Mathematical Analysis of Malaria-Schistosomiasis Coinfection Model
We formulated and analysed a mathematical model to explore the cointeraction between malaria and schistosomiasis. Qualitative and comprehensive mathematical techniques have been applied to analyse the model. The local stability of the disease-free and endemic equilibrium was analysed, respectively. However, the main theorem shows that if , then the disease-free equilibrium is locally asymptotically stable and the phase will vanish out of the host and if , a unique endemic equilibrium is also locally asymptotically stable and the disease persists at the endemic steady state. The impact of schistosomiasis and its treatment on malaria dynamics is also investigated. Numerical simulations using a set of reasonable parameter values show that the two epidemics coexist whenever their reproduction numbers exceed unity. Further, results of the full malaria-schistosomiasis model also suggest that an increase in the number of individuals infected with schistosomiasis in the presence of treatment results in a decrease in malaria cases. Sensitivity analysis was further carried out to investigate the influence of the model parameters on the transmission and spread of malaria-schistosomiasis coinfection. Numerical simulations were carried out to confirm our theoretical findings.
Malaria is highly endemic in various parts of sub-Saharan Africa in which 85% of global malaria cases and 90% of malaria deaths occur . Schistosoma mansoni (the causative agent of intestinal schistosomiasis) is also prevalent in many sub-Saharan African countries [2, 3], accounting for approximately one-third of the total cases of schistosomiasis in the region . The disease is a major contributor to disease burden globally and affects low income countries with climates suitable for transmission seriously. It is a life-threatening disease caused by parasites that are transmitted to people by the bites of infected mosquitoes [4, 5]. The bites by mosquitoes have resulted in the death of a child from malaria every 30 secs according to the report by the World Health Organization (WHO) fact sheet (2009) [4, 5]. Plasmodium falciparum and Plasmodium vivax are the two common species and Plasmodium falciparum is the most deadly. Plasmodium falciparum malaria remains a major cause of mortality and morbidity in the tropics and subtropics areas of the globe [4, 6]. According to the 2009 world report, half of the world’s population is at risk of malaria, with an estimated 247 million cases that led to about 863,000 death in 2008 mostly among African children, a slight drop from 2006 statistics with the estimation that over 2000 young are lost every day across the globe . This population made malaria the dominant parasitic disease of the tropics and one of the top three killer communicable diseases [4, 5, 7]. Malaria makes development to be very slow in several ways; it affects fertility, population growth, savings and investments, and worker productivity and causes absenteeism and premature mortality [4, 5, 7]. Malaria also affects fetal development during early stage of pregnancy in women due to loss of immunity. However, malaria is preventable and curable when treatment and prevention measures are sought early [4, 5, 7].
The disease, schistosomiasis, also known as bilharziasis or snail fever, is a parasitic disease that was first named bilharzia [8, 9] and it is prevalent in several regions of the developing world, predominantly Africa, South America, and Asia, with about 650 million people living in the endemic areas . It is known that estimated 207 million people are infected, where 85 percent lives in underdeveloped areas of Africa , resulting in about 15,000 deaths annually presently . Children below the age of 14 are the major victims of schistosomiasis infection in many parts of the world . The basis of illness in victims is the eggs laid by the parasitic flat worms, that is, blood flukes of the genus Schistosoma . The species of the water-borne flatworm or blood flukes known as schistosomes is the main type that initiates the human schistosomiasis, but Schistosoma mansoni, Schistosoma japonicum, and Schistosoma haematobium are the three major species that are found everywhere . The urinary tract and kidneys as well as the reproductive systems are affected by the Schistosoma haematobium, and they are intense in Africa and the Middle East . The most widely spread species is the Schistosoma mansoni while Schistosoma japonicum is chiefly found in Asia and these two cause chronic hepatic and intestinal fibrosis [8, 10]. When skin comes in contact with contaminated freshwater in which certain types of snails that carry the parasite are living then the infection can be established . Whenever infected people urinate or defecate in the water, freshwater becomes contaminated by Schistosoma eggs . The eggs hatch, and the parasites infect, mature, and reproduce inside the snails when the