Pathophysiology and Clinics of Malaria

## **Chapter 3** Pathology of Malaria

*Pier Paolo Piccaluga and Wanyonyi Ignatius*

#### **Abstract**

Malaria is an acute febrile illness that is caused by infection with *Plasmodium spp*. parasites. Malaria is a serious illness and sometimes it may be fatal resulting in mortality and morbidity. The clinical picture painted in patients with malarial infection occurs following the release of the merozoites into the bloodstream following the rupture of infected red cells. In the infection with the *P. falciparum*, the commonest form affecting humans, all stages of red cells are infected making the infection quite severe as compared to infection with other species which infects the old and young red cells only which contributes to a small percentage of red cells. In this chapter, the Authors review the current knowledge about Malaria epidemiology, pathogenesis and anatomic pathology. The diverse clinical pictures as well as the association with genetic conditions and diseases are discussed.

**Keywords:** malaria, plasmodium malariae, pathology, histology, gross and microscopic pathology, cerabral malaria, pathogenesis

#### **1. Introduction**

Malaria is an acute febrile illness that is caused by infection with *Plasmodium spp*. parasites. Malaria is a serious illness and sometimes it may be fatal resulting in mortality and morbidity.

There were an estimated 241 million malarial cases in 2020 higher than malarial cases in 2019 (227 million cases), a difference of 14 million cases. The rise in malarial cases is thought to be associated with the COVID-19 pandemic, that led to service disruption. The case incidence of malaria from 2000 has declined from 81 per 1000 populations at risk to 56 per 1000 populations at risk. However, in 2020 the case incidence has increased to 59 per 1000 populations at risk. Globally, death due to malaria in 2020 is reported at 627000 which is a rise as compared to the trends in malaria death that has a seen a drop from 896,000 cases in 2000 to 558,000 cases in 2019. It is estimated that 68% (47,000 cases) of the additional deaths due to Malaria are attributed to disruption of services following COVID-19 pandemic. Globally, the malaria mortality rate as of 2020 is 15 per 100,000 population at risk which is higher compared to 2019 which was 14 per 100,000 population at risk. Among children under 5 years, the mortality rate in 2020 increased to 77% as compared to 76% in 2019. Globally, 96% of the malaria cases and deaths were from 29 out of 85 countries with malaria-endemic. Nigeria (26.8%), the Democratic Republic of Congo (12.0%), Uganda (5.4%), Mozambique (4.2%), Angola (3.4%), and Burkina Faso (3.4%) account for 55% of the cases globally. Half of the mortality cases were from

Nigeria (31.9%), the Democratic Republic of Congo (13.2%), The United Republic of Tanzania (4.1%), and Mozambique (3.8%). The WHO African region accounts for 95% of the malarial cases and 96% of the malarial deaths. In the African region, 80% of the cases are among children under 5 years. In Africa, the number of malaria cases in 2020 is 233 per 1000 populations at risk higher than in 2019 at cases per 1000 population cases. The South-east Asia region accounts for 2% of the malaria cases globally among the nine malaria-endemic countries in 2020. India accounts for 83% of malaria cases due to infection by *Plasmodium vivax*. The WHO Eastern Mediterranean Region accounted for 5.7 million cases in 2020 with the increase seen in Sudan, Somalia, and Djibouti. *P. vivax* accounted for about 18% of the cases mainly in Afghanistan and Pakistan. In WHO western Pacific region contributed to 1.7 million cases with an increase of 19% from 2019 cases. The malaria death increased to 3200 death in 2020 as compared to 2600 in 2019. WHO region of Americas registered a reduction in the case incidences by 58% between 2000 and 2020 with a resultant reduction in malaria death by 56% since 2000. Most of the cases in this region were due to *P. vivax*. In WHO European region, there has been free of malaria cases since 2015 with the last case reported in 2014 in Tajikistan with no malaria death from 2000 to 2020.

There are attempts globally to eliminate malaria in different countries. According to WHO a country is declared malaria-free following at least 3 consecutive years with zero indigenous cases. Islamic Republic of Iran and Malaysia reported zero indigenous malaria cases for the third consecutive year and while Belize and Cabo Verde for the second consecutive time. China and El Salvador were certified malaria-free in 2021 following 4 years of zero malaria cases [1].

#### **1.1 Plasmodium**

Plasmodium was first described in the late nineteenth century by Charles Laveran and over time, many species have been discovered. Plasmodium is a unicellular eukaryote that cannot survive outside the host (obligate parasite). The parasite affects different hosts such as reptiles, birds, and mammals and it requires an insect host of the genera Culex and Anopheles. In humans*, P. vivax, Plasmodium falciparum, P. malariae, P. ovale* and *Plasmodium knowlesi* are the most common species affecting humans. *P. falciparum* is the most lethal infection in humans resulting in thousands of deaths.

#### **1.2 Lifecycle of palsmodium**

The life cycle of plasmodium involves distinct stages in the mosquito and vertebrate host. The life cycle can be divided into the sexual phase that occurs in the insect and the asexual phase while inside the vertebrate host.

Upon taking the blood meal, the female anopheles mosquito releases the sporozoites into the bloodstream of an individual. The sporozoites enter the circulation and they attach and enter the hepatocytes through receptors for serum proteins thrombospondin and properdin. In the hepatocytes, sporozoites mature into schizonts. Schizonts are the multinucleated staged cells that form during asexual reproduction. The schizonts in the hepatocytes mature and enlarge which causes the hepatocytes to rupture and release thousands of merozoites into the circulation. For *P. vivax* and *P. ovale*, some schizonts remain in liver cells in the dormant stage called hypnozoites that are responsible for relapses by invading the bloodstream weeks or years later. Upon release of the merozoites into the bloodstream, they invade the erythrocytes for the formation of erythrocytic schizogony. The merozoites bind to sialic acid residue on the glycophorin

#### *Pathology of Malaria DOI: http://dx.doi.org/10.5772/intechopen.110579*

molecule on the surface of the red cells and enter through active membrane penetration aided by a lectin-like molecule. The merozoites undergo asexual multiplication in the infected red cells to form trophozoites characterized by a single chromatin mass. The trophozoite develops into ring-shaped trophozoites and later on forms schizonts with multiple chromatin masses. Some trophozoites differentiate forming male and female gametocytes. The infected red cells rupture to release the merozoites into the bloodstream alongside the products of the red cells that are responsible for some of the clinical presentation. The female anopheles mosquito as it feeds on human blood to nourish its eggs, it ingests the merozoites and gametocytes from the infected host. The male and female gametocytes in the mosquito are known as the sporogonic cycle. In the mosquito stomach, the male and female gametocytes (microgamete and macrogametes) form the zygote. The zygote after some time becomes motile and elongated forming ookinetes. The ookinetes invade the midgut wall of the mosquito to transform it into an oocyst. The oocyst grows over time and ruptures to release sporozoites which migrate into the salivary gland awaiting inoculation into the new host following a blood meal.

#### **2. Malaria pathogenesis and pathophysiology**

The clinical picture painted in patients with malarial infection occurs following the release of the merozoites into the bloodstream following the rupture of infected red cells. The malaria parasite has a wide variety of symptoms which ranges from absent or very mild symptoms to severe cases and even death. Owing to this the infection is categorized into uncomplicated malaria and severe malaria. The incubation period varies depending on the individual factors and the species of plasmodium. The incubation period is the period of the introduction of the sporozoites to the development of the first symptoms. The incubation period varies from 7 to 30 days. However, *P. falciparum* has a short incubation period while *P. malariae* has a longer period.

The release of the merozoites from the rupture of the liver cells infects the red cells. In the infection with the *P. falciparum*, all stages of red cells are infected making the infection quite severe as compared to infection with other species which infects the old and young red cells only which contributes to a small percentage of red cells. The malarial parasites while in the red blood cells derive their energy primarily from the red cells. They can metabolize 70 times faster as compared to the red cells resulting in hypoglycemia and lactic acidosis. The rupture of the infected red cells releases the merozoites alongside the waste and toxic substances from the red cells. The released merozoites infect other healthy red cells in the circulation. Since the infected red cells are destroyed, and red cells are destroyed every time the merozoites infect new red cells, the patient experiences intravascular hemolysis. Intravascular hemolysis results in worsening anemia requiring immediate treatment and transfusion in a patient with very low hemoglobin levels (<5 g/dL). The intravascular hemolysis results in the production of hemoglobin that causes the urine to change and appear as dark colored in the patient. The intravascular hemolysis also causes an increased amount of yellow discoloration due to overloading the ability of the body to conjugate bilirubin causing jaundice on the sclera and mucous membrane and skin. The released products and toxic substances in ruptured red cells cause the body to release cytokines such as tumor necrosis factor (TNF) which contributes to fevers. An increase in body temperature results in sweating and chills. The merozoite surface antigen being foreign to the body as well evokes immune responses that lead to the production of the cytokines. These

cytokines such as TNF, Interferon-gamma, and Interleukin-1 have the potential of suppressing the red blood cell production thus reducing the restoration of hemoglobin concentration, increasing fever, stimulating reactive nitrogen species production that causes tissue damage, and inducing endothelial receptor expression of *P. falciparum* erythrocyte membrane protein 1 (PfEMP1) which is responsible for sequestration. These cytokines as well irritate the vomiting centers resulting in the vomiting as part of clinical presentation.

The infected red cells appear sticky and adhered to the blood vessels causing obstruction of the blood flow and causing ischemia in the affected areas and organs. The sticky red cells as well clump together forming a rosette that if large enough have the potential to obstruct blood flow in major blood vessels. In addition, several proteins such as PfEMP1 contribute to the attachment of red cells on the endothelium leading to sequestration. The PfEMP1 bind to the endothelial cells through CD36, thrombospondin, VCAM1, ICAM1, and E-selectin. The sequestration of the red cells in the blood vessel endothelium and the ability of the body to take away damaged red cells mostly occur in the spleen contributing to splenomegaly in some patients. Sequestration of the red cells, and destruction of the red cells leading to anemia coupled with obstruction of blood flow results in the reduction of tissue perfusion causing fatigue, general body weakness, and ischemia.

#### **2.1** *P. falciparum*

*P. falciparum* is responsible for most malarial cases, especially in the WHO African region. The parasite can infect red blood cells of all ages thus resulting in high parasitemia as compared to *P. ovale* and *P. vivax* which infect only young red cells. High parasitemia results in severe hemolysis that causes hemoglobinuria which has the potential of damaging the kidneys and causing renal failure worsening the prognosis. Infection with *P. falciparum* has the specific property of sequestration. The organisms exhibit adherence properties that result in sequestration of the parasites in small post-capillary vessels. Through sequestration of the parasite, the patient may develop an altered mental state and even a coma. Infection with *P. falciparum* is associated with a high burden of cytokines released by the body which in addition to high parasitemia resulting in end-organ failures. End organ disease specifically involves the central nervous system, lungs, and kidneys.

#### **2.2** *P. vivax*

*P. vivax* infects the immature red blood cells only thus associated with limited parasitemia. However, due to the presence of hypnozoites in the liver lying dormant, 50% of the infected patients experience relapses within a few weeks to 5 years following the initial illness. The relapses decrease in periodicity and intensity over time.

#### **2.3** *Plasmodium ovale*

*P. ovale* infects only the immature red cells and thus has limited parasitemia than *P. falciparum*. The infection is less severe as compared to *P. vivax* and often resolves without treatment. Due to the presence of hypnozoites in the liver, relapses may occur.

#### **2.4** *Plasmodium malariae*

*P. malariae* species have a longer incubation period and therefore the patient may remain asymptomatic for a longer duration. Recrudescence is quite common in a patient infected with *P. malariae*. In addition, *P. malariae* is associated with the deposition of the antibody–antigen complexes on the glomeruli damaging the basement membrane to cause nephrotic syndrome.

#### **2.5** *P. knowlesi*

The infection of *P. knowlesi* should be treated aggressively just as patients with *P. falciparum* as it causes fatal disease. It was identified in Malaysian Borneo, Thailand, Myanmar, Singapore, the Philippines, and neighboring countries.

#### **3. Genetic factors and malaria**

Genetic factors play a crucial role in influencing malaria infection. Two biological characteristics identified in protecting the certain type of malaria are sickle cell trait and negative for Duffy blood group [2].

Individuals with sickle cell traits are relatively protected from infection by *P. falciparum*. Sickle cell traits are individuals with heterozygous for abnormal hemoglobin gene S. The individuals have one copy of the abnormal hemoglobin gene HbS and a copy of normal hemoglobin gene HbA. *P. falciparum* cannot survive in the red cells with HbS conferring protection to individuals. The presence of sickle cell traits is mainly in areas with a high prevalence of *P. falciparum* in the WHO African region. According to the cohort study, the presence of the sickle cell trait confers 60% of the protection against overall mortality with the protection occurring between 2 and 16 months of life, before the development of the immunity.

