0

1198753 (1198961 1198962 1198963) 11990411989611 11990411989622 11990411989633 (14)













1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)



(18)


M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)




























(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018

















1198663 (1199041 1199042 1199043) = 12057311990411199043 + 1205781120573 + 1205781 (17)



(18)


M =(((((

12059711986611205971199041 12059711986621205971199041 1205971198663120597119904112059711986611205971199042 12059711986621205971199042 1205971198663120597119904212059711986611205971199043 12059711986621205971199043 12059711986631205971199043)))))=((((

1205721120574 + 1205831 + 1205721 1205882120574 + 1205831 + 1205721 1205721120572 + 120574 + 12058311205741205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882 12057221205831 + 1205722 + 1205881 + 1205882120573120573 + 1205781 0 120573120573 + 1205781)))) (19)




























(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018



































(34)


1

1

2

1

2

1







R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018









R0

16

15

14

13

12

11

10

09

08

07

2

0 002 004 006 008 010


R0

22

20

18

16

14

12

10

08

06

04

0 002 004 006 008 010


R0

10

09

08

07

06

1

0 002 004 006 008 010






















Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018























Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)



Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)Iℎ(t)

Rℎ(t)


Mos

quito

pop

ulat

ion

sizes

900

800

700

600

500

400

300

200

100

Days0 100 200 300 400 500

Sm(t)

Im(t)






Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018









Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)


Hum

an p

opul

atio

n siz

es

90

80

70

60

50

40

30

20

10

0

Days0 100 200 300 400 500

Sℎ(t)

Iℎ(t)

Rℎ(t)






Notations









References





































Journal of












Journal of







Volume 2018






Volume 2018











References





































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








researcharticle a stochastic model for malaria...

Documents