Analysis of a Malaria Transmission Model in Children

In this paper, a nine compartmental model for malaria transmission in children was developed and a threshold parameter called control reproduction number which is known to be a vital threshold quantity in controlling the spread of malaria was derived. The model has a disease free equilibrium which is locally asymptotically stable if the control reproduction number is less than one and an endemic equilibrium point which is also locally asymptotically stable if the control reproduction number is greater than one. The model undergoes a backward bifurcation which is caused by loss of acquired immunity of recovered children and the rate at which exposed children progress to the mild stage of infection. DOI: https://dx.doi.org/10.4314/jasem.v24i5.9 Copyright: Copyright © 2020 Eguda et al. This is an open access article distributed under the Creative Commons Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dates: Received: 01 March 2020; Revised: 22 April 2020; Accepted: 14 May 2020

In most developing countries in Africa, Asia, Central America and South America, Malaria constitutes a major public health challenge for children. It is reported that from 40% to 50% of the world's population lives in malaria endemic areas (Portugal et al., 2014;WHO, 2010).Malaria is a catastrophic infection with approximately 300-500 million cases yearly resulting in 1-2 million deaths, mostly among young children. Children of all ages living in areas where malaria is non endemic are equally susceptible to the malaria. Studies revealed that children under five years of age (mostly between six months and five years) in endemic areas are at the highest risk of malaria infection than other age groups. Furthermore, an estimated 660,000 malaria deaths were recorded among children around the world in 2010 and approximately 86% of these cases were less than five years of age. In high malaria transmission areas, young children with severe forms of malaria who have not acquired immunity to malaria can rapidly die of malaria. Children with malaria experience high fever which may be accompanied by chills, sweats, and headaches and other common symptoms include abdominal pain, diarrhea, vomiting, weakness, myalgia, and pallor. In children, these symptoms are frequently misdiagnosed with a viral syndrome or acute gastroenteritis. Also, in endemic areas, children with partial immunity frequently present the following symptoms: hepatosplenomegaly, anemia, and jaundice. However, the use of intravenous treatment is an appropriate plan for the medical care of children since the clinical condition of children younger than 5 years old with malaria can worsen rapidly (Metanat, 2005). In areas where malaria is not widespread, health specialists are frequently unfamiliar with the disease, and delays in detection and treatment are common. Increased awareness and knowledge of proper management strategy becomes a necessity considering the severity of this illness. The occurrence of malaria is common in patients who have a history of recent or ongoing use of a malaria chemoprophylactic agent.
This incident can be attributed to factors such as drug resistance, noncompliance with treatment, or inadequate or inappropriate administration (especially in children, because of the difficulties in administering bitter medications). Treatment must include careful supportive care, and intensive care measures should be available for treating children with complicated Plasmodium falciparum malaria. Drug regimens can include mefloquine, atovaquone-proguanil, sulfadoxine-pyrimethamine, quinine or quinidine, clindamycin, doxycycline, chloroquine, and primaquine (Stauffer and Fischer, 2003).
Mathematical models have been widely accepted as vital tools for studying the dynamics of the spread of communicable diseases. Over the years, several researchers and mathematicians have applied mathematical models to study mosquito related diseases. The present work therefore seeks to analyze the equilibrium states of a malaria model for stability and investigate the existence of backward bifurcation.      (Ducrot, et al., 2009) c  Rate at which exposed children progress to the mild stage of infection 0.122 (Ducrot, et al., 2009) (Augusto et al., 2017) , c c   Treatment rates for children at mild and severe stage 0.0082,0.011 (Ducrot, et al., 2009) c  Rate at which recovered children lose their immunity 0.046 Estimated v  Rate at which exposed mosquitoes become infectious 0.18 (Ducrot, et al., 2009) (Ducrot, et al., 2009)

Analysis of the model: Boundedness of solutions
Consider the region can be shown that the set 2 D is positively invariant and an attractor of all positive solutions of the model (1).

Lemma 1: The region D2 is positively invariant for the model (1).
Proof: The rate of change of the total children population is given thus The rate of change of total vector population gives Hence 2 D is a positively invariant set and the solution enter 2 as t .   Hence it is sufficient to consider the dynamics of the model (1) in 2 D .In this region, the model (1) is seen as being mathematically and epidemiologically well posed.

t E t I t I t R t Q t S t E t I t with positive initial data
will remain positive for all time t > 0. Proof: Similarly, it can be shown that all state variables of the model remain positive for all time, t > 0 so that

Local Stability of Disease Free Equilibrium (DFE)
The model (1) has a disease -free equilibrium obtained by setting the right hand sides to zero and all the disease classes to zero to give The stability of   is established using the next generation operator method by using the notation in (Van Den Driessche and Watmough, 2002), so that the matrices F and V are computed as; From model (1) Where, 2 ( The value E R is the effective reproduction number. (1)
The following are the transformed equation for model (1) The Jacobian of the transformed equation (26), evaluated at the DFE, is given as: Since the bifurcation coefficient b is positive. It follows from theorem 2 of (Castillo-Chavez and Song, 2004) that the transformed model (27) Table 2

RESULTS AND DISCUSSION
Epidemiologically, malaria can be eliminated from the children population if the initial size of the population is small enough such that the control reproduction number can be brought below unity.The model (1) will undergo a backward bifurcation whenever a stable disease free equilibrium point coexists with a stable endemic equilibrium point when the associated reproduction number is less than unity. The epidemiological implication of the backward bifurcation of the model (1) is that the classical requirement of the reproduction number being less than unity becomes only a necessity, but not sufficient condition for malaria control. Thus, it follows from the Castillo Chavez theoremin (Castillo-Chavez and Song, 2004) that model (1) does not undergo the phenomenon of backward bifurcation if 0 C c     . Hence, this study shows that the loss of acquired immunity of recovered children ( ) C  and the rate at which exposed children progress to the mild stage of infection ( ) C  are the causes of backward bifurcation in the malaria transmission model.

Conclusion:
A vector-borne compartmental model was formulated to control the spread of malaria among children. The model was seen to exhibit the disease free equilibrium state which is locally asymptotically stable whenever the control reproduction number is less than unity and unstable otherwise. The endemic equilibrium state was proved to be locally asymptotically stable if the control reproduction number was greater than one. The model undergoes the phenomenon of backward bifurcation whenever the stable DFE coexists with a stable endemic equilibrium.