appropriate species of snails exist in the water . The parasite eventually leaves the snail and go into the water where it can persist for about 48 hours [8, 12]. Schistosoma parasites, when wading, swimming, bathing, or washing, can enter the skin of anyone who comes in contact with contaminated freshwater [8, 12]. The parasites migrate through host tissue and develop into adult worms inside the blood vessels of the body for over numerous weeks [8, 12]. The worms mate and females produce eggs after maturity [8, 12]. Several of these eggs eventually travel to the bladder or intestine and are finally passed into the urine or stool [8, 12]. The schistosomiasis symptoms are caused by the body’s reaction to the eggs but not by the worms themselves [12, 13]. Eggs shed by the adult worms that do not pass out of the body can become lodged in the intestine or bladder, causing inflammation or scarring [12, 13]. Repeatedly infected children can acquire anemia, malnutrition, and learning difficulties [12, 13]. The parasite can as well damage the liver, intestine, spleen, lungs, and bladder even several years after infection [12, 13]. It is known at present that both malaria and intestinal schistosomiasis contribute to common epidemiological distributions and are currently posing a great task to public health and socioeconomic development throughout the tropical region . The interactive pathology between malaria and S. mansoni has received increased investigation in the recent time, as a result of their coendemicities [1, 11, 15, 16]. It has been discovered that considerable S. mansoni infections are linked with a major increase in the incidence of malaria among school-age children . In individuals infected with S. mansoni the technique responsible for the magnification of malaria is not yet fully understood [1, 9]. Thus, it is observed that the interface between the two diseases is perhaps set in motion by contradicting effects; the parasites possess the immunological cytokines; that is, the balance between Th1 and Th2 type immune responses which reduces immunological control of malaria may be altered by S. mansoni, while other methods are probable [1, 15, 17–19].
It is our view that this study represents the very first modeling work that presents a mathematical analysis of the qualitative dynamics of malaria-schistosomiasis coinfection. There are few studies done on the malaria-schistosomiasis coinfection model so far. In , a coepidemic model of malaria and S. mansoni transmission dynamics is established, where the model reports major epidemiological coupling between the two diseases in terms of aggravated malaria incidence among individuals with S. mansoni extreme egg output. Their model was factored for S. mansoni extreme-risk endemic areas, applying epidemiological and clinical data of the relationship between S. mansoni and malaria among children in sub-Saharan Africa. They also assessed the potential influence of the S. mansoni malaria interface and mass treatment of schistosomiasis on malaria prevalence in coendemic areas.
In this paper, we develop a mathematical model of the interplay between malaria and S. mansoni in which we have modeled the malaria transmission and the S. mansoni together as coendemic deterministic model. Our aim here is to study and analyse a mathematical model of malaria-schistosomiasis transmission model. Additionally, there are some important differences between the model in  and the one in this paper. This paper is organized as follows: we present a malaria-schistosomiasis coinfection transmission model formulation in Section 2, where the general mathematical framework, notations, and model equations were analysed with the basic properties of the models and their analysis. In Section 3, we present the existence of steady state solution. In Section 4, the basic reproduction number and stability were derived and carried out. Sensitivity analysis of the model was performed to determine the most important parameters that influence in Section 5. In Section 6, we show our numerical simulation results while, in Section 7, we discussed our conclusions and recommendations.
2. Model Formulation
In this model, we denote the total human population by and subdivide it into the following subclasses of individuals who are susceptible , individuals with malaria symptoms only (i.e., who are already infected and infective with malaria parasite) , individuals infected with schistosomiasis only , individuals infected with both malaria and schistosomiasis , individuals who recovered from malaria only , individuals who recovered from schistosomiasis only , and individuals who recovered from both malaria and schistosomiasis such that . The total snail population is denoted by , which comprises susceptible snails and infected as well as infectious snails . That is, . The total mosquito population is denoted by , which comprises susceptible mosquitoes and infected as well as infectious mosquitoes . That is, .