People who are negative for the Duffy blood group have shown to be resistant to infection with *P. vivax*. The *P. vivax* infection is rare in West Africa because of the presence of Duffy-negative individuals that confer protection against the infection. *P. ovale* which can infect the Duffy negative red cells predominates in this setting.

Blood cell dyscrasias such as hemoglobin *C. thalassemia* and G6PD deficiency are prevalent in the malaria areas which confer protection against the infection.

#### **4. Acquired immunity against malaria infection**

Acquired immunity influences malaria infection in individuals and communities. Through repeated infection, an individual develops partially protective immunity. Despite the acquired immunity, the individuals may get infected by the malaria parasite but the severity of the illness will be less ad they may lack the typical symptoms. The acquired immunity allows the individual to mount the immune response to the presence of the parasite in the circulation leading to a reduction in the severity of the infection. The areas that have a high malarial infection such as *P. falciparum* in the south Saharan region in Africa have individuals with acquired immunity. The mother who has repeated infection with *P. falciparum* can confer protection to the newborn. While the newborn is in-utero, the maternal antibodies are transferred through the placenta to the fetus which confers protection to the newborn up to the first few months of life. After the reduction of the maternal antibodies in the circulation, the young children become susceptible to infection leading to an increase the infection. If the children survive the repeated infection to an older age of about 2–5 years, they attain acquired immunity. Owing to the acquired immunity in the areas of high transmission such as the WHO African region,

there is increased mortality affecting the young population (90% of the mortalities were children in 2020 mortality cases) with less mortality affecting the adults. In areas where there is lower transmission such as Asia and Latin America, infection is minimal and few individuals are exposed to repeated infection. The individuals do not develop acquired immunity. Following the lack of acquired immunity, a larger proportion of the older children are as well affected by the epidemic of malaria infection.

#### **5. Sickle cell disease and malaria**

Despite the presence of sickle cell trait conferring protestation against malarial infection over the healthy population, the presence of sickle cell disease does not confer any protection. Sickle cell disease occurs when the individual has a mutation in both alleles resulting in homozygous HbS. Sickle cell disease carries the worst prognosis with infection by the malarial parasite, especially *P. falciparum*. The worst prognosis contributed to the reduced half-life of sickled red cells and increased ability to undergo sequestration. Coupled with hemolysis and reduced hemoglobin concentration among the sicklers, they easily undergo severe anemia and severe hypoxia which worsen sickling and obstruction of the blood vessel resulting in end-organ disease. Sickle cell disease does not tolerate infection with malaria infection requires immediate and aggressive management to improve the prognosis [2–4].

#### **6. Clinical features of cerebral malaria**

Cerebral malaria is the most severe neurological complication that arises following infection with *P. falciparum*. It is estimated that annually, 575,000 cases of children in the sub-Saharan region are affected by cerebral malaria. According to WHO, cerebral malaria refers to the clinical syndrome where there is coma at least 1 hour after termination of seizure or correction of hypoglycemia and presence of parasitemia for *P. falciparum* on the peripheral blood smear with other causes evident to causing coma. Loss of consciousness is the hallmark of cerebral malaria with coma being the most severe presentation. However, for a diagnosis of cerebral malaria to be made, the patient should have no other cause of coma such as viral encephalitis, metabolic disorder, and poisoning among others. Apart from the altered mental status, cerebral malaria may be followed by progressive weakness and prostration. In adults, cerebral malaria presents as one of the organ failures following end-organ disease with severe *P. falciparum* infection.

The development of cerebral malaria following infection with *P. falciparum* occurs in cases of high parasitemia. It is thought that infected red cells are sequestrated through the attachment on the endothelium that causes occlusion of the cerebral capillaries. This coupled with sequestration of the parasites cause reduction in blood circulation in the microvasculature resulting in a decreased supply of nutrients and oxygen. The obstruction of the blood flow shuts down energy production which is needed to maintain the blood–brain barrier. As a result, ischemia ensues and there is a neuronal alteration that causes cerebral swelling and increased the cerebral intracranial pressure. This leads to altered consciousness and retinal changes. The damage to the blood vessels as well leads to intracranial hemorrhage the leads to altered mental status. Cytokines and chemokine are also described as through a complex role in protecting and posing harmful effects in the pathogenesis of cerebral malaria. Schizogony triggers pro-inflammatory cytokines that may contribute to cerebral malaria while

anti-inflammatory cytokine poses protective benefits. Unlike in bacterial and viral infections, altered mental status following *P. falciparum* infection does not result from the presence of the parasite in the brain parenchyma but due to cerebral occlusion that causes end ran tissue damage due to disruption of brain parenchyma.

Cerebral malaria is fatal and without treatment, the mortality rates increase exponentially. In children treated with intravenous antimalarial medication, the mortality is about 5–20%. However, despite the recovery, many children sustain significant brain injuries. About 11% have gross neurological deficits which may improve with time while 25% have long-term impairments, especially in cognition, motor, and behavior domain. About 10% develops epilepsy. The risk factors include seizures, deep and prolonged coma, intracranial hypertension, and hypoglycemia.

#### **7. Congenital malaria**

Congenital malaria refers to the infection with malaria parasite present in the peripheral smear of the newborn from 24 hours to 7 days of life. It is thought that the parasite can infect the placenta and gain access to fetal circulation. Congenital malaria is rare but it is fatal if not detected earlier in newborns. However, in areas where malaria is endemic, congenital malaria is rare as compared to other areas. This is because, in malaria-endemic areas, the maternal have had repeated attacks thus developing acquired immunity. The antibodies in the maternal circulation are therefore protected against the development of congenital malaria and protection in the early stages of life. The occurrence of congenital malaria is at 0.3% in the immune mothers and 7.4% in the non-immune mothers. The symptoms of congenital malaria occur within 10 to 30 days of life. They include fever, anemia, and splenomegaly in most cases. Other presentations may include hepatomegaly, jaundice, loose stool, and poor feeding among others. Since this presentation mimics most conditions in the neonates such as neonatal jaundice and neonatal sepsis, the diagnosis is often missed leading to mortality and morbidity [5, 6].

Congenital malaria is of concern among neonates. Given that the parasite is present in utero, the human body learns to recognize the parasite as self, contrary to the normal case where the parasite is considered an antigen to cause activation of immune responses. The parasite, therefore, can multiply and cause damage to red cells without evoking immune responses as in the case of a healthy individual. Therefore, the newborn experience excessive hemolysis and resultant organomegaly. Lack of the ability to recognize the parasite as an antigen coupled with an immature immune system allows the parasite to multiply necessitating immediate treatment to improve the outcome among the neonates. Owing to intravascular hemolysis, the neonates present with anemia and splenomegaly which are the most common forms of presentation. Intravenous antimalarial medication is essential in management [5, 6].

#### **8. Gross and microscopic pathology**

#### **8.1 Bone marrow**

*P. falciparum* infection demonstrates dyserythropoietic with iron sequestration and erythrophagocytosis, especially in the acute phase. The dysthrombopoeisis has large atypical megakaryocytes and giant platelets. The defect is present for up to 3 weeks after infection resolution.

#### **8.2 Spleen**

Splenic enlargement is one of the findings of all types of malaria. However, with *P. falciparum*, the spleen may not be palpable even though splenomegaly is one of the earliest signs of malaria. In the initial stage, grossly, the spleen enlarges, round and hard on palpation but tender. As the disease progresses, it becomes larger and harder but less tender. Splenic enlargement results from the engorgement of blood vessels initially, with oedema of the pulp in the initial cases. However, later there is lymphoid and reticuloendothelial hyperplasia and increased hemolytic and phagocytic function. Pulp sclerosis and dilated sinuses occur due to relapses. Splenic rupture occurs in some cases of malarial infection due to rapid and considerable enlargement, and it poses a serious complication, especially in primary malaria episodes. Splenic rupture has been found less likely in chronic splenomegaly due to fibrosis and peri splenitis.

Tropic splenomegaly syndrome is characterized by marked enlargement of the spleen with a spleen weighing 2–4.4 kgs. Usually seen among the adult population in Africa and India, presenting with a huge spleen. The spleen demonstrates marked hyperplasia of lymphoid with dilated sinuses. With splenomegaly, there is increased phagocytosis of red and white cells, and the patient has a picture of anemia, leucopenia and thrombocytopenia, though the general health is well maintained. There may be concomitant enlargement of the liver with lymphoreticular infiltration of sinusoids. The patients have high levels of IgG and M against malaria. However, the use of antimalarial medication reduces the splenic size [7, 8].

Treatment of malarial infection results in splenic regression, usually within weeks, but the duration may be protracted with large fibrotic spleen secondary to repeated malaria, though complete involution is common. The patient undergoing splenectomy has a risk of latent infection reactivation since the spleen plays a crucial role in the immune response against malaria.

#### **8.3 Liver**

In malarial infection, hepatomegaly also occurs in the early stages. The liver is enlarged, firm and tender. It may appear brown, gray or black following malaria pigment deposition. Microscopically, there is oedema and dilatation of hepatic sinusoids containing hypertrophied Kupffer cells and parasitized red cells. In severe malarial cases, shock and disseminated intravascular coagulation result in small areas of centrilobular necrosis. Prolonged infection results in stromal induration and diffuse proliferation of fibrous connective tissue, but changes of cirrhosis are absent [7, 8].

*P. falciparum* affects mesenchyma and hepatocytes, causing malarial hepatitis due to functional changes. Malarial hepatitis is characterized by conjugated hyperbilirubinemia, increased levels of transaminase and alkaline phosphatase. Severe infection with *P. falciparum* is due to hepatocellular damage due to impaired local microcirculation.

In repeated infection, there is significant hepatomegaly with associated splenomegaly, but no functional abnormality exists. Despite prolonged malarial infection having diffuse fibrous connective tissue proliferation, malaria is not a proven cause of liver cirrhosis.

#### **8.4 Lungs**

Acute pulmonary oedema may occur in a patient with *P. falciparum* infection, presenting as a rare but fatal complication. It occurs due to impairment in

#### *Pathology of Malaria DOI: http://dx.doi.org/10.5772/intechopen.110579*

microcirculation, with capilarying showing endothelial lesions and perivascular oedema. The edematous endothelium results in the narrowing of the lumen. In addition, microscopically, pulmonary vascular, capillaries, and venule show diffuse inflammatory and parasitized red cells. Interstitial oedema and hyaline membrane formation may also be seen [7, 8].

#### **8.5 Cardiovascular system**

Cardiovascular functioning is deranged with malarial infection, especially during the paroxysmal episode. There is peripheral vasodilation causing decreased blood pressure and postural hypotension, tachycardia, transient systolic murmur, muffled heart sound and occasional cardiac dilatation. In the patient with pre-existing cardiac dysfunction, malarial infection aggravates the dysfunction leading to fatal cardiac failure. The microcirculation changes involve myocardial capillaries congestion with lymphocytes, plasma cells and parasitized red cells. Pigment-laden macrophages are also seen microscopically [7, 8].

#### **8.6 Gastrointestinal system**

Malaria may manifest with abdominal symptoms such as nausea, vomiting, anorexia, abdominal distension and epigastric pain in the acute phase. Nausea and vomiting are usually central in origin. In addition, the patient may have watery diarrhea, mimicking gastroenteritis or cholera. Some patients may experience severe abdominal colic mimicking appendicitis and acute abdomen.

*P. falciparum* infection results in impaired splanchnic microcirculation, causing bowel ischemia, necrosis and ulceration. In addition, there is mucosal oedema which hampers absorption. These changes may also lead to the absorption of toxins and precipitate septic shock [7, 8].

#### **8.7 Kidneys**

Kidneys may be affected during malarial infection. Acute diffuse malarial nephritis rarely occurs with patients exhibiting hypertension, oedema and albuminuria. However, albuminuria alone is common during an acute attack. *P. malariae* may lead to nephrotic syndrome, described as Quartan malaria nephropathy. It is an immunemediated nephropathy characterized by oedema, hypertension and albuminuria, occurring weeks after infection [7, 8].

Severe disease with *P. falciparum* causes acute renal failure in 0.1–0.6% of the patients. Impairment in microcirculation associated with severe *P. falciparum* infection leads to anoxia, sebsequate glomeruli, and renal tubule necrosis. Disseminated intravascular coagulation may worsen acute renal failure.

#### **8.8 Central nervous system**

As mentioned, the CNS presentation is due to the dissemination of malarial parasites into the brain, paroxysmal fever and side effects of antimalarial drugs. The patient presents with headache, vomiting, delirium, anxiety and restlessness during fever paroxysms, with symptoms resolving once the fever normalizes. Chloroquine, quinine, mefloquine and halofantrine can lead to vertigo, restlessness, hallucinations, confusion, delirium, tinnitus, dizziness, convulsions and sometimes frank psychosis.

