What happens in a brain of a person suffering with apathy?
Starting your comment, first, apathy in itself is not a disease; it's not categorized as such by any medical or psychiatric organization. It is part of the symptoms of some disorders, including depression, schizophrenia and Alzheimer's.

Furthermore, apathy is not a simple symptom, but rather a constellation. That also makes it more difficult to address its brain correlates. From a review,  Moretti &amp; Signori (2016):


  Apathy is an uncertain nosographical entity, which includes reduced motivation, abulia, decreased empathy, and lack of emotional involvement; it is an important and heavy-burden clinical condition which strongly impacts in everyday life events, affects the common daily living abilities, reduced the inner goal directed behavior, and gives the heaviest burden on caregivers. Is a quite common comorbidity of many neurological disease. [...] As a general observation, the occurrence of apathy is connected to damage of prefrontal cortex (PFC) and basal ganglia; “emotional affective” apathy may be related to the orbitomedial PFC and ventral striatum; “cognitive apathy” may be associated with dysfunction of lateral PFC and dorsal caudate nuclei; deficit of “autoactivation” may be due to bilateral lesions of the internal portion of globus pallidus, bilateral paramedian thalamic lesions, or the dorsomedial portion of PFC. On the other hand, apathy severity has been connected to neurofibrillary tangles density in the anterior cingulate gyrus and to gray matter atrophy in the anterior cingulate (ACC) and in the left medial frontal cortex, confirmed by functional imaging studies. These neural networks are linked to projects, judjing and planning, execution and selection common actions, and through the basolateral amygdala and nucleus accumbens projects to the frontostriatal and to the dorsolateral prefrontal cortex. Therefore, an alteration of these circuitry caused a lack of insight, a reduction of decision-making strategies, and a reduced speedness in action decision, major responsible for apathy. Emergent role concerns also the parietal cortex, with its direct action motivation control.