The population of susceptible humans is generated through birth (at a constant per capita rate ), by the loss of immunity to the malaria disease only (at a constant per capita rate ), loss of immunity to the schistosomiasis disease only at a rate , and loss of immunity to malaria and schistosomiasis disease at a rate . It is reduced by natural death (at a rate ) and through the rate of acquiring malaria through contact with infectious mosquitoes (at a rate ), where is the transmission probability per bite, is the per capita biting rate of mosquitoes, and is the contact rate of mosquito per human per unit time. It is also reduced by rate of acquiring schistosomiasis through contact with infected snails (at a rate ). Hence, the rate of change of population of susceptible humans is given by The rate of change of the population of individuals with malaria only is increased by the rate of acquiring malaria through contact with infectious mosquitoes (at a rate ) and by the rate of acquiring schistosomiasis through contact with infectious snail (at a rate ) It is also reduced by human spontaneous recovery (at a rate ). It is also reduced by the disease induced death rate (at per capita rate ) and by the natural death rate (at per capita rate ). Hence it is given by
The rate of change of the population of individuals infected with schistosomiasis only is increased by the rate of acquiring schistosomiasis through contact with infectious snail (at a rate ) and decreased by infected mosquitoes (at a rate ) and by human spontaneous recovery from schistosomiasis only (at a rate ). It is also reduced by the disease induced death rate (at per capita rate ) and by the natural death rate (at per capita rate ). Hence it is given by
The rate of change of the population of individuals infected with schistosomiasis and malaria is increased by the rate of acquiring malaria by infected mosquitoes (at a rate ) and schistosomiasis through contact with infectious snails (at a rate ) and reduced by human spontaneous recovery from schistosomiasis only (at a rate ). It is also reduced by the malaria disease induced death rate (at per capita rate ) and schistosomiasis induced death rate (at per capita rate ) and by the natural death rate (at per capita rate ). Hence it is given by
The individuals who recovered from malaria only is generated following a human spontaneous recovery (at a rate ) and by the dually infected individuals who recovered from malaria only at a rate () decreased by loss of immunity (at a rate ) and by natural death (at a rate ). Then
The individuals who recovered from schistosomiasis only are generated following a human spontaneous recovery (at a rate ) and by the dually infected individuals who recovered from schistosomiasis only at a rate () decreased by loss of immunity (at a rate ) and by natural death (at a rate ). Then
The individuals who recovered from malaria and schistosomiasis are generated following a human spontaneous recovery (at a rate ) decreased by loss of immunity (at a rate ) and by natural death (at a rate ). Then
Susceptible snail population is generated by the birth of snails (at a per capita rate of ). It is reduced by rate of acquiring schistosomiasis through contacts with infected humans at a rate , where is a modification parameter. It is also reduced by natural death (at a rate ). Thus, The population of infected snail is increased by rate of acquiring schistosomiasis through contacts with infected humans at a rate and decreased by the natural death rate (at a rate ), where is a modification parameter. Hence, it is given by
Susceptible mosquito population is generated by the birth of mosquitoes (at a per capita rate of ). It is reduced by rate of acquiring malaria through contacts with infected humans at a rate , where is probability for a vector (mosquito) to get infected by an infectious human, where is a modification parameter. It is also reduced by natural death (at a rate ). Thus, The population of infected mosquito is increased by rate of acquiring malaria through contacts with infected humans at a rate and decreased by the natural death rate (at a rate ), where is a modification parameter. Hence, it is given by
2.1. The Full Schistosomiasis-Malaria Coinfection Model
Bringing the above formulation and assumptions together leads to the following set of ordinary differential equations which may be a new malaria-schistosomiasis coinfection model:subject to the initial conditions , , , , , , , , , , and .