#### **Figure 1.**

*Liver and Spleen Malaria. Gross pathology of the liver and spleen At the top of this picture are the spleen (left) and liver (right) from an autopsy of a child. They show a normal spleen and liver appearance. At the bottom of the picture are the spleen and liver from the autopsy of a child who died of malaria, which are a darker color than the normal organs. The dark color comes from extreme congestion and heavy deposition of haemozoin. (From: P. falciparum malaria: liver and spleen. Welcome Collection. CC0 1.0 Universal).*

#### **Figure 2.**

*Gross pathology of cerebral malaria. The vast majority of cases, regardless of diagnosis, showed brain swelling with flattened gyri and narrowed sulci (A). In this example, the brain has the classic "slate gray" to "purple" appearance of CM which is possibly due to malaria pigment within vessels (A). The cerebellum had petechial hemorrhages in both the gray and white matter and thus, visible on the surface grossly (B). In the classic CM2 appearance, petechial hemorrhages are seen diffusely in the white matter throughout the brain (C). A higher magnification demonstrates the abrupt transition from white to gray matter and the lack of hemorrhages in the gray (D). From Milner et al. [9].*

#### *Pathology of Malaria DOI: http://dx.doi.org/10.5772/intechopen.110579*

Quinine is associated with hypoglycemic coma, while artemisinin cause brainstem dysfunction based on animal studies [7, 9].

*P. falciparum* in CNS causes cerebral hypoxia and anoxia through impairing microcirculation as the parasitized red cells have decreased deformability and increased cytoadherence, causing occlusion of microcirculation. Malarial encephalitis and meningoencephalitis arise due to cerebral anoxia, the development of malarial granulomas and punctate hemorrhages [9].

Macroscopically, the brain is usually edematous during the autopsy, appearing leaden or plum colored with a cut surface slatey gray hue. The sulci are narrowed, and the gyri flattened due to brain swelling. The small blood vessels are congested with parasitized red cells. Mature forms of parasites including schizonts are found in brain biopsies. The large vessels demonstrate evidence of margination, where the parasites are arranged in a layer along the endothelium. Despite this, the

#### **Figure 3.**

*Histological features of cerebral malaria. The abrupt transition from gray to white matter (A) and the presence of ring hemorrhages are demonstrated in this classic case of CM (CM2, 100X, H&E). The cerebellum (B) with ring hemorrhages in all levels including white and gray matter are shown (100x, H&E). Visibly congested blood vessels (C) even at low power may be the result of dense sequestration downstream; these vessels can contain both parasitized and uninfected red bloods (200X, H&E). The classic appearance of a ring hemorrhage with fibrin (D) is shown; these hemorrhages can also include pigmented parasites, free pigment, and admixed fibrin within the microvessel at the nexus of the lesion; uninfected erythrocytes constituting the surrounding hemorrhage are seen (400X, H&E). Two examples of sequestration showing predominantly early (less pigmented) parasites (E), and late stage (more pigmented) parasites (F) densely packing vessels (1000X, H&E). From Milner et al. [9].*

endothelium also shows pseudopodial projections, which may be in close apposition to the knobs on the surface of parasitized red cells. There is numerous petechial hemorrhage proximal to the occlusive plug of end arterioles in the white matter. The ring hemorrhage is diffuse in the brain with hemorrhage containing fibrin, pigmented parasites, free pigments and admixed fibrin. The uninfected red cells are seen in surrounding hemorrhage. Diirck's granulomata may be seen in areas of hemorrhage characterized by a small collection of microglial cells surrounding an area of demyelination. In addition, the brain demonstrates the abrupt transition from gray to white matter which, together with ring hemorrhage, are classical findings of cerebral malaria (**Figures 1**–**3**) [9].

### **Author details**

Pier Paolo Piccaluga1,2,3,4\* and Wanyonyi Ignatius4

1 Biobank of Research, IRCCS S. Orsola-Malpighi Academic Hospital, Bologna, Italy

2 Department of Experimental, Diagnostic, and Specialty Medicine, University of Bologna School of Medicine, Bologna, Italy

3 Institute of Hematology and Medical Oncology "L. and A. Seràgnoli", Italy

4 Department of Pathology, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

\*Address all correspondence to: pierpaolo.piccaluga@unibo.it

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[3] Balla G, Vercellotti GM, Muller-Eberhard U, Eaton J, Jacob HS. Exposure of endothelial cells to free heme potentiates damage mediated by granulocytes and toxic oxygen species. Laboratory Investigation; A Journal of Technical Methods and Pathology. 1991;**64**(5):648-655

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[5] Thapar RK, Saxena A, Devgan A. Congenital malaria. Medical Journal, Armed Forces India. 2008;**64**(2):185

[6] Lesko CR, Arguin PM, Newman RD. Congenital malaria in the United States: A review of cases from 1966 to 2005. Archives of Pediatrics & Adolescent Medicine. 2007;**161**(11):1062-1067

[7] Mackintosh CL, Beeson JG, Marsh K. Clinical features and pathogenesis of severe malaria. Trends Parasitol. Dec 2004;**20**(12):597-603. DOI: 10.1016/j. pt.2004.09.006. PMID: 15522670

[8] Rowe JA, Claessens A, Corrigan RA, Arman M. Adhesion of *Plasmodium falciparum*-infected erythrocytes to human cells: Molecular mechanisms and therapeutic implications. Expert Reviews in Molecular Medicine. 2009;**11**:e16. DOI: 10.1017/S1462399409001082

[9] Milner DA Jr, Whitten RO, Kamiza S, Carr R, Liomba G, Dzamalala C, et al. The systemic pathology of cerebral malaria in African children. Frontiers in Cellular and Infection Microbiology. 2014;**4**:104. DOI: 10.3389/fcimb.2014. 00104

#### **Chapter 4**

## The Submicroscopic *Plasmodium falciparum* Malaria in Sub-Saharan Africa – Current Understanding of the Host Immune System and New Perspectives

*Kwame Kumi Asare*

#### **Abstract**

The bottlenecks in malaria infections affect malaria control and eradication programs. The gaps in the relationships between stages specific parasites molecules and their effects in the various stages of malaria development are unknown. The challenge hampers the wholesome understanding of policies and programs implemented to control and eliminate malaria infections in the endemic areas. Submicroscopic malaria and its transmission dynamisms are of interest in malaria control programs. The role of various stages of natural protective immunity in submicroscopic malaria infections and the insight into the collaborative role of antibodies from antigens for maintaining lower and submicroscopic malaria could provide a relevant guideline for vaccine developments. The chapter discusses the roles of mosquito and malaria antibodies in maintaining submicroscopic *P. falciparum* infection and its transmission potentials in malaria-endemic areas and the new perspectives on the inter-relatedness of stagespecific antibodies to improve malaria control programs in Sub-Saharan Africa.

**Keywords:** Malaria, Plasmodium falciparum, Anopheles gambiae, stage specific immunity, Submicroscopic infection, gSG6-P1, PfCSP, PfEBA175, Pfs230

#### **1. Introduction**

Malaria is a protozoan disease of global health importance. Malaria affects about 241 million people with 627,000 deaths [1–3]. The majority of malaria cases and deaths occur in Sub-Saharan Africa [4]. The malaria eradication campaign has marked reductions in malaria morbidity and mortality [2, 5]. The eradication is challenged by the recent stall gains achieved in malaria control programs [6, 7].

Malaria infection is initiated and transmitted during a blood meal by an infected female Anopheles mosquito [8]. The mosquito injects saliva containing sporozoites for liver invasion and liver-stage merozoites development and subsequent rapturing and infection of the red blood cells (RBCs) [9, 10]. The blood-stage merozoites

undergo several rounds of replication and reinvasion of RBCs; a small proportion of the merozoites form the sexual stage gametocytes (male and female gametocytes) are required for malaria transmission from humans to mosquitoes to complete the parasite life cycle (**Figure 1**) [11–15].

The malaria parasites employ several parasite antigens (parasite proteins) for initial recognition and reversible attachment, reorientation and irreversible attachment, and final tissue or cell invasion and development [16–18]. The circumsporozoite protein (CSP), erythrocyte membrane protein 1 (PfEMP-1), repetitive interspersed family proteins (RIFINs) and ookinete surface antigen (Pfs230) are exposed to the host immune system to induce complex immune response at each stage of the parasite development [19–21]. Other factors such as malaria control strategies (treatment, vector control strategies), malaria distribution, epidemiological factors and chronic malaria infections account for low parasite density and submicroscopic malaria infections in malaria-endemic populations [22–27].

#### **Figure 1.**

*Life cycle of malaria parasite. In the pre-erythrocytic stage:* Plasmodium falciparum*-infected Anopheles mosquito bites a human and transmits sporozoites into the bloodstream. Sporozoites migrate through the dermis and bloodstream to invade hepatocytes; divide to form multinucleated schizonts. Erythrocytic phase: The liver stage schizonts rupture and release merozoites into the circulation and subsequent invasion into red blood cells. The merozoites mature from ring forms to trophozoites to multinucleated schizonts. Some merozoites differentiate into male or female gametocytes. Anopheles mosquito ingests gametocytes into the midgut, where it develops into sporozoites.*

*The Submicroscopic* Plasmodium falciparum *Malaria in Sub-Saharan Africa – Current… DOI: http://dx.doi.org/10.5772/intechopen.105086*

#### **2. The submicroscopic** *Plasmodium falciparum* **malaria**

Submicroscopic malaria is *Plasmodium* parasite infections below the detection limit of microscopy [28]. Large numbers of falciparum malaria go undetected by the current point-of-care diagnostic (POC) techniques due to the low sensitivity of microscopy and rapid diagnostic test (RDTs) [28–30].

Human malaria, *Plasmodium falciparum*, *Plasmodium vivax*, *Plasmodium malariae*, *Plasmodium ovale* and *Plasmodium knowlesi*; *Plasmodium falciparum* malaria is the most virulent species that subvert the host physiology, results in severe complications such as cerebral malaria, severe anaemia, and respiratory distress and causes most deaths [31]. However, estimating the malaria burden and transmission intensity caused *by P. falciparum* in highly endemic Sub-Saharan Africa is essential for malaria control and eradication efforts [32, 33].

The undetected malaria infections serve as a reservoir and prominent contributor to malaria transmission [34, 35]. The underlying mechanisms for residual submicroscopic parasitaemia or gametocytaemia and its transmission dynamics are poorly understood. Thus, submicroscopic *P. falciparum* gametocyte densities could result in mosquito infections and maintain malaria transmission in the endemic communities [35]. A recent study showed that diagnosed and treated malaria-infected individuals had higher submicroscopic prevalence 30 days after treatment [36]. The low *P. falciparum* malaria infections in endemic Sub-Saharan Africa are associated with partial humoral immunity in the population [37–39].

#### **3. Humoral antibody immunity to malaria**

Acquisition of antibodies in *P. falciparum* malaria infection reduces parasitaemia and the risk of severe and mild malaria cases [40, 41]. The mechanisms such as antibody-dependent cellular inhibition (ADCI) and antibody-dependent cellular

cytotoxicity (ADCC) play a role in malaria infection [42]. The hyperimmune immunoglobulin G (IgG) and the predominant isotypes (IgG1, IgG2 and IgG3) in malaria-endemic areas have lower parasitaemia and lower risk of malaria attack [43–45]. The natural immunity to malaria develops gradually and is hyper-polymorphic [44]. The functionality of parasite antigen and antibodies interactions determines the quality of protection against malaria [46]. The robust immune response elicited at the various developmental stages of *P. falciparum* is poorly understood. Several of the parasite variant antigens are vaccine candidates. Thus, the IgG antibodies reduce parasitaemia and clinical symptoms in malaria-endemic communities. The insight into the collaborative role of antibodies from various antigens for maintaining lower and submicroscopic malaria could provide a relevant guideline for vaccine developments (**Figure 2**).

#### **4. Pre-erythrocytic antibodies in** *P. falciparum* **endemic areas**

#### **4.1** *An. gambiae* **salivary gland protein-6 peptide 1 (gSG6-P1) antibody**

The active micro-components in Anopheles saliva prevent blood coagulation, induce complement activation, elicit and modify the immune response to influence the vector-malaria transmission [47]. *An. gambiae* salivary gland protein 6 (gSG6) has recently attracted scientific interest as a biomarker to estimate the intensity of exposure to mosquito bites and the risk of malaria infection [48–50]. The IgG immune antibody against gSG6-P1 has high sensitivity and specificity to determine the entomological inoculation rate (EIR) and overcome some of the challenges associated with EIR [51, 52]. The IgG antibody is an essential estimation of the risk of malaria transmission in the endemic areas [53]. Aside using gSG6-P1 IgG antibodies to estimate the risk of infection or vector-malaria transmission, does gSG6-P1 play any role in *Plasmodium falciparum* invasion and development? A question that will prove vital to the control efforts of malaria if rightly answered.

#### **4.2 The circumsporozoite protein (CSP) antibody**

The female Anopheles mosquito inoculates saliva containing sporozoites which migrate from the dermis into the hepatocytes [53, 54]. *P. falciparum* sporozoites are highly susceptible to the induced antibody against the most abundant sporozoite surface protein, the CSP [55–57]. The prolonged exposure of anti-CSP against sporozoites severely affects the development of the liver stage infections, thus reducing the chances of successful blood-stage of *P. falciparum* infections [56]. Currently, CSP based RTS, S anti-malaria vaccine is the only malaria vaccine that has been rolled out with efficacy of approximately 78% [58, 59]. These favourable safety profiles and protection-inducing immunity do not interfere with the general immune response mechanisms in a paediatric population. Although the RTS, S malaria vaccine has achieved a milestone, it is not without several concerns in high parasitaemia levels (>5000 parasites/ul) among subjects considered to be protected [60]. The inexplicable variations in the protection-induced immune response by CSP antibody in asymptomatic and symptomatic malaria require further understanding.

#### **5. The erythrocytic stage antibodies in** *P. falciparum* **endemic areas**

#### **5.1** *P. falciparum* **erythrocyte binding antigen-175**

The merozoites invade red blood cells through parasite ligands-host receptors interactions to facilitate initial attachment, apical reorientation, tight junction formation and final entry into the red blood cells [61]. The exposed parasite ligands to the host immune system induce antibodies against parasite ligands [62–64]. Naturally acquired PfEBA-175 antibodies bind *P. falciparum* erythrocyte binding antigen-175 (PfEBA-175) to prevent red blood cell invasion [65, 66]. The PfEBA-175 engagement with Glycophorin A (GPA) remains one of the major pathways for red blood cells invasion and establishment of clinical disease in malaria [67, 68]. Therefore, blocking this critical pathway of parasite invasion into RBCs has become an approach to controlling malaria. The PfEBA-175-RII IgG-mediated antibody offers protection by binding to the functional R217 loop region of the PfEBA-175 inhibiting *P. falciparum* invasion of RBCs [69, 70]. Although the malaria parasite uses PfEBA-175-GPA invasion pathway, it can also adopt alternative ways to invade the PfEBA-175 antibody RBCs invasion blockage [71]. The high levels of PfEBA-175 antibody among the population in malaria-endemic areas and high-level submicroscopic malaria infection suggest a role of the PfEBA-175-GPA pathway in malaria infections [66, 72]. This is backed by the high relative avidity of IgG antibodies against EBA175RIII-V in low malaria-endemic communities [66, 73, 74].

#### **5.2 Pfs230 gametocyte antibody**

The sustained reservoirs of submicroscopic or asymptomatic malaria infections are not well understood [75–77]. The *P. falciparum* mature stage gametocytes could persist at submicroscopic levels for effective transmission to the mosquito [78, 79]. The exposure of gametocyte antigen Pfs230 expressed on the mature gametocyte surface and the gamete surface induces an antibody response in the human host [80–82]. There is a natural antibody response against the Pfs230 in malaria-endemic populations [83, 84]. The Pfs230 antibody blocks the fusion of gametes in the mosquito and prevents transmission [85, 86]. The anti-gametocytes IgG antibodies against Pfs230 reduce oocyst intensity by 55–70% and up to 44% reduction in proportions of infected mosquitoes [87]. The Pfs230 antibody has been associated with recent and concurrent high-density gametocyte exposure and impacts the dynamism of transmission by significantly reducing the infectiousness of high gametocyte density infections.

#### **5.3 The missing links and the inter-relationships between stage-specific**  *P. falciparum* **antibodies**

There are several missing links in the association between the effects of various stage-specific antibodies and how they subsequently affect the development of the other parasites' stages [88]. For instance, there are several unknown points in the sexual stage immunity which could aid in the reduction of transmission to mosquitoes and elimination of malaria; such as how the Pfs230 antibody develops in either microscopic or submicroscopic gametocyte carriage, and the subsequent impacts on parasite development in mosquitoes and reinfection of the human host is still obscure. The current

#### **Figure 3.**

*A strong negative correlation between PfCSP IgG antibody and anopheles gSG6-P1 IgG antibody. A. Correlation between IgG antibodies concentration of PfCSP and gSG6-P1. B. Correlation between IgG antibodies concentration of Pfs230 and gSG6-P1. C. Correlation between IgG antibodies concentration of PfEBA-175 and gSG6-P1. D. Correlation between IgG antibodies concentration of PfEBA-175 and Pfs230. The statistical analysis was performed using spearman correlation. The IgG antibody concentration of PfCSP and IgG antibody concentration of anopheles gSG6-P1 showed a significant negative correction r = −0.1571 (−0.2704 to −0.03944), p = 0.0073. The observed strong negative correlation between IgG antibodies of PfCSP and gSG6-P1 is an indication that salivary gland proteins may play a role in the invasion and development of malaria sporozoites in the liver. Further studies are required to ascertain the specific role of gSG6-P1 in liver-stage malaria infection.*

knowledge of the functionality of the Pfs230 antibody has shown to be dependent on density [89–92]. Also, factors that reduce or prevent blood-stage parasitaemia development have the potential to reduce gametocyte development and reduce malaria transmission [93, 94]. Other immune factors such as antibodies against Anopheles salivary proteins and their relationship to the development of the liver-stage malaria parasites, erythrocytic phase development or sexual stage development are poorly understood.

However, current study has revealed a strong negative correlation between IgG antibodies of PfCSP and gSG6-P1. There is no evidence of an association between IgG antibodies of gSG6-P1, PfEBA175, Pfs230 or IgG antibodies of PfEBA175 and IgG antibodies of Pfs230 (**Figure 3**). Although the exact role of gSG6-P1 is unknown, this finding suggests either gSG6-P1 is involved in sporozoites invasion or the liver stage development.

#### **6. The new perspectives**

The salivary gland molecules may play a role in sporozoite survival and invasion of hepatocytes. However, there is no identified role of salivary gland molecules on sporozoite migration through the skin to the liver or development in the liver.

Previous studies have reported a putative mucin-like protein, the anti-platelet protein, the long-form D7 salivary protein, the putative gVAG protein precursor,

*The Submicroscopic* Plasmodium falciparum *Malaria in Sub-Saharan Africa – Current… DOI: http://dx.doi.org/10.5772/intechopen.105086*

the D7-related 3.2 protein, gSG7 salivary proteins, and the gSG6 protein may be involved in sporozoite maturation and transmission [95]. The authors of this study could not establish the direct association or role of the individual salivary gland proteins in the development of malaria parasites.

Also, the salivary gland molecules and underlying mechanisms for sporozoites recognition and invasion of the mosquito salivary gland are poorly understood. Until recently, the unique sporozoite ligand-salivary gland receptor interaction and molecules involved in triggering the mosquito salivary gland invasion were unknown. The discovery of *Anopheles* salivary gland protein, the CSP-binding protein (CSPBP) and its role in the mosquito salivary glands invasion has opened a new understanding of the invasion mechanism of sporozoites [96]. The antibodies raised against the CSPBP reduced sporozoites load by 25% and 90% in 14 and 18 days after the infected blood meal by mosquitoes, respectively [96].

In the hepatocyte invasion, region II of the C-terminal region of the sporozoite CSP attaches to the liver cells through the heparan sulfates proteoglycans (HSPG) [97, 98]. The conformations of CSP play a role in the sporozoite migration through different tissues in mosquito and human hosts [99]. However, factors involved in the sporozoite migration, invasion and development in hepatocytes remain largely unknown. The mosquitoes' gamma interferon-inducible thiol reductase (mosGILT) negatively influences sporozoite speed and cell traversal movement in the host [100–102]. Thus, the mosquito salivary gland proteins could either enhance or reduce the transmission of malaria parasites. The interaction of the PfCSP and gSG6-P1 proteins may play an essential role in sporozoites development in the human host. This finding is new information that requires further research into the role of gSG6-P1 in the development of pre-erythrocytic phase parasites.

In conclusion, submicroscopic malaria infection and transmission is one of the bottlenecks in malaria control and elimination strategy in malaria-endemic areas. The inoculation of the female Anopheles salivary gland content and sporozoites initiate the entire malaria human host developmental process. However, there are several missing links to how vector and parasite molecules influence the development and establishment of malaria infection. The observed strong negative correlation between IgG antibodies of PfCSP and gSG6-P1 is an indication that salivary gland proteins may play a role in the invasion and development of malaria sporozoites in the liver. Further studies are required to ascertain the specific role of gSG6-P1 in liver-stage malaria infection.

*Malaria - Recent Advances and New Perspectives*

#### **Author details**

Kwame Kumi Asare

Department of Biomedical Science, University of Cape Coast, School of Allied Health Sciences, College of Allied Health Sciences, Cape Coast, Ghana

\*Address all correspondence to: kwamsare@hotmail.com; kwame.asare@ucc.edu.gh

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*The Submicroscopic* Plasmodium falciparum *Malaria in Sub-Saharan Africa – Current… DOI: http://dx.doi.org/10.5772/intechopen.105086*

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#### **Chapter 5**

## Estimation of Malaria Mortality in Developing Countries

*Sepiribo Lucky Braide*

#### **Abstract**

This chapter considered monitoring human health condition as vital variable for well-being of man/society required input data for effective daily planning. Researchers have contributed to prediction of incidence/recovery rate for malaria mortality. Modified state-estimation model based (matrix-formulation, weighted sum of squares of errors) was applied. The instrument (sphygmomanometer, etc.) is manipulated for study under investigation to examined existing state of system. Four (4) measurements data were analyzed from different geographical locations for patients with malaria endemic cases. Physician measurement data are implemented into modified state-estimation equations to estimate degree of error (s) to classify as bad measurement. Results shows bad data estimation attributed to poor instrument calibration, aging, and poor physician measurement. These reveal discrepancies between actual (true-measurement) and patient-physical measurements. Four vital measurements include blood pressure (Bp), blood sugar level (BSL), body temperature (BT), and Plasmodium ViVax with relied validation test following chi-square distribution for 2-degree freedom with 99% significance level suspected as error measurement. Model-matrix coded in MATLAB gives state-estimation results ð Þ *x***<sup>1</sup>** ¼ **8***:***5225** *and x***<sup>2</sup>** ¼ **13***:***235** , indicating strong variation between actual and physical measurements for some patients having low pulse rate under the measurement of blood pressure (Bp). Essentially, physicians' measurements must be revalidated for accuracy before drugs prescription/administration to avoid under- or over-dose since patients' body chemistry varies significantly for different persons.

**Keywords:** measurement, matrix formulations, malaria, mortality, state-estimates, distribution

#### **1. Introduction**

#### **1.1 Improved state estimation model techniques and governing equations**

The analysis and investigation of basic human body diagnosis uses the vital statistics variables of physical measurement because it deals with scientific application to the life history of communities and nation. That is historical development which is a

function of human society can be numerically be expressed qualitatively or quantitatively. The purpose of this study is to find out the changing composition of communities or nations with reference to sex, age, education, economic and civic status which is considered under study [1].

Evidently the state estimation model considered the numerical records of marriage, births, sickness and deaths by which the health and growth of community may be studied. The vital statistics variable provides the results of the biometry which deals with data and the laws of human mortality, morbidity and demography [2].

This technical contents considered the data or the laws which explain the phenomenon of birth, deaths, health and longevity, composition and concentration etc.

This study can also be extended to solve information particularly on fertility, mortality, maternity, urban density which is indispensable for planning and evaluation of the schemes of health, family planning and other vital amenities for purpose of efficient planning, evaluation and analysis of various economic and social policies [3].

Importantly, data collected on life expectancy at various age levels is helpful in actual calculations of risk on life policies, this means that life table called the biometer of public health and longivity helps in fixing the life insurance premium rates at various age level.

The modified state estimation model described the estimation strategy using different mathematical matrics operations to estimates the physical measurements carried out by a physician/medical doctors, on the view to determine the estimated physical measurements for malaria mortality in a developing countries, the question is how true? is the physical measurement taken by the medical doctor for purpose of placing drug dispensation and administration evidently, research studies has provided with results the measurement equations characterizing the meter readings with the view to the addition of error terms to the system model for purpose of validating physical measurement by medical practitioners [4].

Several measurement instruments and devices are used in the measurements of vital statistics variable (state-variables) of the human life expectancy. We have the sphygmomanotreter (device that measures blood pressure) thermometer (device that measures temperature), (blood sugar level), (plasmodium viviax: parasite for malaria) etc. [5].

The physical measurements data are collected from patients in the health/clinic and hospital which are not always true to some extend, the acquired data from the measurements device/instruments contains inaccuracies which is unavoidable since physical measurements cannot be entirely free from random errors or noise. These errors can be quantified in a statistical sense and the estimated values of the quantities being measured are then either accepted as reasonable or not, if certain measurement of accuracy are exceeded because of noise the true values of physical quantities are never known and we have to consider how to calculate the best possible estimates of the unknown quantities [6].