We describe the associated model variables and parameters in the following list and Table 1.
Variables in the Model : susceptible human : human infected with malaria only : human infected with schistosomiasis only : human infected with malaria and schistosomiasis only : human recovered from malaria only : human recovered from schistosomiasis only : human recovered from malaria and schistosomiasis only : susceptible snail : infected snail : susceptible mosquito : infected mosquito
2.2. Qualitative Analysis of the Full Schistosomiasis-Malaria Coinfection Model
The schistosomiasis-malaria coinfection model (12) will be analysed in a biologically feasible region for both humans, snail, and mosquito populations. Hence, for it to be epidemiologically well posed, it is necessary to show that all its state variables are nonnegative for all time .
2.3. Positivity of the Solution
Theorem 1. If the initial value for , , , , , , , , , , and then the solutions of the schistosomiasis only model (16) are nonnegative for all .
Proof. We let , .
Since the variables , , , , , , , , , , and then, . If , then are equal to zero at . It follows from the first equation of the system (12) that Therefore,HenceAlsoNow,And then Now, Furthermore, Now, Also Therefore Hence so thatAlso, Therefore Hence so thatAlso, Then, Hence so thatAlso, Therefore, Then, we have Similarly, Therefore, so thatfor all .
Also, Therefore, Then, we have Similarly, Therefore, so thatfor all .
2.4. Boundedness of the Solution
Theorem 2. All solutions of the malaria-schistosomiasis coinfection model (12) are bounded. Therefore, from (12) if then , , and .
Proof. For the proof of boundedness, we note that , , , , and . We add the first eight equations and the last two equations of the malaria-schistosomiasis coinfection model (12) and yieldAll solutions of model (12) are bounded. The feasible region for the human population is given by + ≤ . And the feasible region for the snail population is given byAnd the feasible region for the mosquito population is given byTherefore, Hence,
2.5. Invariant Region
Here, we analysed the malaria-schistosomiasis only model (12) in a biologically feasible region. Hence the system of (12) is split into three parts, namely. We have the human population (: with ), the snail (vector) population (: with ), and the mosquito (vector) population (: with ). Let us consider the feasible region as with We take the following steps to reaffirm the positive invariance of (i.e., solution in remains in for all ). The rate of change of humans, snails, and mosquitoes population is given in (12) and it follows that By the standard comparison theorem we obtain In particular, if , if , and if . Therefore, the region is positively invariant. Hence, it is enough to consider the dynamics of the flow generated by (12) in . Thus, the model can be considered to be epidemiologically and mathematically well posed in this region. Therefore every solution of the malaria-schistosomiasis coinfection model (12) with initial data (conditions) in remains in . The summary of this result is given below.
Theorem 3. The region is positively invariant for the malaria-schistosomiasis coinfection model (12) with nonnegative initial conditions in .
3. Existence of Steady States Solution
We analysed the model equation (12) in this section qualitatively to investigate the condition of existence of equilibrium points. We would like to know what will eventually happen to the disease in the long run. The question that will arise is (i) will the disease (malaria-schistosomiasis disease) die out? Or (ii) will it be established in the population and become endemic? In order to answer these questions we have to investigate the long-term behaviour of the solutions. This behaviour depends largely on the equilibrium points, that is, time-independent solutions of the system. Since these solutions do not depend on time, we set Therefore, we have the equilibrium solution This equilibrium exists for all values of the parameters. Since it implies that the disease will disappear from the population. Thus, this equilibrium is referred to as the disease-free equilibrium (DFE).
3.1. Existence of Disease-Free Equilibrium Point (DFE)
We analysed the system qualitatively by studying the system of equation in closed setA qualitative study of the system exists in two forms, namely,(i)the disease-free (dies out),(ii)endemic.