Incidentally, the techniques of least square is often used to best fit measurement data relating to two or more quantities. However, modified equations are developed and formulated to characterize measurements of the physical data collected or acquired from patients for purpose of analysis with the integration of errors terms in the system model. The best estimates are choosen which minimizes the weighted sum of squares of the measurement errors [7].

#### **2. Background of study**

State estimation techniques is the process of determining a set of values for a set of unknown systems (Engineering systems, human being systems, medical systems, biological systems, machine/instruments system etc). State variables are based on certain criterion making use of physical measurements made which are not precise due to inherent error associated with controllers/transducers and sometimes creates redundancy of measurement that does not require assessment of the true-value of the systems state variables [8]. Statistical methods are also used to estimate the true value of the system state variables with the minimum number of variables required to analyze the system in all aspects. The measurements taken by a medical practitioner are required to meet the basic records for a given circumstances intended to address particularly to the engineering systems, medical system, biological systems and related systems in order to provide good quality of reliable physical measurements for system variables [9]. Evidently, the necessary conditions for vital statistics variables analysis for administering drugs dispension to patients ensure and declared measurements records conformity to calculation of the measuring instruments, precision and tolerance for purpose of standard practice for world health organization (WHO) [10].

Importantly, if the measurement instruments introduces errors while undertaking details information variables statistics about patients status, this may results into wrong presentation of recorded data required for prescription/dispensing of drugs this will eventually results into gradual breakdown of human body system, immunity, antibodies vital tissues and sensitive organs like liver, kidney etc. in a colossal decay condition that may leads to early mortality for associated emergence of infected and transmitted diseases [11].

Essentially for purpose of good working condition of human lives, standard practice for healthy living, is strongly monitored in order to regularly checked for recalibration of measuring instruments, for purpose of high precision, tolerance and accuracy [12].

#### **3. Materials and method**

#### **3.1 Model formulation of modified state estimation equations**

The measurement set used by medical practioners consists of thermometer reading ð Þ *z*<sup>1</sup> , sphygmomanometer ð Þ *z*<sup>2</sup> , Blood-sugar level ð Þ *z*<sup>3</sup> and plasmodiums vivax parasite transmission measurement ð Þ *z*<sup>4</sup> etc. Where *z*1, *z*2, *z*<sup>3</sup> *and z*4, are mathematical measurement parameter sowing to the physical measurements/quantities respectively.

Similarly, the symbol *x* represents physical quantities being estimated.

Then the measurement equations that is characterizing the meter readings/measured terms are found by adding error terms (*e*) to the system model as:

$$Z = H\mathfrak{x} + \mathfrak{e} \tag{1}$$

*Z* : Measurement variables.

*H* : rectangular matrice operations.


The numerical coefficient are determined by the human body circulatory components and the error terms *e*1, *e*2, *e*<sup>3</sup> *and e*<sup>4</sup> which represents errors in measurement *z*1, *z*2, *z*<sup>3</sup> *and z*<sup>4</sup> and respectively.

If the errors terms *e*1, *e*2, *e*3, *e*<sup>4</sup> are zero, then it is an ideal condition which mean that the meter reading of the physical measurement taken by the vital statistics variable of the patients are exact and true measurements ð Þ*z* , which gives exact values of *χ*<sup>1</sup> *and χ*<sup>2</sup> *of* ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup> which could be determined.

From the relationship of eq. (1) it can be obtained as:

$$Z\_1 = h\_{11}\mathbf{x}\_1 + h\_{12}\mathbf{x}\_2 + e\_1 = Z\_{1,true} + e\_1 \tag{2}$$

$$Z\_2 = h\_{21}\varkappa\_1 + h\_{22}\varkappa\_2 + e\_2 = Z\_{2,true} + e\_2 \tag{3}$$

$$Z\_3 = h\_{31} \mathbb{1}\_1 + h\_{32} \mathbb{1}\_2 + e\_3 = Z\_{3,true} + e\_3 \tag{4}$$

$$Z\_4 = h\_{41} \mathbf{x}\_1 + h\_{42} \mathbf{x}\_2 + e\_4 = Z\_{4,true} + e\_4 \tag{5}$$

Where; *Zjtrue*: True value of the measured quantity *Zj:* Reformulating eq. (2)–(5) into vector form to obtained as;

$$
\begin{bmatrix} e\_1 \\ e\_2 \\ e\_3 \\ e\_4 \end{bmatrix} = \begin{bmatrix} z\_1 \\ z\_2 \\ z\_3 \\ z\_4 \\ z\_4 \end{bmatrix} - \begin{bmatrix} z\_{1\text{true}} \\ z\_{2\text{true}} \\ z\_{3\text{true}} \\ z\_{4\text{true}} \end{bmatrix} = \begin{bmatrix} z\_1 \\ z\_2 \\ z\_3 \\ z\_4 \end{bmatrix} - \begin{bmatrix} h\_{11} & h\_{12} \\ h\_{21} & h\_{22} \\ h\_{31} & h\_{32} \\ h\_{41} & h\_{42} \end{bmatrix} \begin{bmatrix} \chi\_1 \\ \chi\_2 \end{bmatrix} \tag{6}
$$

In compact form representation given as:

$$\mathbf{z} = \mathbf{z} - \mathbf{z}\_{\text{true}} = \mathbf{z} - \mathbf{H}\mathbf{x} \tag{7}$$

Which represents the error term between the actual measurement ð Þ*z :* and the true (but unknown) values *ztrue*<sup>≜</sup>*Hx* of the measurement quantities. The true values of *x*<sup>1</sup> *and x*<sup>2</sup> cannot be determined but we can calculate the estimates ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup>

Substituting these estimates ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup> ð Þ into eq. (6) gives the estimated values of the errors in the form as:

$$
\begin{bmatrix}
\hat{e}\_1 \\ \hat{e}\_2 \\ \hat{e}\_3 \\ \hat{e}\_4
\end{bmatrix} = \begin{bmatrix}
z\_1 \\ z\_2 \\ z\_3 \\ z\_4
\end{bmatrix} - \begin{bmatrix}
h\_{11} & h\_{12} \\ h\_{21} & h\_{22} \\ h\_{31} & h\_{32} \\ h\_{41} & h\_{42}
\end{bmatrix} \begin{bmatrix}
\hat{\alpha}\_1 \\ \hat{\alpha}\_2
\end{bmatrix} \tag{8}
$$

The quantitie are estimates of the corresponding quantities.

The vector, ^*e* which represents the differences between the actual measurement ð Þ*z* and their estimated values ^*ztrue*<sup>≜</sup>*Hx*, thus in compact form is given as:

$$
\hat{\mathbf{e}} = \mathbf{z} - \hat{\mathbf{z}} = \mathbf{z} - H\hat{\mathbf{x}} = \mathbf{e} - H(\hat{\mathbf{x}} - \mathbf{x})\tag{9}
$$

The criterion for calculating the estimates ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup> ð Þ can be determined as:

$$
\hat{\mathbf{e}} = \begin{bmatrix} \hat{\mathbf{e}}\_1 \hat{\mathbf{e}}\_2 \hat{\mathbf{e}}\_3 \hat{\mathbf{e}}\_4 \end{bmatrix}^T \tag{10}
$$

and

*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

$$\hat{\mathbf{z}} = \begin{bmatrix} \hat{\mathbf{z}}\_1 \hat{\mathbf{z}}\_2 \hat{\mathbf{z}}\_3 \hat{\mathbf{z}}\_4 \end{bmatrix}^T \tag{11}$$

Eq. (10) and (11) have to be computed for purpose of estimation analysis.

The techniques can be used to describe algebraic sum of errors to minimize positive and negative operations which may not be acceptable.

However, it is preferable to minimize the direct sum of squares of errors.

To ensure that measurement from meters or data collected of known greater accuracy are considered more favorably than less accurate measurements instrument/ device, that is in each term the sum of squares is multipled by an appropriate weighting factor (w) to give the objective function as:

$$f = \sum\_{j=1}^{4} w\_j e\_j^2 = w\_1 e\_1^2 + w\_2 e\_2^2 + w\_3 e\_3^2 + w\_4 e\_4^2 \tag{12}$$

We select the best estimates of the state variables as those values ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup> which cause the objectives functions ð Þ*f* to take on its minimum value.

In accordance to the application of necessary conditions for minimizing function ð Þ*f* , the estimates ^*χ*<sup>1</sup> *and* ^*χ*<sup>2</sup> are those values of *χ*<sup>1</sup> *and χ*<sup>2</sup> which satisfy the equations as;

$$\left|\frac{\partial f}{\partial \mathbf{x}\_{1}}\right| = 2\left[w\_{1}e\_{1}\frac{\partial e\_{1}}{\partial \mathbf{x}\_{1}} + w\_{2}e\_{2}\frac{\partial e\_{2}}{\partial \mathbf{x}\_{2}} + w\_{3}e\_{3}\frac{\partial e\_{3}}{\partial \mathbf{x}\_{3}} + w\_{4}e\_{4}\frac{\partial e\_{4}}{\partial \mathbf{x}\_{4}}\right]\Big| = \mathbf{0} \tag{13}$$

Similarly,

$$\frac{\partial \hat{f}}{\partial \mathbf{x}\_2} \bigg| = 2 \left[ w\_1 e\_1 \frac{\partial \mathbf{e}\_1}{\partial \mathbf{x}\_1} + w\_2 e\_2 \frac{\partial \mathbf{e}\_2}{\partial \mathbf{x}\_2} + w\_3 e\_3 \frac{\partial \mathbf{e}\_3}{\partial \mathbf{x}\_3} + w\_4 e\_4 \frac{\partial \mathbf{e}\_4}{\partial \mathbf{x}\_4} \right] \bigg| = \mathbf{0} \tag{14}$$

That is, the notation: j*x*^ ¼ 0 which, indicates that the equations have to be evaluated from the state estimates *x*^ ¼ ½ � *x*^1*x*^<sup>2</sup> *<sup>T</sup>* since the true values for the states variable are not known.

The unknown actual error *ej* � � are then replaced by estimated errors ^*ej* which can be calculated once the true estimate *x*^*<sup>i</sup>* are known.

Eq. (14) and (15) can be represented in vector form as:

$$
\begin{bmatrix}
\frac{\partial \mathbf{e}\_1}{\partial \mathbf{x}\_1} & \frac{\partial \mathbf{e}\_2}{\partial \mathbf{x}\_1} & \frac{\partial \mathbf{e}\_3}{\partial \mathbf{x}\_1} & \frac{\partial \mathbf{e}\_4}{\partial \mathbf{x}\_1} \\
\frac{\partial \mathbf{e}\_1}{\partial \mathbf{x}\_2} & \frac{\partial \mathbf{e}\_2}{\partial \mathbf{x}\_2} & \frac{\partial \mathbf{e}\_3}{\partial \mathbf{x}\_2} & \frac{\partial \mathbf{e}\_4}{\partial \mathbf{x}\_2}
\end{bmatrix}
\begin{bmatrix}
w\_1 & \mathbf{x} & \mathbf{x} & \mathbf{x} \\
\mathbf{x} & w\_2 & \mathbf{x} & \mathbf{x} \\
\mathbf{x} & \mathbf{x} & w\_3 & \mathbf{x} \\
\mathbf{x} & \mathbf{x} & \mathbf{x} & w\_4
\end{bmatrix}
\begin{bmatrix}
e\_1 \\ e\_2 \\ e\_3 \\ e\_4
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \tag{15}
$$

Where w is the diagonal matrix of weighting factor which have special significance to the accuracy of measurement.

The partial derivatives for substitution in eq. 15 are found from eqs. (2) through (5) to be constant given by the elements (H) given as:

$$
\begin{bmatrix} h\_{11} & h\_{21} & h\_{31} & h\_{41} \\ h\_{12} & h\_{22} & h\_{32} & h\_{42} \end{bmatrix} \begin{bmatrix} w\_1 & \times & \times & \times \\ \times & w\_2 & \times & \times \\ \times & \times & w\_3 & \times \\ \times & \times & \times & w\_4 \end{bmatrix} \begin{bmatrix} \hat{e}\_1 \\ \hat{e}\_2 \\ \hat{e}\_3 \\ \hat{e}\_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \tag{16}
$$

Using the operation of the compact notation process of eq. (9) can be applied to also represent eq. (17) as:

$$H^T W \hat{e} = H^T W (z - H\hat{\mathbf{x}}) = \mathbf{0} \tag{17}$$

Multiplying through this eq. (17) and solving for the estimates (*x*^Þ as:

$$
\hat{\mathfrak{X}} = \begin{bmatrix} \hat{\mathfrak{x}}\_1 \\ \hat{\mathfrak{x}}\_2 \end{bmatrix} = \underbrace{\left(H^T \mathbf{W} \mathbf{H}\right)^{-1}}\_{\mathbf{G}} = H^T \mathbf{W} \mathbf{Z} = \mathbf{G}^{-1} H^T \mathbf{W} \mathbf{Z} \tag{18}
$$

Where *x*^<sup>1</sup> *and x*^<sup>2</sup> are the weighted least square estimates of the state variables, It is the rectangular matrix component

The symmetrical matrix (called the gain-matrix, G) must be inverted as a single entity to give as:

$$\mathbf{G}^{-1} = \left(\mathbf{H}^T \mathbf{W} \mathbf{H}\right)^{-1} \tag{19}$$

Which is also symmetrical in the mathematical operations.

Essentially, the weighted least-squares procedure are required to estimates *x*^*<sup>i</sup>* how close to the true value *xi* of the state variables,

The expression for the difference ð Þ *xi* � *x*^*<sup>i</sup>* is found my substituting for;

$$Z = H\mathfrak{x} + e \text{ to obtain as}\tag{20}$$

$$\hat{\boldsymbol{x}} = \boldsymbol{G}^{-1} \boldsymbol{H}^{T} \boldsymbol{W} (\boldsymbol{H} \boldsymbol{x} + \boldsymbol{e}) = \boldsymbol{G}^{-1} \underbrace{\left(\boldsymbol{H}^{T} \boldsymbol{W} \boldsymbol{H}\right) \boldsymbol{x}}\_{\boldsymbol{G}} = \boldsymbol{G}^{-1} \boldsymbol{H}^{T} \boldsymbol{W} \boldsymbol{e} \tag{21}$$

It is an important consideration to check the useful operation of the dimensions of each term in the matrix product of eq. (21) which is an important idea for developing properties of the weighted least-square estimations. In this analysis we have the matrix operation: *G*�<sup>1</sup> *<sup>H</sup>TW* which as an overall operation of row � column dimensions of 2 � 4, which means that any one or more of the four errors ð Þ *e*1, *e*2, *e*3, *e*<sup>4</sup> an influence the difference between each state estimates.

The weighted least squares calculations spread the effects of the errors in any one measurements to some or all other estimates the characteristics is the basis for detecting bad data measurement.

In a similar manner we can compare the calculated values of ^*z* ¼ *Hx*^ of the measured quantities with their actual measurements (z) by substituting for *x*^ ¼ *x* from eq. (21) into eq. (7) to obtain as:

$$\hat{\mathbf{e}} = \mathbf{z} - \hat{\mathbf{z}} = \mathbf{e} - \mathbf{H} \mathbf{G}^{-1} \mathbf{H}^{T} \mathbf{W}\_{\varepsilon} = \left[ \mathbf{I} - \mathbf{H} \mathbf{G}^{-1} \mathbf{H}^{T} \mathbf{W} \right]\_{\varepsilon} \tag{22}$$

Where; I: unit or identity matrix.

#### **4. Statistical analysis, errors and estimates**

In reality we know absolutely true state of a physical operating system. Great care is taken to ensure accuracy. Unavoidable random noise enters into the measurements process to distort more or less the physical results. Repeated measurements of the

same quantity under careful controlled conditions reveals certain statistical properties from which the true value can be estimated [13].

If the measured values are plotted as a function of their relative frequency of occurrence, a histogram is obtained to which a continues curve can be fitted as the number of measurement increases theoretically to any number [14].

The continuous curve commonly encountered is a ball-shaped function *p*(*z*). This is the Gaussian or normal probability density function gives as:

$$p(\mathbf{z}) = \frac{1}{5\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\mathbf{z}-\mu}{\delta}\right)^2} \tag{23}$$

The probability that z-takes on values between the point a and b in **Figure 1** is the shaded area given as;

$$pr = (a < z < b) = \int\_{a}^{b} p(z)dz = \frac{1}{\sigma\sqrt{2\pi}} \int\_{a}^{b} e^{-\frac{1}{2}} = \frac{1}{\sigma\sqrt{2\pi}}\tag{24}$$

The total area under the curve p(z) between 0 and equals to 1. The value of z is certain with probability equal to 1 or 100% [15].

The Gaussian distribution plays a very important role in the measurement statistic because the probability function density cannot be directly integrated. For transfer from z to y using change of variable formula given as (**Tables 1**–**3**) (**Figure 2**):

**Figure 1.** *Gaussian normal distribution.*

#### *Malaria - Recent Advances and New Perspectives*


#### **Table 1.**

*Normal chi-square distribution table (values of area α to the right of χ*<sup>2</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>χ</sup>*<sup>2</sup> *<sup>k</sup>*,*<sup>α</sup>.*


*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*


#### **Table 2.**

*Gaussian distribution for chi-square distribution.*


#### **Table 3.**

*Data collected from geographical location about laboratory confirmed cases.*

#### **5. Results and discussion**

The data collected from different geographical location for patients by a physician on malaria epidemic cases were estimated using the developed modified state estimation model for malaria mortality. The measurement statistics are *<sup>z</sup>* : *<sup>z</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> <sup>¼</sup> <sup>120</sup> <sup>80</sup> *mmHg* ,

**Figure 2.** *Gaussian distribution skewed to the right.*

#### **Figure 3.**

*A bar chart showing clinical/laboratory confirmed cases.*

*<sup>z</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> <sup>¼</sup> <sup>120</sup> <sup>80</sup> *mmHg* , *<sup>z</sup>*<sup>3</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>8</sup> <sup>¼</sup> <sup>140</sup> <sup>80</sup> *mmHg* , *<sup>z</sup>*<sup>4</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>9</sup> <sup>¼</sup> <sup>150</sup> <sup>80</sup> *mmHg* while the estimated measurement the model ^*z* : ð Þ ^*z*<sup>1</sup> ¼ 1*:*4759 ,ð Þ ^*z*<sup>2</sup> ¼ 1*:*4759 , and ð Þ ^*z*<sup>4</sup> ¼ 1*:*87295 . The state estimates of the body blood pressure were determined as *x*1,*x*<sup>2</sup> : 8*:*5225 *and* 13*:*235 for the respective patients under study, which indicate low-pulse rate or heart rate that required attention, which is also agree with the validated results of the

coded matrix laboratory (matlab results) (**Figure 3**).

#### **5.1 Expected values of measurements**

We assume that the noise terms *e*1, *e*2, *e*3, *and e*<sup>4</sup> are independent Gaussian random variables with zero means and the respective variances *σ*<sup>2</sup> 1, *σ*<sup>2</sup> 2, *σ*<sup>2</sup> <sup>3</sup> *and σ*<sup>2</sup> <sup>4</sup> of the physical *Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

#### **Figure 4.**

*A chart showing positive prediction for clinical/laboratory cases.*

#### **Figure 5.**

*A line chart showing distribution of clinical/laboratory cases.*

measurement are considered (**Figure 4**). Two variables are termed independent when *E eiej* � � <sup>¼</sup> <sup>0</sup> *for i* 6¼ *<sup>j</sup>:* The zero mean assumption implies that the error in each measurement has equal probability of taking on a positive and negative values of a given magnitudes (**Figure 5**).

The analysis of the vector ð Þ*<sup>e</sup>* and its transpose *<sup>e</sup><sup>T</sup>* <sup>¼</sup> ½ � *<sup>e</sup>*1*e*2*e*3*e*<sup>4</sup> can be represented as (**Figure 6**):

$$\begin{aligned} e^{T} = \begin{bmatrix} e\_1 \\ e\_2 \\ e\_3 \\ e\_4 \end{bmatrix} \begin{bmatrix} e\_1 e\_2 e\_3 e\_4 \end{bmatrix} = \begin{bmatrix} e\_1^2 & e\_1 e\_2 & e\_1 e\_3 & e\_1 e\_4 \\ e\_2 e\_1 & e\_2^2 & e\_2 e\_3 & e\_2 e\_4 \\ e\_3 e\_1 & e\_3 e\_2 & e\_3^2 & e\_3 e\_4 \\ e\_4 e\_1 & e\_4 e\_2 & e\_4 e\_3 & e\_4^2 \end{bmatrix} \tag{26}$$

Analysis consideration:

*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

#### **Figure 6.**

*Algorithms (flow-chart for modified state estimation model for malaria mortality showing different measurement system plan.*


$$E\left[e e^{T}\right] = R = \begin{bmatrix} E\left[e\_1^2\right] & \cdot & \cdot & \cdot \\ \cdot & E\left[e\_2^2\right] & \cdot & \cdot \\ \cdot & \cdot & E\left[e\_3^2\right] & \cdot \\ \cdot & \cdot & \cdot & E\left[e\_4^2\right] \end{bmatrix} \tag{27}$$

$$= \begin{bmatrix} \sigma\_1^2 & \cdot & \cdot & \cdot \\ \cdot & \sigma\_2^2 & \cdot & \cdot \\ \cdot & \cdot & \sigma\_3^2 & \cdot \\ \cdot & \cdot & \cdot & \sigma\_4^2 \end{bmatrix} \tag{28}$$

Evidently, the statistical properties of the weighted least-square estimation provides solution to the measurement ð Þ*z* , which is the sum of the Gaussian random variables ð Þ *e*<sup>1</sup> and the constant term ð Þ *h*11*x*<sup>1</sup> þ *h*12*x*<sup>2</sup> which represents the true values *Z*1*true* of *Z*1. The addition of the constant term to shift the curve of *e*<sup>1</sup> of the right by the amount of the true value.

Weighting Factor ð Þ *w*

The analysis of formulating the objective function ð Þ*f* , the preferential weighting ð Þ *wi* is given to be more accurate measurement by choosing the weight ð Þ *wi* as the reciprocal of the corresponding to the variance *δ*<sup>2</sup> *j* � �. This means that an errors of smaller variance have greater weight ð Þ *w* , hence we can specify the weighting matrix, ð Þ *w* as:

$$\mathcal{W} = \boldsymbol{R}^{-1} = \begin{bmatrix} \mathbb{1}\_{\delta\_1^2}^2 & \cdot & \cdot & \cdot \\ \cdot & \mathbb{1}\_{\delta\_2^2}^2 & \cdot & \cdot \\ \cdot & \cdot & \mathbb{1}\_{\delta\_3^2}^2 & \cdot \\ \cdot & \cdot & \cdot & \mathbb{1}\_{\delta\_4^2}^2 \end{bmatrix} \tag{29}$$

The, gain matrix becomes

$$\mathbf{H}^{\prime}(\mathbf{G}) = H^{T} \mathbf{R}^{-1} \mathbf{H} \tag{30}$$

The weighted sum of square of error as the objective function ð Þ*f* , determination. The critical value of the statistics f can be determined using the tabulated values of *χ*2 *<sup>k</sup>*,*<sup>α</sup>* for a quantifiable level of significance, *k* : degree of freedom ð Þ *Nm* � *Ns* . That is the calculated value can be compared to the tabulated value to measure accuracy which is given as:

$$p\_r(f \le \chi^2\_{k,a}) = (1 - a) \tag{31}$$

The weighted sum of squares equation of errors with weight ð Þ *wi* choosen equals to the reciprocal of corresponding error *δ*<sup>2</sup> *<sup>j</sup>* variance which is given as:

$$\begin{split} f &= \sum\_{j=1}^{N} \frac{\sigma\_j^2}{\sigma\_j^2} = \frac{\begin{pmatrix} \mathbf{z}\_1 - h\_1(\mathbf{x}\_1, \ \mathbf{x}\_2) \end{pmatrix}^2}{\delta\_1^2} + \frac{\begin{pmatrix} \mathbf{z}\_2 - h\_2(\mathbf{x}\_1, \ \mathbf{x}\_2) \end{pmatrix}^2}{\delta\_2^2} + \frac{\begin{pmatrix} \mathbf{z}\_3 - h\_3(\mathbf{x}\_1, \ \mathbf{x}\_2) \end{pmatrix}^2}{\delta\_3^2} \\ &+ \frac{\begin{pmatrix} \mathbf{z}\_4 - h\_4(\mathbf{x}\_1, \ \mathbf{x}\_2) \end{pmatrix}^2}{\delta\_1^2} \end{split} \tag{32}$$

Where: *h*1, *h*2, *h*<sup>3</sup> and *h*<sup>4</sup> are function that expresses the measured quantities in terms of the state variable while *e*1, *e*2, *e*<sup>3</sup> *and e*<sup>4</sup> are Gaussian random variables terms. The true value of *χ*<sup>1</sup> *amd χ*<sup>2</sup> and not known and have to be estimated from measurement: *Z*1, *Z*2, *Z*<sup>3</sup> *and Z* respectively.

Chi-square test statistics (*f*), as a sum of squares of error terms for purpose of validation The analysis of physical measurement, the chi-square distribution test and validation are required to the check the presence of bad data. Then eliminate any bad data detected and recalculate subsequent resultant state estimation from the determined results.