When the malaria-schistosomiasis disease dies out naturally, the solution asymptotically tends to a disease-free equilibrium of the form = . Hence, the threshold that determines the stability of this equilibrium is the reproduction number (which is the average number of secondary infections caused by a single infectious individual during their entire infectious lifetime).
4. The Basic Reproduction Number
Whenever , each individual releases on average less than one infected individual and therefore the disease (malaria-schistosomiasis only) dies out. Whenever , each individual release more than one new infected individual and therefore the disease is able to invade the susceptible population. This allows us to determine the effectiveness of control measures. Epidemiologically, the reproductive number tells us how many secondary cases will one infected individual or vector produce in an entirely susceptible population of hosts and vectors, in other words, to determine the number of infected people that are generated by the introduction of a single infected person into a susceptible population. We need the computation of the basic reproduction number to assess the stability of the disease-free equilibrium (DFE) and the endemic equilibrium point (EEP). The basic reproduction number for particular infections is dependent on the biological characteristics of the disease and on the behavioural pattern of the population. When there is a high transmission of the disease per unit time and durations of the infectious period, the basic reproduction number increases.
Proposition 4. The value of the basic reproduction number is .
Proof. The argument uses the approach of the next generation matrix (see ). Supposewhich results intowhere is the rate of appearance of new infections in one compartment and is the transfer of individuals and mosquitoes into one compartment which is a Jacobian matrix evaluated at and the Jacobian matrix of is evaluated at to giveThe product of matrix and givesFinding the eigenvalue of , we obtain for the malaria-schistosomiasis coinfection model (12) is given byConsidering the parameters for our model from Table 1, the value of appears to be far greater than 1. This implies that the disease (malaria-schistosomiasis coinfection model) will not die out but remains endemic in the population until some control strategies are applied properly. The disease-free equilibrium is locally stable if . We can achieve the global stability for the disease-free equilibrium by following the Lyapunov argument for . Hence, when the disease-free equilibrium is unstable, there exists an endemic equilibrium. Next we discuss the local stability of the disease-free equilibrium.
4.1. Local Stability of the Disease-Free Equilibrium of Malaria-Schistosomiasis Coinfection Model
The disease-free equilibrium for the malaria-schistosomiasis coinfection model given by = exists for all values of , , and . If the threshold value is , then the disease-free equilibrium is locally asymptotically stable and the disease cannot invade or spread in the population or community. The summary of the result is given by Theorem 5.
Theorem 5. The disease-free equilibrium point is locally asymptotically stable if and unstable if .
It means that the malaria-schistosomiasis coinfection can be eliminated from the population whenever . ; then, averagely an infected person produces less than one newly infected person over the entire period of his infectiousness and malaria-schistosomiasis coinfection dies out. For to be less than one, the , , , and (rates of transmission) must decrease without bounds. Otherwise, if , then each infected person produces less, averagely more than one new infection, and the malaria-schistosomiasis coinfection can spread or invade the population. Consider the model equation (12). At the equilibrium state , the Jacobian is given bywhere , , , , , , , , , and .
The local stability of is determined by the signs of the eigenvalues of the Jacobian matrix (65). The disease-free equilibrium point, , is said to be locally asymptotically stable if the real parts of the eigenvalues of the Jacobian matrix (65) are all negative; otherwise it is said to be unstable. By considering the Jacobian matrix (65), we obtainwhere , , , , , , , , , and , such that
The remaining characteristic polynomial corresponding to iswhereThe Routh-Hurwitz conditions to establish that all roots of () have negative real parts are , , , and . Note that all model parameters are positive. For to be positive set which implies that This leads to .which is true. Thus, by the Routh-Hurwitz condition, all the eigenvalues have negative real parts whenever , such that is locally asymptotically stable. , where The Routh-Hurwitz conditions to make certain that all roots of () have negative real parts are , , , and . Note that all model parameters are positive. For