The objective function (*f*) for chi-square distribution can be presented as:

$$\hat{f} = \sum\_{j=1}^{N} \frac{\hat{e}\_j^2}{\sigma\_j^2} = w\_1 \hat{e}\_1^2 + w\_2 \hat{e}\_2^2 + w\_3 \hat{e}\_3^2 + w\_4 \hat{e}\_4^2 \tag{33}$$

where;

*w* ¼ <sup>1</sup>*=σ*2

*σ*: variance of measurement error

*α*: quantifiable level of significance for accuracy

*f* ≤*χ*<sup>2</sup> *<sup>α</sup>*,*<sup>k</sup>*: for a quantifiable level of significant ð Þ *α* for a given degree of freedom ð Þ*k* . If *f* is greater than *χ*<sup>2</sup> *<sup>α</sup>*,*<sup>k</sup>* then there is at least suspected bad data measurements in the analysis which required attention.

This means that the suspected measurement instrument (z) must be recalculated and subjected to test statistics and required re-measurement of the vital statistics variables of the system.

This could be traceable to human error, instrument error due to poor calibration, aging of the instrument and temperature/humidity etc.

If *f* ≤*χ*<sup>2</sup> *<sup>α</sup>*,*<sup>k</sup>* is satisfied it is adequate for purpose of physical measurement accuracy, while comparing calculated and tabulated values for validation.

Standardized error estimates

The standardized error estimations can be determined using diagonal elements as:

$$\begin{aligned} \frac{\hat{\mathbf{e}}\_1}{\sqrt{\mathbf{R}\_{11}'}} &= \mathbf{Sde}\_1\\ \frac{\hat{e}\_2}{\sqrt{\mathbf{R}\_{22}'}} &= \mathbf{Sde}\_2\\ \frac{\hat{\mathbf{e}}\_3}{\sqrt{\mathbf{R}\_{33}'}} &= \mathbf{Sde}\_3\\ \frac{\hat{\mathbf{e}}\_4}{\sqrt{\mathbf{R}\_{44}'}} &= \mathbf{Sde}\_4 \end{aligned} \tag{34}$$

Analysis of measurement for malaria mortality Using modified state estimation model

The measurement equations characterizing the physical status of patient using sphygmomanometer for blood pressure considered the true/actual value of the system model and the addition of error term in the system model were developed. The question is how true is the physical measurement carried out by the physician? Which can be represented as:

$$Z = h\_{\vec{\eta}} \mathbf{x}\_{\mathbf{i}} + \mathcal{E}\_{\mathbf{i}} \tag{35}$$

*Z*: Measurement of the vital statistics variable of the patients.

*hij*: Account for the precision of the measurements to the accuracy of the instruments (how close to the true value).

*xi*: Unknown state variable that need to be determined and accounts for the pulse/health rate for blood pressure measurement.

*ℓ<sup>i</sup>* Error term introduced in the measurement which can be expressed mathematically as:

$$Z = h\_{11}\varkappa\_1 + h\_{12}\varkappa\_2 + e\_1\tag{36}$$

Eq. (36), provided the vital statistics variable measurement which is the weighted sum of squared of errors for the Gaussian distribution. Since the physical measurements of the patients is considered as a function of their relative frequency of occurrence, a histogram may be represented which is a continuous curve in relation to the number of occurrence.

Clinical Diagnosis Measurement, geographical location and distribution.

Data were collected from reputable hospitals and clinics from different location for malaria mortality using blood pressure measurements as one of the key associated parameter for the incidence of malaria parasite transmission variable (**Tables 4**–**6**).


#### **Table 4.**

*Distribution of suspected and confirmed cases of malaria parasites (plasmodium) according to the province in a developing countries.*


**Table 5.**

*Blood pressure measurement (Sphygmomano-meter) for purpose of incidence of malaria parasite transmission.*


#### **Table 6.**

*Sphygmomanometer measurement and precision for true (actual value of measurement) for associated malaria parasite transmission.*

Measurements Matrix, (Z) equation given as:

$$[Z] = \begin{bmatrix} Z\_1 \\ Z\_2 \\ Z\_3 \\ Z\_4 \end{bmatrix} = \begin{bmatrix} \frac{120}{80} mmHg = 1.5 \\ \frac{120}{80} mmHg = 1.5 \\ \frac{140}{80} mmHg = 1.8 \\ 80 \\ \frac{150}{80} mmHg = 1.9 \end{bmatrix} \tag{37}$$

Measurements error ð Þ*e* given as:

$$\begin{aligned} [\boldsymbol{e}] &= \begin{bmatrix} \boldsymbol{e}\_1 \\ \boldsymbol{e}\_2 \\ \boldsymbol{e}\_3 \\ \boldsymbol{e}\_4 \end{bmatrix} = \text{To be determined} \tag{38} \end{aligned} \tag{38}$$

True value of system model term given as:

$$\left[h\_{\vec{\eta}}\mathbf{x}\_{\vec{\eta}}\right] = h\_{11}\mathbf{x}\_1 + h\_{12}\mathbf{x}\_2 + e\_1\tag{39}$$

Measurement instruments precisions to the accuracy of true/actual value give as:

$$Z\_1 = 8\mathfrak{W}\mathbf{x}\_1 + 6\mathfrak{W}\mathbf{x}\_2 + \mathbf{e}\_1\tag{40}$$

$$Z\_2 = 8\mathfrak{G}\mathfrak{\kappa}\_1 + \mathfrak{G}\mathfrak{\ll}\mathfrak{\kappa}\_2 + \mathfrak{e}\_2\tag{41}$$

$$Z\_3 = \Re \mathfrak{G} \mathbf{x}\_1 + \Re \mathfrak{G} \mathbf{x}\_2 + \mathbf{e}\_3 \tag{42}$$

$$\mathbf{Z}\_4 = 8\mathfrak{W}\mathbf{x}\_1 + \mathfrak{P}\mathfrak{W}\mathbf{x}\_2 + \mathbf{e}\_4 \tag{43}$$

Eqs. (40), (41), (42), (43) can be represented as:

$$\mathbf{Z}\_1 = \mathbf{0}.08\mathbf{x}\_1 + \mathbf{0}.06\mathbf{x}\_2 + \mathbf{e}\_1\tag{44}$$

$$Z\_2 = 0.08x\_1 + 0.06x\_2 + e\_2\tag{45}$$

$$\mathbf{Z}\_3 = \mathbf{0}.09\mathbf{x}\_1 + \mathbf{0}.08\mathbf{x}\_2 + \mathbf{e}\_3\tag{46}$$

$$\mathbf{Z}\_4 = \mathbf{0}.08\mathbf{x}\_1 + \mathbf{0}.09\mathbf{x}\_2 + \mathbf{e}\_4\tag{47}$$

Where; *x*<sup>1</sup> *and x*<sup>2</sup> and are the state variable that is unknown that needed to be determined

Case 1: Formulations of the Coefficient matrix ð Þ *H* which can be represented as:

$$H = \begin{bmatrix} 0.08 & 0.06 \\ 0.08 & 0.06 \\ 0.09 & 0.08 \\ 0.08 & 0.09 \end{bmatrix} \tag{48}$$

Case 2: Weighting factor for measuring instrument ð Þ *wi* which can be presented in diagonal matrix form as:

$$w\_i = \begin{bmatrix} \mathbf{100} & \cdot & \cdot & \cdot \\ \cdot & \mathbf{100} & \cdot & \cdot \\ \cdot & \cdot & \mathbf{50} & \cdot \\ \cdot & \cdot & \cdot & \mathbf{50} \end{bmatrix} \tag{49}$$

Where

$$w\_i = w\_1, w\_2, w\_3, w\_4$$

The weighing factor ð Þ *w* for the measurement instrument *w*1, *w*2, *w*3, *w*<sup>4</sup> are 100, 100, 50, 50

Case 3: Determine nation of transpose of coefficient matrix (H) given as:

$$H^T = \begin{bmatrix} 0.08 & 0.08 & 0.09 & 0.08\\ 0.06 & 0.06 & 0.08 & 0.09 \end{bmatrix} \tag{50}$$

Case 4: Compute the matrix operation *HTW* as:

$$H^T W = \begin{bmatrix} 0.08 & 0.08 & 0.09 & 0.08 \\ 0.06 & 0.06 & 0.08 & 0.09 \end{bmatrix} \begin{bmatrix} 100 & \cdot & \cdot & \cdot \\ \cdot & 100 & \cdot & \cdot \\ \cdot & \cdot & 50 & \cdot \\ \cdot & \cdot & \cdot & 50 \end{bmatrix} \tag{51}$$
 
$$= \begin{bmatrix} 8 & 8 & 4.5 & 4 \\ 6 & 6 & 4 & 4.5 \end{bmatrix} \tag{52}$$

Case 5: Determination of gain matrix (G) given as:

$$\mathbf{G} = \mathbf{H}^{\mathrm{T}} \mathbf{W} \mathbf{H}$$

Thus

$$G = \begin{bmatrix} 8 & 8 & 4 & 4 \\ 6 & 6 & 4 & 4.5 \end{bmatrix} \begin{bmatrix} 0.08 & 0.06 \\ 0.08 & 0.06 \\ 0.09 & 0.08 \\ 0.08 & 0.09 \end{bmatrix} \tag{53}$$

This implies;

$$\mathbf{G} = \mathbf{H}^T \mathbf{W} \mathbf{H} = \begin{bmatrix} 2.005 & 1.68 \\ 1.68 & 1.445 \end{bmatrix} \tag{54}$$

Case 6: Determination of Inverse of gain matrix (G) given as:

$$G^{-1} = \frac{I}{|G|} = \begin{bmatrix} 2.005 & 1.68\\ 1.68 & 1.445 \end{bmatrix} \tag{55}$$

$$|\mathbf{G}| = 0.074825$$

$$G^{-1} = \frac{I}{0.074825} = \begin{bmatrix} 1.445 & -1.68\\ -1.68 & 2.005 \end{bmatrix} \tag{56}$$

$$\text{or}$$

$$\mathbf{G}^{-1} = \begin{bmatrix} 19.3 & -22.45 \\ -22.45 & 26.9 \end{bmatrix} \tag{57}$$

Case 7: Determination of multiplication of inverse operation of gain matrix *G*�<sup>1</sup> � � and *HTW* given as:

$$\mathbf{G}^{-1}\mathbf{H}^{T}\mathbf{W} = \begin{bmatrix} 19.3 & -22.45 \\ -22.45 & 26.8 \end{bmatrix} \begin{bmatrix} 8 & 8 & 4.5 & 4 \\ 6 & 6 & 4 & 4.5 \end{bmatrix} \tag{58}$$

This implies

$$G^{-1}H^{T}W = \begin{bmatrix} 19.7 & 19.7 & -2.95 & -23.825 \\ -18.8 & -18.8 & 6.175 & 30.8 \end{bmatrix} \tag{59}$$

Case 8: Determination of matrix operation *G*�<sup>1</sup> *HTWZ* � � given as:

$$\begin{bmatrix} G^{-1}H^{T}WZ = \begin{bmatrix} 19.7 & 19.7 & -2.95 & -23.825 \\ -18.8 & -18.8 & 6.175 & 30.8 \end{bmatrix} \begin{bmatrix} 1.5 \\ 1.5 \\ 1.8 \\ 1.9 \end{bmatrix} \tag{60}$$

Where; *G*�<sup>1</sup> *HTWZ* is represented as the matrix operations determination of the state variables estimate *x*^<sup>1</sup> *and x*^<sup>2</sup> given as:

$$
\begin{bmatrix}
\hat{\mathfrak{x}}\_1 \\
\hat{\mathfrak{x}}\_2
\end{bmatrix} = \mathbf{G}^{-1} H^T W \mathbf{Z} = \begin{bmatrix}
\mathbf{8.5225} \\
\mathbf{13.235}
\end{bmatrix} \tag{61}
$$

Case 9: Determination of the matrix operation *Hx* as a function of the estimate of the physical measurements ð Þ *Z*1*Z*2*Z*3*Z*<sup>4</sup> given as:

$$
\hat{Z} = H\hat{X}\tag{62}
$$

$$\begin{aligned} \text{Similarly,} \begin{bmatrix} \hat{Z}\_1 \\ \hat{Z}\_2 \\ \hat{Z}\_3 \\ \hat{Z}\_4 \end{bmatrix} = H \begin{bmatrix} X\_1 \\ X\_2 \end{bmatrix} \end{aligned} \tag{63}$$

Where;

$$\begin{aligned} \mathbf{H} = \begin{bmatrix} \mathbf{0.08} & \mathbf{0.06} \\ \mathbf{0.08} & \mathbf{0.06} \\ \mathbf{0.09} & \mathbf{0.08} \\ \mathbf{0.08} & \mathbf{0.09} \end{bmatrix} \hat{\boldsymbol{\mu}} = \begin{bmatrix} \mathbf{X}\_1 \\ \mathbf{X}\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{8.5225} \\ \mathbf{13.235} \end{bmatrix} \end{aligned} \tag{64}$$

Thus; the estimates of the measurement *Z*^ � ) becomes as:

$$
\begin{bmatrix}
\hat{Z}\_1\\ \hat{Z}\_2\\ \hat{Z}\_3\\ \hat{Z}\_4
\end{bmatrix} = \begin{bmatrix}
\text{0.08} & \text{0.06} \\ \text{0.08} & \text{0.06} \\ \text{0.09} & \text{0.08} \\ \text{0.08} & \text{0.09}
\end{bmatrix} \begin{bmatrix}
\text{8.5225} \\ \text{13.235}
\end{bmatrix} \tag{65}
$$

This implies as:

$$
\begin{bmatrix}
\hat{Z}\_1\\ \hat{Z}\_2\\ \hat{Z}\_3\\ \hat{Z}\_4
\end{bmatrix} = \begin{bmatrix}
1.4759\\ 1.4759\\ 1.825825\\ 1.87295
\end{bmatrix} \tag{66}
$$

Case 10: Determination of number of physical measurement ð Þ *Nm* and states variable ð Þ *Ns* on the view to determine redundancy ð Þ *K* for a quantifiable level of significance.

*Nm* ¼ *Z*1, *Z*2, *Z*<sup>3</sup> *and Z*<sup>4</sup> ¼ 4 (four physical measurements) *Ns* ¼ *x*1&*x*<sup>2</sup> ¼ 2 (blood pressure for pulse/heart rate)

$$\text{Redundancy}(K) = N\_m - N\_s = 4 - 2 = 2\tag{67}$$

Case 11: Determination of errors estimatesð Þ^*e* of the physical measurements and estimated measurementsð Þ^*z* are presented as:

$$
\hat{e} = \begin{bmatrix} \text{Physical\ measurement} \ (\mathbf{Z}) \end{bmatrix} - \begin{bmatrix} \text{estimated\ measurement} \ (\hat{\mathbf{z}}) \end{bmatrix} \tag{68}
$$

That is;

$$
\begin{bmatrix}
\hat{\boldsymbol{e}}\_1 \\
\hat{\boldsymbol{e}}\_2 \\
\hat{\boldsymbol{e}}\_3 \\
\hat{\boldsymbol{e}}\_4
\end{bmatrix} = \begin{bmatrix}
\mathbf{Z}\_1 \\
\mathbf{Z}\_2 \\
\mathbf{Z}\_3 \\
\mathbf{Z}\_4
\end{bmatrix} - \begin{bmatrix}
\hat{\boldsymbol{Z}}\_1 \\
\hat{\boldsymbol{Z}}\_2 \\
\hat{\boldsymbol{Z}}\_3 \\
\hat{\boldsymbol{Z}}\_4
\end{bmatrix} \tag{69}
$$

Where

$$
\begin{bmatrix} Z\_1 \\ Z\_2 \\ Z\_3 \\ Z\_4 \end{bmatrix} = \begin{bmatrix} 1.5 = 120/80 mmHg \\ 1.5 = 120/80 mmHg \\ 1.8 = 140/80 mmHg \\ 1.9 = 180/80 mmHg \end{bmatrix}; \begin{bmatrix} \hat{Z}\_1 \\ \hat{Z}\_2 \\ \hat{Z}\_3 \\ \hat{Z}\_4 \end{bmatrix} = \begin{bmatrix} 1.4759 \\ 1.4759 \\ 1.825825 \\ 1.87295 \end{bmatrix} \tag{70}
$$

Thus,

$$
\begin{bmatrix}
\hat{e}\_1\\ \hat{e}\_2\\ \hat{e}\_3\\ \hat{e}\_4
\end{bmatrix} = \begin{bmatrix}
0.0241\\ 0.0241\\ -0.025525\\ 0.02705
\end{bmatrix} \tag{71}
$$

Case 12: Determination of sum of Squared of Errors ð Þ*e* Measurement

If calculated weighted sums of squares of errors calculated *<sup>f</sup> calculated* � � is greater than chi-square distribution it can be deduced that there is suspected error (or bad data in the measurement sets).

Similarly,

If calculated objective function *<sup>f</sup> calculated* � � is less than chi-square distribution value, it is concluded or declared a violations which does not satisfy the measurements criteria for decision making for purpose of diagnosing human body conditions.

#### **6. Results and discussion**

Case 13. Estimate of measurements error ð Þ*e*

The objectives function ð Þ*f* of the weighted sum of squares of errors can be presented as:

$$\mathbf{f} = \sum\_{j=1}^{n} \frac{\mathbf{e}\_j^2}{\sigma\_j^2} = \mathbf{100}(\mathbf{e}\_1)^2 + \mathbf{100}(\mathbf{e}\_2)^2 + \mathbf{50}(\mathbf{e}\_3)^2 + \mathbf{50}(\mathbf{e}\_4)^2$$

$$\text{where}: \mathbf{e}\_1 = 0.0241, \mathbf{e}\_2 = 0.0241, \mathbf{e}\_3 = -0.025525, \mathbf{e}\_4 = 0.02705$$

*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

$$\mathbf{f} = \sum\_{j=1}^{n} \frac{\mathbf{e}\_j^2}{\sigma\_j^2} = 100(0.0241)^2 + 100(0.0241)^2 + 50(-0.025525)^2 + 50(0.02705)^2$$

¼ 100 0ð Þþ *:*00058081 100 0ð Þþ *:*00058081 50 0ð *:*000651525625Þ þ 50 0ð Þ *:*0007317025

¼ 0*:*058081 þ 0*:*058081 þ 0*:*03257628125 þ 0*:*036585125

¼ 0*:*185323125

Similarly,

The chi-square distribution from tabulated results given as: *χ*<sup>2</sup> *<sup>k</sup>*,*<sup>α</sup>* <sup>¼</sup> <sup>9</sup>*:*<sup>21</sup> � �

$$\mathbf{f}\_{\text{calculated}} \le \chi^2\_{\mathbf{k},\alpha}$$

where;

$$\mathbf{K} = \mathbf{N\_m} - \mathbf{N\_s} = \mathbf{4} - \mathbf{2} = \mathbf{2}$$

For, α = 1% quantifiable level of significant = 0.01ð Þ *α* ¼ 0*:*01

(Significance level, *α* ¼ 0*:*01)

The probability of 1 ðð Þ¼ � *α* ð Þ 1 � 0*:*01 Þ ¼ 99% declared confidence level for a degree of freedom, *K* ¼ *Nm* � *Ns* ¼ 2

From the tabulated result obtained for a given degree of freedom and a quantifiable level of significance of *α* ¼ 0*:*01, for *k* ¼ 2 given as 9.21 which means that the calculated value of *fcalculated* � � is greater than the critical value of the chi-square distribution value. Thus, the chi-square of *f* provides a test statistics for validating the error measurements of bad data or probably suspected adverse health condition that required attensions for purpose of ensuring good status healthy conditions.

```
%MATTIX LABORATORY
%PROGRAM: MATLAB FOR MALARIA MORTALITY
%AUTHUR: BRAIDE, SEPIRIBO LUCKY
%TECHNICAL PAPER PORESENTAION
%First iteration calculation using a flat start values (,k,t which
%represent the static variables x1 and x2);
k = 1;
t = 1;
% measurement instrument represented as z: z1 z2 z3 z4
z1 = 1.5;
z2 = 1.5;
z3 = 1.8;
z4 = 1.9;
z = [1.5; 1.5; 1.8; 1.9];
% estimated measurement represented as zk1 zk2 zk3 zk4
zk1 = 1.4759;
zk2 = 1.4759;
zk3 = 1.825825;
zk4 = 1.87295;
% measurement error represented as e1 e2 e3 e4
e1 = z1-zk1;
e2 = z2-zk2;
```

```
e3 = z3-zk3;
e4 = z4-zk4;
% Jacobianmatrix represented as H
H = [0.08 0.06;0.08 0.06;0.09 0.08;0.08 0.09];
%Transpose of jacobian matrix T represented as H
T=H0
% Diagonal matrix of the weighting factor as W
W = [100,0,0,0; 0,100,0,0;0,0,50,0;0,0,0,50];
```

```
% MUutiply Transpose of jacobian matrix H0 and weighting factor W as
D=H0
     *W
```
% Multiply the matrix operation D and jacobian matrix H to obtain gain matrix represented as

```
G = D*H
% The inverse of gain matrix G represented as F
F = inv.(G)
% multiply the matrix operation F and matrix D be represented as
M = F*D
```

```
% multiply matrix M and Z representation as N = M*Z state variable [xi x2]
```

```
% Determination of matrix operation of state estimation [x1 x2] = N
for z = [1.5; 1.5; 1.8; 1.9];
v = M*z
end
T =
0.0800 0.0800 0.0900 0.0800
0.0600 0.0600 0.0800 0.0900
D =
8.0000 8.0000 4.5000 4.0000
6.0000 6.0000 4.0000 4.5000
G =
2.0050 1.6800
1.6800 1.4450
F =
19.3117–22.4524
�22.4524 26.7959
M =
19.7795 19.7795–2.9068-23.7888
�18.8440-18.8440 6.1477 30.7718
v =
8.9075
13.0003
```
#### **7. Recommendation**

Considering the fact that health is wealth the relationship between the two is frequently misconstrued. That is there is strong need for healthy living to create wealth for man and society.

*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

Measurement of malaria endemictiy is basically on vector or parasites measurement especially for infected cases on transmission of plasmodium parasite etc. which may seriously leads to some vital sign and symptom: Temperature rise beyond normal level, headache, blood pressure become abnormal, blood-sugar concentration level may also be affected medically etc.

Evidently, the mapping of malaria transmission have demonstrated a wider geographical distribution of the parasite (plasmodium, etc). The number of clinical report cases due to malaria infection resulted into early mortality which is becoming an increase on daily basis across the world. Particularly in south America, where *P. Vivax* is currently the most predominant malaria species. Although control measures and programs have had a significant impact on malaria associated cases. Consequently the centers for disease control and prevention (CDC) Saving lives, protecting people in various states of America have considered regular, rapid and accurate diagnosis of malaria which is an integral part to the appropriate treatment of affected individuals. It is also requested that the CDC: Health care providers should always obtain a travel history form patients, especially person who traveled in a malaria epidemic area, must be evaluated for check using appropriate tests (for malaria) for purpose of healthy free environment. This technical paper having carried out thorough extensive investigation especially to this robust medically affected case of earthly malaria mortality for man which may also be associated to bad measurement for vital statistics variables about a patients. The detail history of patients measured must be checked for accurate, precision, tolerance, error etc. to the true value declared by standard world health organization (like WHO etc). Therefore physical measurements taken by medical practioner from patient must:


The transition rates: *h and r* can be estimated from transition frequencies: *α and β* with

$$
\hat{\mathbf{h}} = \frac{\alpha}{\mathbf{t}(\alpha + \beta)} \ln \frac{\mathbf{1}}{\mathbf{1} - (\alpha - \beta)}
$$

$$
\hat{\mathbf{r}} = \frac{\beta}{\mathbf{t}(\alpha - \beta)} \ln \frac{\mathbf{1}}{\mathbf{1} - (\alpha + \beta)}
$$

Where;

t : time interval for estimation

h : incidence rate

r : recovery rate

#### **8. Conclusion**

The analysis of modified state estimation model is developed for estimated measurement of malaria mortality in a case of a developing countries. This model help to identify detect and flag measurement error which is classified as bad data measurement; whose result may be received on drug prescription and administration for early malaria mortality rate especially when measurement instruments are not regularly calibrated for efficiency. This technical paper for purpose of analysis and scope considered the associated vital statistic variables like blood-pressure (BP) using sphygmomanometer as a key driver for this analysis. The techniques establishes a mathematical modified state estimation model that characterizes the physical measurement of the patient to the acceptable value of *BP* <sup>¼</sup> <sup>120</sup> <sup>80</sup> *mmHg* in order to satisfy the operating normal conditions.

The true value of the system model equations and error term addition are considered for purpose of estimating measurements made by physician instrument. Data were collected from four (4) geographical area in a developing countries for purpose of validation of the modified estimation model. The techniques allow for an algebraic summation operation of measurement for the weighted sum provided the declared standard measurement of blood pressure of <sup>120</sup>*mm* <sup>80</sup> *Hg is not violated* .

The analysis technical paper identified bad data measurement which required urgent attention for check: The surrounding circumstance of the patients, human error, calibration error etc. for the purpose of determining the state of the body system configuration.

The paper also validated this work research via the objective function ð Þ*f* as the calculated value versus chi-square distribution (tabular form) for 99% level of significance with 2 degree of freedom in quantitative measures.

This paper also establishes a string conformity, uniquencie and synergy between measured ad estimated measurements which eventually validated with objective function calculated ð Þ*f* and chi-square distribution and between matrix operations and matrix laboratory code programs for a test statistics.

*Estimation of Malaria Mortality in Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.107059*

#### **Author details**

Sepiribo Lucky Braide Department of Electrical Engineering, Rivers State University, Port Harcourt, Rivers State, Nigeria

\*Address all correspondence to: braidesepiribo@yahoo.com; sepiribo.braide@ust.edu.ng

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 4
